--- title: "Lach-Space: A Unified Geometric Programme for Quantum Gravity, Matter, and the Standard Model" subtitle: "Monograph, third edition, with complete proofs" author: "Datacraft Theoretical Physics Group" date: "August 2026" version: "v3.0" --- # Preface This is the third edition of the Lach-Space monograph. It differs from v2.x in five ways. First, the separate *Rigorous Proofs Companion* no longer exists as a distinct document. Every proof it contained has been absorbed into the body of the present text, rewritten where necessary so that the hypotheses actually consumed by each step are the hypotheses stated in the theorem. A reader of v3 needs no other Lach-Space document. Second, the status tag *Heuristic* has been abolished. In v2.x it sheltered several statements whose logical content was unclear. Each of them has been examined and either upgraded to *Conditional* on a named hypothesis that makes the argument a proof, or demoted to *Conjecture* or *Phenomenological*. No statement in this edition carries a tag stronger than what its written proof establishes. Third, two chapters are new. Chapter 20 constructs explicit reference families of Quantum Spacetime Networks whose continuum limits are Minkowski space, spatially flat Friedmann-Robertson-Walker spacetimes, the Schwarzschild exterior, and de Sitter space, and verifies the structural hypotheses on each. Chapter 21 makes the passage between the compact real form used in the representation-theoretic chapters and the Lorentzian real form used in the geometric chapters into a theorem about holonomies rather than a remark. Fourth, the hypothesis schedule has been refined. Two hypotheses used silently in v2.x are now named: H8, a density hypothesis on the inductive system of edge-mode algebras, and H9, a random-tensor-network scrambling hypothesis on which the Page-time estimate rests. A tenth hypothesis, H10, names the regularity of the continuum limit that the geometric chapters require beyond what the synthetic-geometry theorems deliver. The isoperimetric hypothesis H4' has been given the two-sided form that the volume-doubling argument in fact uses. Fifth, several technical assertions inherited from v2.x have been corrected. The most consequential are recorded at the places where they arise, and collected in Appendix A.5. Among them: the v2.x fixed-point exponents $\theta_{G}\approx2.3$, $\theta_{\Lambda}\approx-0.3$ are withdrawn in favour of the complex pair $1.4753\pm3.0432i$; the v2.x Standard-Model-shifted fixed point $(1.05,0.21)$ and the stability count "$52<100$" are withdrawn, the matter combination entering the type-Ia beta functions being $M_{\mathrm{mat}}=N_{S}-N_{D}-N_{V}$; the v2.x discreteness coefficient $-2/3$ is withdrawn in favour of $+1/12$; the v2.x Page time $t_{\mathrm{evap}}/2$ is replaced by $(1-2^{-3/2})t_{\mathrm{evap}}$; and the Henneaux-Teitelboim theorem is no longer called unconditional, since it rests on Theorem 8.1. Three further corrections deserve mention here because they affect the reading of Part II. Beyond them, the final referee passes removed every discrete-to-continuum curvature transfer from Part II: the continuum limits are now Conditional on curvature-dimension conditions assumed on the limit (H-BE, H-TCD), the Lorentzian precompactness on a causal-fraction hypothesis (H-cf), Theorem 7.1 and Lemma 5.2 are Conjectures, the timelike Ollivier hypothesis is an upper bound (H6$^{+}$), and the $\Gamma$-convergence of the action to Einstein-Hilbert (Claim B) is withdrawn as a theorem, surviving only as a Riemannian-sector conditional statement. (i) A Poisson sprinkling into all of Minkowski space has, almost surely, infinitely many links at every element; the consistency model of Chapter 3 therefore sprinkles into a bounded globally hyperbolic region, and the reference families of Chapter 20 are built accordingly. (ii) Ollivier curvature at step size $\varepsilon$ on a smooth manifold is of order $\varepsilon^{2}$; a uniform lower bound along a scaling sequence must therefore be stated for the rescaled curvature $\varepsilon_{n}^{-2}\mathcal{K}_{n}$, and this is how hypothesis H2 is applied to sequences throughout. (iii) The synthetic regularity theorems for non-collapsed limits with Ricci bounded below give a topological manifold with a continuous metric on an open regular set, not a smooth manifold; smoothness of the limit is the content of H10. A fourth point is a limitation rather than a correction: the reference families of Chapter 20 do not yet furnish a sequence satisfying the hypotheses of the Lorentzian continuum limit (Theorem 5.6) with a Lorentzian manifold as limit; at fixed degree bound the limit is Lorentz-Finsler (Proposition 20.9). The Lorentzian theorems of Part II are stated for a hypothesis class whose non-emptiness with a Lorentzian target is an open problem, recorded as such in Chapter 1 and Appendix A. **Status vocabulary.** Every numbered result carries exactly one of the following tags. * $\boxed{\text{Theorem}}$: proved in the text from the axioms, the stated hypotheses, and standard mathematics whose use is routine. * $\boxed{\text{Theorem modulo Classical}(X)}$: proved in the text conditional on the named classical inputs from the list in §2.9, each of which is a published theorem whose hypotheses are verified in the QSN setting where it is invoked. Where a classical input is itself flagged as active research, the tag records that dependence. * $\boxed{\text{Proposition}}$: a subsidiary proved statement. * $\boxed{\text{Conditional (on H$\cdots$)}}$: proved in the text under the listed hypotheses, which are not themselves proved. * $\boxed{\text{Conjecture}}$: a statement believed true, motivated in the text, not proved. * $\boxed{\text{Phenomenological}}$: an empirical proposal or estimate, not derived from prior results. If a passage reads stronger than its tag, the tag governs. **Conventions.** $\hbar=c=k_{B}=1$ unless restored. Signature $(-,+,+,+)$. The spacetime dimension is $d$, equal to the isoperimetric parameter $N_{0}$ of H4', with $d=4$ physical. $\ell_{P}$ is the Planck length. Chapters are numbered globally; results are numbered consecutively within each chapter across all kinds; equations are labelled by chapter and sequence. No em or en dashes are used. Nairobi, August 2026. --- # Table of Contents * Part I. Foundations * Chapter 1. Motivation * Chapter 2. Mathematical Preliminaries and the Classical Inputs * Chapter 3. The Axioms of Lach-Space * Part II. The Quantum Spacetime Network and Emergent Geometry * Chapter 4. The QSN: Nodes, Edges, Hilbert Space * Chapter 5. Discrete Curvature, Volume Doubling, and the Continuum Limits * Chapter 6. The Proper-Time Postulate * Chapter 7. Timelike Curvature-Dimension and the Hauptvermutung * Part III. Classical and Quantum Dynamics * Chapter 8. Emergence of Einstein's Equations * Chapter 9. Chiral Fermions and the Ginsparg-Wilson Relation * Part IV. Functional Renormalization Group and Asymptotic Safety * Chapter 10. The Wetterich Flow on the QSN * Chapter 11. Gravitational Beta Functions and the Vacuum Fixed Point * Chapter 12. Standard-Model-Shifted Fixed Point and Truncation Persistence * Chapter 13. The Cosmological Constant: Volume Quantisation and Unimodularity * Part V. Operator-Algebraic Holography * Chapter 14. Edge-Mode Algebras and the Hyperfinite $\mathrm{II}_{1}$ Factor * Chapter 15. The Page Curve * Part VI. The $q$-Deformed Standard Model * Chapter 16. Quantum-Group Representation Theory on the QSN * Chapter 17. The Grand Unified Quantum Group and Anomaly Cancellation * Part VII. Phenomenology, Numerics, and Reference Constructions * Chapter 18. Phenomenological Proposals * Chapter 19. Numerical Methods * Chapter 20. Reference QSN Families and Their Continuum Limits * Chapter 21. Wick Rotation of Holonomies * Appendix A. Axiom Schedule, Hypothesis Inventory, and Corrections * Appendix B. Notation and Units * Appendix C. Master Index of Results * Appendix D. Bibliography --- # Part I. Foundations --- ## Chapter 1. Motivation ### 1.1. The ultraviolet obstruction General relativity and the quantum theory of matter have empirical domains separated by some twenty-five orders of magnitude in energy. Every attempt to combine them as a quantum field theory on a smooth background meets one obstruction: the perturbative expansion of the Einstein-Hilbert action about Minkowski space is not renormalizable. At one loop the pure-gravity divergences are removable by a field redefinition ('t Hooft, Veltman, *Ann. Inst. Henri Poincaré* **20** (1974) 69); at two loops a counterterm cubic in the Riemann tensor appears with a divergent coefficient that no bare parameter of the Einstein-Hilbert theory can absorb (Goroff, Sagnotti, *Nucl. Phys. B* **266** (1986) 709). With matter, the obstruction appears already at one loop. The difficulty is ultraviolet. A candidate theory must say what happens to the geometry at the Planck length $\ell_{P}=\sqrt{\hbar G/c^{3}}\approx 1.6\times 10^{-35}$ m. Three strategies dominate. String theory replaces the point particle by an extended object and trades the divergences of point-like gravity for the modular invariance of worldsheet amplitudes (Polchinski, *String Theory*, Cambridge, 1998); it has not produced a fully realistic low-energy vacuum. Loop quantum gravity quantises the Ashtekar variables and predicts discrete spectra for geometric operators (Ashtekar, *Phys. Rev. Lett.* **57** (1986) 2244; Thiemann, *Modern Canonical Quantum General Relativity*, Cambridge, 2007); it has not derived the classical Einstein equations in a controlled limit of its dynamics. Asymptotic safety conjectures an interacting ultraviolet fixed point of the exact renormalization group with finitely many relevant directions (Weinberg, in *General Relativity: An Einstein Centenary Survey*, Cambridge, 1979; Reuter, *Phys. Rev. D* **57** (1998) 971); its evidence comes from truncations whose error is not controlled. Causal set theory (Bombelli, Lee, Meyer, Sorkin, *Phys. Rev. Lett.* **59** (1987) 521), causal dynamical triangulations (Ambjørn, Jurkiewicz, Loll, *Phys. Rep.* **519** (2012) 127), and group field theory (Oriti, *Stud. Hist. Phil. Mod. Phys.* **46** (2014) 186) supply discrete substrates with partial results on the recovery of continuum geometry. ### 1.2. The Lach-Space proposition Lach-Space commits to a single physical proposition: the microstates of quantum gravity are not manifolds but countable combinatorial-algebraic objects, and smooth spacetime is a coarse-grained observable on those microstates, defined only in a scaling limit. If this is right, the ultraviolet problem is dissolved rather than solved. There is no small-distance limit at the microscopic level because there is no small distance. The burden shifts to the infrared: one must prove that Lorentzian manifolds re-emerge. The fundamental object is the Quantum Spacetime Network (QSN): a locally finite directed acyclic graph whose nodes carry Planck-scale Hilbert factors and whose edges carry a gravitational holonomy valued in $\mathrm{Spin}(4,1)$ together with a Standard Model holonomy valued in $\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}$. A single energy functional built from an edge tension and Ollivier's coarse Ricci curvature governs the graph. What distinguishes the programme from causal sets and from loop quantum gravity is the conjunction of three commitments: the causal order is microscopic and primitive; every edge carries algebraic data for both gravity and gauge fields; and the bridge from the discrete substrate to continuum geometry is synthetic Ricci curvature in the sense of Lott, Sturm, Villani, and, in Lorentzian signature, Cavalletti and Mondino. ### 1.3. What is established and what is not The programme's results fall into four groups, and the reader should hold the grouping in mind throughout. The first group is proved outright: the QSN with its longest-chain proper time is a Lorentzian pre-length space (Lemma 5.1); under H1' and H4' alone, with no curvature input, the graph balls satisfy volume doubling (Lemma 5.3); the Lorentzian shell expansion of the causal random walk holds as a one-sided inequality (Lemma 7.4); the node volume expectation is rigidly Planckian (Lemma 13.1, unconditional); and the inductive-limit edge-mode algebra is a hyperfinite finite von Neumann algebra (Propositions 14.2, 14.3, 14.4). The second group is proved modulo named classical theorems whose hypotheses are verified in the text. The continuum limits no longer belong here as transfer theorems: Part II proves no discrete-to-continuum curvature transfer. The Riemannian limit (Theorem 5.5) is Conditional on H-conv and on H-BE, where H-BE now means that the mGH limit is assumed to be an $\mathrm{RCD}(K,N_{0})$ space; the Lorentzian limit splits into (a) GHL precompactness (Theorem 5.6(a)), Conditional on the causal-fraction hypothesis H-cf modulo Classical A-Lor, and (b) the regular-set statement (Theorem 5.6(b)), Conditional on H-TCD (the timelike curvature-dimension condition assumed on the limit) and on Conjecture C-Lor; the sprinkling curvature estimate (Conjecture 5.2), the timelike doubling lemma (Lemma 5.4, Conditional on H-cf), and the timelike curvature transfer (Conjecture 7.1) are no longer theorems. What the second group does contain is: the overlap Dirac operator on the QSN (Theorem 9.1); the fixed point of the Einstein-Hilbert truncation (Theorem 11.1, modulo Classical (Reuter), the type-Ia Litim background-field beta functions) at $(g^{*},\lambda^{*})=(0.70732088,\,0.19320051)$, the unique real root of $14x^{3}-41x^{2}+59x-24$ in $x=1-2\lambda$, with the complex-conjugate pair of critical exponents $\theta=1.4753\pm3.0432i$, so that the fixed point is a spiral ultraviolet attractor with both directions relevant (Theorem 11.2, modulo Classical (Reuter)); the Standard-Model-shifted fixed point (Theorem 12.1, modulo Classical (Reuter), type-Ia scheme, matter combination $M_{\mathrm{mat}}=N_{S}-N_{D}-N_{V}$) at $(g^{*},\lambda^{*})=(0.8632,-1.2394)$ with real exponents $(1.969,3.848)$, and its persistence under truncation deformation (Theorem 12.2); and the modular tensor category of the $q$-deformed $\mathfrak{spin}(5)$ (Theorem 16.1). One classical input, the regularity of non-collapsed Lorentzian curvature-dimension limits, is active research; the Lorentzian results that use it say so. A further caveat governs the reading of Theorem 5.6: no physically constructed sequence of QSNs has yet been shown to lie in its hypothesis class. The bounded-degree window sprinklings of Chapter 20 converge, at fixed degree bound $D$, to a Lorentz-Finsler time separation strictly smaller than the target Lorentzian one, and recover it only as $D\to\infty$ (Theorem 20.5, Proposition 20.9); uniform H1' and convergence to a Lorentzian manifold have not been achieved simultaneously. Theorem 5.6 is therefore, at present, a theorem about a class whose non-emptiness is open. The third group is proved under hypotheses internal to the programme. Claim B, $\Gamma$-convergence of the rescaled QSN action to the Einstein-Hilbert action, is withdrawn as a theorem. Theorem 8.1 survives only in its Riemannian compact sector, Conditional on H10, H12, H13, H14 (a recovery sequence satisfying H1' and H2 uniformly), and H15(c), with a Lorentzian reading available only through the holonomy continuation of Theorem 21.1 and Proposition 21.2 under H-hol; the window-sprinkling recovery sequences give a Lorentz-Finsler $\Gamma$-limit (Propositions 8.11, 8.12), Conditional in addition on H-conn and Conjecture 20.8, and the curvature term supplies only an upper bound on the discrete action because Lemma 7.4 is one-sided. The Henneaux-Teitelboim decomposition (Theorem 13.2) inherits the status of Theorem 8.1 and yields the constraint $\frac{d-2}{2}R+8\pi GT=d\Lambda_{0}$ with $\Lambda_{0}$ an integration constant; the gravitational anomalous dimension $\eta^{\mathrm{grav}}_{A}=-5g/(3\pi(1-2\lambda)^{2})$ (Theorem 11.3, with the coefficient imported from the matter-gravity literature); the form of the discreteness correction $\delta\beta=\delta\beta_{\mathrm{cont}}[1+\alpha(k\ell_{P})^{2}+O((k\ell_{P})^{4})]$ with $\alpha\geq0$ for every random-walk regulator (Proposition 11.4, a theorem in its form), with $\alpha=+1/12$ in the isotropic model so that couplings increase at trans-Planckian scale; the discrete quantum extremal surface formula and the Page-curve mechanism (Theorems 15.1, 15.2), the Page-time estimate $t_{\mathrm{Page}}=(1-2^{-3/2})\,t_{\mathrm{evap}}$ (Proposition 15.3), the $q$-anomaly invariance and the anomaly cancellation of the embedded Standard Model (Theorems 16.2, 17.1). The fourth group is open: the sprinkling curvature estimate (Conjecture 5.2), the timelike curvature transfer (Conjecture 7.1), the equality case of the Lorentzian shell expansion (Conjecture E-Lor), the regularity of Lorentzian curvature-dimension limits (Conjecture C-Lor), the Hauptvermutung analogue (Conjecture 7.2), the spectral rigidity of the proper-time assignment (Conjecture 6.2), the identification of the edge-mode factor with the Chandrasekaran-Penington-Witten crossed product (Conjecture 14.6), edge-mode completeness (Conjecture 14.7), the Type $\mathrm{II}_{1}$ refinement of the Page curve (Conjecture 15.4), the existence of a sequence of QSNs satisfying uniform H1'-H7 whose GHL limit is a Lorentzian, rather than Lorentz-Finsler, manifold (Proposition 20.9); and, outside the programme's remit, the existence of a fixed point of the exact Wetterich flow. Nothing in this monograph asserts a historical provenance for the name Lach-Space beyond its use as a project name. ### 1.4. Hypothesis discharge ledger | Result | Classical inputs | Hypotheses used | Steps | |---|---|---|---| | (none: Chapter 1 states no results) | | | | --- ## Chapter 2. Mathematical Preliminaries and the Classical Inputs This chapter fixes definitions and states precisely every external theorem the monograph invokes. The external theorems are collected in §2.9 as Classical A through Classical V; later chapters cite them by letter and verify their hypotheses at the point of use. ### 2.1. Graphs, orders, and measures A *directed graph* is a pair $\mathsf{Q}=(N,E)$ with $N$ a set and $E\subseteq N\times N$ with no loops $(n,n)$. For $n\in N$ write $N^{+}(n)=\{m:(n,m)\in E\}$ and $N^{-}(n)=\{m:(m,n)\in E\}$ for the out- and in-neighbourhoods, and $\deg^{\pm}(n)=|N^{\pm}(n)|$. The graph is *locally finite* if $\deg^{+}(n)+\deg^{-}(n)<\infty$ for every $n$, and of *bounded degree* $D$ if $\deg^{+}(n)+\deg^{-}(n)\leq D$ for every $n$. A *directed path* from $n$ to $m$ is a finite sequence $n=n_{0},n_{1},\dots,n_{k}=m$ with $(n_{i-1},n_{i})\in E$; its *length* is $k$. The graph is *acyclic* if no directed path of positive length returns to its starting node. The *causal order* is $n\preceq m$ iff $n=m$ or there is a directed path from $n$ to $m$; on an acyclic graph this is a partial order. The *graph distance* $d_{G}(n,m)$ is the minimum length of a path in the underlying undirected graph, $+\infty$ if none exists. The *counting measure* $\mu$ on $N$ assigns mass one to each node. Graph balls are $B(n,r)=\{m:d_{G}(n,m)\leq r\}$, and for $B\subseteq N$ the *edge boundary* is $\partial B=\{(n,m)\in E: n\in B, m\notin B\text{ or } n\notin B, m\in B\}$. A *Poisson sprinkling* of intensity $\rho$ into a Lorentzian manifold $(M,g)$ is a Poisson point process $\Pi_{\rho}$ on $M$ with intensity measure $\rho\,\mathrm{vol}_{g}$. Its *Hasse diagram* (or *link graph*) has node set $\Pi_{\rho}$ and an edge $(x,y)$ whenever $x\prec y$ causally and no $z\in\Pi_{\rho}$ satisfies $x\prec z\prec y$. **Remark 2.1.** For $(M,g)$ all of Minkowski space $\mathbb{R}^{1,d-1}$, the expected number of links from a given element of $\Pi_{\rho}$ is infinite for every $\rho>0$, and the number of links is almost surely infinite: the region of points $y\succ x$ with empty causal interval $(x,y)\cap\Pi_{\rho}=\emptyset$ has infinite volume because it is invariant under boosts about $x$ and contains a neighbourhood of the future light cone of $x$ of infinite volume (Bombelli, Henson, Sorkin, *Mod. Phys. Lett. A* **24** (2009) 2579). The Hasse diagram of a sprinkling into unbounded Minkowski space is therefore not locally finite. Sprinklings into bounded globally hyperbolic regions have almost surely finitely many points, and a finite graph is locally finite. This is the form used in Proposition 3.1 and throughout Chapter 20. ### 2.2. Optimal transport and Ollivier curvature Let $(X,d)$ be a complete separable metric space and $\mathcal{P}_{p}(X)$ the Borel probability measures with finite $p$-th moment. The $p$-Wasserstein distance is $$ W_{p}(\mu,\nu)=\Big(\inf_{\pi\in\Pi(\mu,\nu)}\int_{X\times X}d(x,y)^{p}\,\mathrm{d}\pi(x,y)\Big)^{1/p}, \tag{2.1} $$ the infimum over couplings with marginals $\mu,\nu$. Kantorovich-Rubinstein duality gives $W_{1}(\mu,\nu)=\sup\{\int f\,\mathrm{d}(\mu-\nu):\mathrm{Lip}(f)\leq 1\}$ (Villani, *Optimal Transport: Old and New*, Springer, 2009, Particular Case 5.16 of Thm 5.10). Let $(X,d)$ be a metric space and $x\mapsto m_{x}\in\mathcal{P}_{1}(X)$ a Markov kernel. Ollivier's *coarse Ricci curvature* along $(x,y)$, $x\neq y$, is $$ \kappa(x,y)=1-\frac{W_{1}(m_{x},m_{y})}{d(x,y)} \tag{2.2} $$ (Ollivier, *J. Funct. Anal.* **256** (2009) 810). On a locally finite graph with $m_{x}$ the uniform measure on a prescribed neighbour set of $x$ and $d=d_{G}$, (2.2) is a computable rational number. Ollivier's basic theorem is that $\kappa(x,y)\geq\kappa>0$ for all $x\neq y$ implies $W_{1}(\mu P,\nu P)\leq(1-\kappa)W_{1}(\mu,\nu)$ for the induced transition operator $P$ (loc. cit., Prop. 20). The continuous-time refinement of Lin, Lu, and Yau replaces $m_{x}$ by the lazy kernel $m_{x}^{\alpha}=\alpha\delta_{x}+(1-\alpha)m_{x}$ and sets $\kappa_{\mathrm{LLY}}(x,y)=\lim_{\alpha\to1}\kappa^{\alpha}(x,y)/(1-\alpha)$ (Lin, Lu, Yau, *Tohoku Math. J.* **63** (2011) 605). ### 2.3. Gromov-Hausdorff convergence For compact subsets $A,B$ of a metric space $(Z,d)$ the Hausdorff distance is $d_{H}(A,B)=\max\{\sup_{a\in A}d(a,B),\sup_{b\in B}d(b,A)\}$. The Gromov-Hausdorff distance between compact metric spaces is $$ d_{GH}(X_{1},X_{2})=\inf_{Z,\iota_{1},\iota_{2}}d_{H}^{Z}(\iota_{1}X_{1},\iota_{2}X_{2}) \tag{2.3} $$ over isometric embeddings into a common space (Burago, Burago, Ivanov, *A Course in Metric Geometry*, AMS, 2001, §7.3). It is a metric on isometry classes of compact metric spaces. A *metric measure space* is a triple $(X,d,\mathfrak{m})$ with $\mathfrak{m}$ a Borel measure. A sequence of compact metric measure spaces with finite total mass converges in the *measured Gromov-Hausdorff* (mGH) sense to $(X,d,\mathfrak{m})$ if there are a compact $Z$ and isometric embeddings $\iota_{n},\iota$ with $d_{H}^{Z}(\iota_{n}X_{n},\iota X)\to0$ and $(\iota_{n})_{\#}\mathfrak{m}_{n}\to\iota_{\#}\mathfrak{m}$ weakly (Fukaya, *Invent. Math.* **87** (1987) 517). The *pointed measured Gromov* (pmG) convergence of Gigli, Mondino, Savaré is the corresponding notion for possibly non-compact spaces with reference points, defined through a common ambient space and weak convergence of measures on bounded sets (Gigli, Mondino, Savaré, *J. Reine Angew. Math.* **705** (2015) 233, Def. 3.9); on compact spaces with probability measures it coincides with mGH. A metric measure space is *doubling* with constant $C_{\mathrm{dbl}}$ if $\mathfrak{m}(B(x,2r))\leq C_{\mathrm{dbl}}\mathfrak{m}(B(x,r))$ for all $x$ and $r>0$. ### 2.4. Synthetic Ricci curvature: Riemannian signature On a metric measure space $(X,d,\mathfrak{m})$ with $\mathfrak{m}$ a probability measure, the relative entropy is $\mathrm{Ent}_{\mathfrak{m}}(\mu)=\int\rho\log\rho\,\mathrm{d}\mathfrak{m}$ for $\mu=\rho\mathfrak{m}$ and $+\infty$ otherwise. The space satisfies $\mathrm{CD}(K,\infty)$ if for all $\mu_{0},\mu_{1}\in\mathcal{P}_{2}(X)$ with finite entropy there is a $W_{2}$-geodesic $(\mu_{t})$ with $$ \mathrm{Ent}_{\mathfrak{m}}(\mu_{t})\leq(1-t)\mathrm{Ent}_{\mathfrak{m}}(\mu_{0})+t\,\mathrm{Ent}_{\mathfrak{m}}(\mu_{1})-\tfrac{K}{2}t(1-t)W_{2}(\mu_{0},\mu_{1})^{2} \tag{2.4} $$ (Sturm, *Acta Math.* **196** (2006) 65; Lott, Villani, *Ann. Math.* **169** (2009) 903). The dimensional condition $\mathrm{CD}(K,N)$ replaces the entropy by the Rényi entropies $\mathcal{U}_{N}$ with distortion coefficients $\tau^{(t)}_{K,N}$ (Sturm, *Acta Math.* **196** (2006) 133). The space is *infinitesimally Hilbertian* if the Cheeger energy $\mathrm{Ch}(f)=\frac12\int|\nabla f|_{w}^{2}\,\mathrm{d}\mathfrak{m}$ is a quadratic form; $\mathrm{RCD}(K,N)$ means $\mathrm{CD}(K,N)$ together with infinitesimal Hilbertianity (Ambrosio, Gigli, Savaré, *Duke Math. J.* **163** (2014) 1405); Erbar, Kuwada, Sturm (*Invent. Math.* **201** (2015) 993) define $\mathrm{RCD}^{*}(K,N)$ through the reduced condition $\mathrm{CD}^{*}(K,N)$, and $\mathrm{RCD}^{*}(K,N)=\mathrm{RCD}(K,N)$ by Cavalletti, Milman, *Invent. Math.* **226** (2021) 1, Thm 1.1. On an infinitesimally Hilbertian space the heat flow $P_{t}=e^{t\Delta}$ is the $L^{2}$-gradient flow of $\mathrm{Ch}$, and the *Bakry-Émery condition* $\mathrm{BE}(K,\infty)$ is the inequality $\frac12\Delta|\nabla f|^{2}-\langle\nabla f,\nabla\Delta f\rangle\geq K|\nabla f|^{2}$ in the weak form $\Gamma_{2}(f)\geq K\Gamma(f)$. On a smooth complete Riemannian manifold with $\mathfrak{m}=\mathrm{vol}_{g}$, $\mathrm{CD}(K,N)$ holds iff $\mathrm{Ric}_{g}\geq Kg$ and $\dim\leq N$ (Sturm; Lott-Villani). Discrete spaces do not satisfy (2.4): $\mathcal{P}_{2}$ over a graph contains no non-trivial $W_{2}$-geodesics. The discrete theory is built instead on the transport distance $\mathcal{W}$ of Maas and the entropic curvature of Erbar and Maas (Erbar, Maas, *Arch. Rat. Mech. Anal.* **206** (2012) 997), or on Ollivier's $\kappa$. No passage from discrete Ollivier bounds to continuum $\mathrm{RCD}$ bounds is proved in this monograph; Classical B and Classical H below supply only heat-flow contraction and Mosco convergence, and the curvature-dimension condition on the limit is hypothesis H-BE. ### 2.5. Synthetic Ricci curvature: Lorentzian signature A *Lorentzian pre-length space* (Kunzinger, Sämann, *Ann. Glob. Anal. Geom.* **54** (2018) 399) is a quintuple $(X,\mathsf{d},\ll,\leq,\tau)$ where $(X,\mathsf{d})$ is a metric space, $\leq$ is a preorder, $\ll$ is a transitive relation contained in $\leq$, and $\tau\colon X\times X\to[0,\infty]$ is lower semicontinuous with $\tau(x,y)=0$ unless $x\leq y$, $\tau(x,y)>0$ iff $x\ll y$, and the reverse triangle inequality $$ \tau(x,z)\geq\tau(x,y)+\tau(y,z)\qquad(x\leq y\leq z). \tag{2.5} $$ The space is *globally hyperbolic* if it is non-totally-imprisoning and all causal diamonds $J^{+}(x)\cap J^{-}(y)$ are compact. A *measured* Lorentzian pre-length space carries in addition a Borel measure $\mathfrak{m}$. A *bounded Lorentzian metric space* in the sense of Minguzzi and Suhr is a set $X$ with $\tau\colon X\times X\to[0,\infty)$ bounded, satisfying (2.5) on its causal relation $x\leq y\iff\tau(x,y)>0\text{ or }x=y$, with a distinguishing property and a compactness property; Minguzzi and Suhr define a Gromov-Hausdorff distance on the isometry classes of such spaces and prove that it is a metric and that bounded Lorentzian metric spaces that are uniformly totally bounded form a precompact family (Minguzzi, Suhr, *Lett. Math. Phys.* **114** (2024) 73). We write GHL for this convergence. Three timelike curvature-dimension conditions are in use, for $p\in(0,1)$, $K\in\mathbb{R}$, $N\in[1,\infty)$, on a measured Lorentzian pre-length space with $\ell_{p}$ the $p$-Lorentz-Wasserstein distance. Let $\sigma^{(t)}_{K,N}(\theta)$ and $\tau^{(t)}_{K,N}(\theta)=t^{1/N}\sigma^{(t)}_{K,N-1}(\theta)^{(N-1)/N}$ be the distortion coefficients of Sturm, $\mathcal{U}_{N}(\mu)=\int\rho^{1-1/N}\,\mathrm{d}\mathfrak{m}$ the Rényi entropy, and $\mathrm{Ent}_{\mathfrak{m}}$ the Boltzmann entropy. *Entropic condition* $\mathrm{TCD}^{e}_{p}(K,N)$ (Cavalletti, Mondino, *Camb. J. Math.* **12** (2024) 417, Def. 3.2): for every timelike $p$-dualisable pair $\mu_{0},\mu_{1}$ with finite entropy there is an $\ell_{p}$-geodesic $(\mu_{t})$ along which $U_{N}=\exp(-\mathrm{Ent}_{\mathfrak{m}}/N)$ satisfies $U_{N}(\mu_{t})\geq\sigma^{(1-t)}_{K/N,1}(\ell_{p}(\mu_{0},\mu_{1}))U_{N}(\mu_{0})+\sigma^{(t)}_{K/N,1}(\ell_{p}(\mu_{0},\mu_{1}))U_{N}(\mu_{1})$. *Reduced condition* $\mathrm{TCD}^{*}_{p}(K,N)$ (Braun, *J. Math. Pures Appl.* **177** (2023) 46, Def. 3.1): the same with $\mathcal{U}_{N'}$ in place of $U_{N}$ and coefficients $\sigma^{(t)}_{K,N'}$, for all $N'\geq N$. *Condition* $\mathrm{TCD}_{p}(K,N)$ (Braun, loc. cit., Def. 3.1): the same with $\mathcal{U}_{N'}$ and the coefficients $\tau^{(t)}_{K,N'}$. Braun proves $\mathrm{TCD}_{p}(K,N)\Rightarrow\mathrm{TCD}^{*}_{p}(K,N)$ (loc. cit., Prop. 3.4) and, on timelike $p$-essentially non-branching spaces, $\mathrm{TCD}^{*}_{p}(K,N)\Leftrightarrow\mathrm{TCD}^{e}_{p}(K,N)$ (loc. cit., Thm 3.35). On a smooth globally hyperbolic spacetime $(M,g)$ with $\mathfrak{m}=\mathrm{vol}_{g}$ and $\dim M=d$, all three hold with $N\geq d$ iff $\mathrm{Ric}(v,v)\geq K|g(v,v)|$ for all timelike $v$; in the signature $(-,+,+,+)$ adopted here this reads $\mathrm{Ric}(v,v)\geq-Kg(v,v)$, and the absolute value is kept throughout to avoid the sign ambiguity between conventions (McCann, *Camb. J. Math.* **8** (2020) 609, Thm 8.5, in signature $(+,-,-,-)$; Mondino, Suhr, *J. Eur. Math. Soc.* **25** (2023) 933, Thm 1.1; Cavalletti, Mondino, loc. cit., Thm 3.10). ### 2.6. von Neumann algebras A von Neumann algebra $\mathcal{M}\subseteq B(\mathcal{H})$ is a unital $*$-subalgebra closed in the weak operator topology; equivalently $\mathcal{M}=\mathcal{M}''$. It is a *factor* if $\mathcal{M}\cap\mathcal{M}'=\mathbb{C}\mathbf{1}$. Factors are classified by Murray and von Neumann into types $\mathrm{I}_{n}$, $\mathrm{I}_{\infty}$, $\mathrm{II}_{1}$, $\mathrm{II}_{\infty}$, $\mathrm{III}$; a factor is of type $\mathrm{II}_{1}$ iff it is infinite-dimensional and admits a faithful normal tracial state. $\mathcal{M}$ is *hyperfinite* (approximately finite-dimensional) if it is the weak closure of an increasing union of finite-dimensional $*$-subalgebras. Tomita-Takesaki theory attaches to a faithful normal state $\omega$ on $\mathcal{M}$ a modular operator $\Delta_{\omega}$ and the modular automorphism group $\sigma^{\omega}_{t}=\mathrm{Ad}\,\Delta_{\omega}^{it}$; $\omega$ satisfies the KMS condition at inverse temperature $1$ with respect to $\sigma^{\omega}$, and $\sigma^{\omega}$ is trivial iff $\omega$ is a trace (Takesaki, *Theory of Operator Algebras II*, Springer, 2003, Thm VIII.1.2). ### 2.7. The functional renormalization group For a field theory with bare action $S$ and infrared regulator $R_{k}$, the effective average action $\Gamma_{k}$ interpolates between $S$ at $k\to\infty$ and the full effective action at $k=0$ and satisfies the Wetterich equation $$ \partial_{t}\Gamma_{k}=\tfrac12\,\mathrm{STr}\big[(\Gamma^{(2)}_{k}+R_{k})^{-1}\partial_{t}R_{k}\big],\qquad t=\log k \tag{2.6} $$ (Wetterich, *Phys. Lett. B* **301** (1993) 90; Morris, *Int. J. Mod. Phys. A* **9** (1994) 2411). For gravity the background-field formulation of Reuter with the Litim regulator $R_{k}(p^{2})=(k^{2}-p^{2})\theta(k^{2}-p^{2})$ is used (Reuter, *Phys. Rev. D* **57** (1998) 971; Litim, *Phys. Rev. D* **64** (2001) 105007). The Einstein-Hilbert truncation is $\Gamma_{k}=\frac{1}{16\pi G_{k}}\int\sqrt{g}(2\Lambda_{k}-R)+S_{\mathrm{gf}}+S_{\mathrm{gh}}$ with dimensionless couplings $\widetilde{G}_{k}=k^{d-2}G_{k}$, $\widetilde{\Lambda}_{k}=\Lambda_{k}/k^{2}$, which for $d=4$ read $k^{2}G_{k}$ and $\Lambda_{k}/k^{2}$. ### 2.8. Quantum groups and modular categories For a complex simple Lie algebra $\mathfrak{g}$ with Cartan matrix $(a_{ij})$ and symmetrising integers $d_{i}$, the Drinfeld-Jimbo algebra $U_{q}(\mathfrak{g})$ is the Hopf algebra over $\mathbb{C}(q)$ generated by $E_{i},F_{i},K_{i}^{\pm1}$ with $K_{i}E_{j}K_{i}^{-1}=q_{i}^{a_{ij}}E_{j}$, $[E_{i},F_{j}]=\delta_{ij}(K_{i}-K_{i}^{-1})/(q_{i}-q_{i}^{-1})$, $q_{i}=q^{d_{i}}$, and the quantum Serre relations (Chari, Pressley, *A Guide to Quantum Groups*, Cambridge, 1995, §9.1). Lusztig's divided-power form allows specialisation at a root of unity $q=e^{i\pi/\ell}$. The category of tilting modules modulo the tensor ideal of negligible morphisms is a ribbon fusion category (Andersen, Paradowski, *Commun. Math. Phys.* **169** (1995) 563). A ribbon fusion category is *modular* if its $S$-matrix is invertible, and *unitary* if it admits a compatible positive dagger structure (Bakalov, Kirillov, *Lectures on Tensor Categories and Modular Functors*, AMS, 2001; Turaev, *Quantum Invariants of Knots and 3-Manifolds*, de Gruyter, 1994). ### 2.9. The classical inputs Each item below is stated as a numbered theorem with the hypotheses and conclusion of the cited source, with the source's theorem number where it has one. Two items (B-Lor, C-Lor) are not theorems in the literature in the form needed and are labelled accordingly. Later chapters cite each input by its letter and discharge its hypotheses in labelled steps. **Theorem 2.2** $\boxed{\text{Theorem}}$ **(Classical A: Gromov precompactness).** Let $\mathcal{F}$ be a family of compact metric spaces with $\mathrm{diam}\leq T$ and such that there is a function $\mathcal{N}\colon(0,\infty)\to\mathbb{N}$ with the property that every $X\in\mathcal{F}$ contains at most $\mathcal{N}(\varepsilon)$ points pairwise at distance $\geq\varepsilon$. Then $\mathcal{F}$ is precompact in the Gromov-Hausdorff topology. If each $X$ is doubling with constant $C_{\mathrm{dbl}}$, then one may take $\mathcal{N}(\varepsilon)=C_{\mathrm{dbl}}^{\lceil\log_{2}(2T/\varepsilon)\rceil+1}$. If in addition each $X$ carries a probability measure, every GH-convergent subsequence has a further mGH-convergent subsequence. (Gromov, *Metric Structures for Riemannian and Non-Riemannian Spaces*, Birkhäuser, 2007, Prop. 5.2 and Thm 5.3; Burago-Burago-Ivanov Thm 7.4.15; Fukaya, *Invent. Math.* **87** (1987) 517, Prop. 2.1 and Prokhorov's theorem for the measured refinement.) The doubling-to-net estimate is proved as follows, assuming as part of the doubling hypothesis that $0<\mathfrak{m}(B(x,r))<\infty$ for every ball: if $x_{1},\dots,x_{m}$ are $\varepsilon$-separated then the balls $B(x_{i},\varepsilon/2)$ are disjoint and contained in $B(x_{1},T+\varepsilon/2)$; iterating doubling $j=\lceil\log_{2}(2T/\varepsilon)\rceil+1$ times gives $\mathfrak{m}(B(x_{1},T+\varepsilon/2))\leq C_{\mathrm{dbl}}^{j}\mathfrak{m}(B(x_{1},\varepsilon/2))$, and the same bound holds with $x_{1}$ replaced by any $x_{i}$ since $B(x_{i},T+\varepsilon/2)\supseteq X$; summing the disjoint balls, $m\min_{i}\mathfrak{m}(B(x_{i},\varepsilon/2))\leq\mathfrak{m}(X)\leq C_{\mathrm{dbl}}^{j}\min_{i}\mathfrak{m}(B(x_{i},\varepsilon/2))$, whence $m\leq C_{\mathrm{dbl}}^{j}$. **Theorem 2.3** $\boxed{\text{Theorem}}$ **(Classical B: Ollivier bounds and heat-flow contraction, three parts).** *(B1)* On a locally finite graph with bounded degree, with the lazy-walk Laplacian $\Delta=P-I$ and $P_{t}=e^{t\Delta}$, the Lin-Lu-Yau curvature satisfies $\kappa_{\mathrm{LLY}}\geq K$ on all pairs iff $\|\nabla P_{t}f\|_{\infty}\leq e^{-Kt}\|\nabla f\|_{\infty}$ for all $t\geq0$ and all bounded $f$, iff $W_{1}(\mu P_{t},\nu P_{t})\leq e^{-Kt}W_{1}(\mu,\nu)$ for all $\mu,\nu\in\mathcal{P}_{1}(N)$ (the moment condition is automatic on finite graphs), where $\|\nabla f\|_{\infty}=\sup_{(x,y)\in E}|f(x)-f(y)|$ (Münch, Wojciechowski, *Adv. Math.* **356** (2019) 106759, Thms 2.1 and 3.1). *(B2)* (Kuwada duality, two forms.) *Global form:* on a complete separable metric space with a Markov operator $P$ acting on bounded measurable functions and dually on probability measures, the estimate $W_{1}(\mu P,\nu P)\leq e^{-Kt}W_{1}(\mu,\nu)$ for all $\mu,\nu$ is equivalent to $\mathrm{Lip}(Pf)\leq e^{-Kt}\mathrm{Lip}(f)$ for all bounded Lipschitz $f$, with $\mathrm{Lip}$ the global Lipschitz constant; this follows from Kantorovich-Rubinstein duality and needs no kernel. *Kuwada pairing:* on a complete separable length space with $P_{t}$ having a kernel, the contraction $W_{p}(\mu P_{t},\nu P_{t})\leq C_{t}W_{p}(\mu,\nu)$ is equivalent to the gradient estimate $|\nabla P_{t}f|(x)\leq C_{t}\big(P_{t}|\nabla f|^{q}\big)^{1/q}(x)$ with $1/p+1/q=1$ and $|\nabla f|$ the local Lipschitz constant (Kuwada, *J. Funct. Anal.* **258** (2010) 3758, Thm 2.2). With $p=1$, $q=\infty$ this is $|\nabla P_{t}f|\leq e^{-Kt}\||\nabla f|\|_{\infty}$, a Lipschitz-constant bound; the pointwise estimate $|\nabla P_{t}f|\leq e^{-Kt}P_{t}|\nabla f|$ is the case $q=1$ and is dual to $W_{\infty}$ contraction. An Ollivier bound therefore yields, through B1 and B2, the $W_{1}$ contraction and the Lipschitz-constant estimate on the limit, and nothing stronger. *(B3)* (Ambrosio-Gigli-Savaré characterisation.) Let $(X,d,\mathfrak{m})$ be a complete separable metric measure space with $\mathfrak{m}$ a Borel measure satisfying $\mathfrak{m}(B(x_{0},r))\leq C e^{cr^{2}}$ for some $x_{0}$, $C,c>0$, and all $r>0$, such that $d$ coincides with the intrinsic distance of the Cheeger energy. Then $(X,d,\mathfrak{m})$ is $\mathrm{RCD}(K,\infty)$ iff it is infinitesimally Hilbertian, has the Sobolev-to-Lipschitz property, and satisfies $\mathrm{BE}(K,\infty)$, i.e. $\Gamma_{2}(f)\geq K\Gamma(f)$ in the weak form for $f$ in the domain of the generator (Ambrosio, Gigli, Savaré, *Ann. Probab.* **43** (2015) 339, Thm 4.17). Under these hypotheses $\mathrm{BE}(K,\infty)$ is equivalent to the $L^{2}$ gradient estimate $|\nabla P_{t}f|^{2}\leq e^{-2Kt}P_{t}|\nabla f|^{2}$ (loc. cit., Cor. 4.18), and the $L^{1}$-type estimate $|\nabla P_{t}f|\leq e^{-Kt}P_{t}|\nabla f|$ is equivalent to these under the same hypotheses (Savaré, *Discrete Contin. Dyn. Syst.* **34** (2014) 1641, Thm 3.4 and Cor. 3.5). None of the equivalent conditions of B3 ($\mathrm{BE}(K,\infty)$, the $L^{2}$ estimate, the $P_{t}|\nabla f|$ estimate) is delivered by B1 and B2, which give only the $W_{1}$ contraction and the Lipschitz-constant bound. On a smooth Riemannian manifold every $W_{p}$ contraction with rate $e^{-Kt}$ is equivalent to $\mathrm{Ric}\geq K$ (von Renesse, Sturm, *Comm. Pure Appl. Math.* **58** (2005) 923, Thm 1), but that equivalence uses smoothness, which the limit does not have. The passage from the $W_{1}$ contraction to $\mathrm{BE}(K,\infty)$ on a non-smooth limit is therefore recorded as a hypothesis: *(H-BE)* The mGH limit $(X,d,\mathfrak{m})$ of Theorem 5.5 is infinitesimally Hilbertian, has the Sobolev-to-Lipschitz property, satisfies the Gaussian volume bound and the intrinsic-distance identity of B3, and its heat semigroup satisfies $|\nabla P_{t}f|^{2}\leq e^{-2Kt}P_{t}|\nabla f|^{2}$ $\mathfrak{m}$-a.e. Even under the gradient estimate above, B3 gives only $\mathrm{RCD}(K,\infty)$, and no step in this monograph upgrades it to $\mathrm{RCD}(K,N_{0})$; the hypothesis H-BE of §3.3 therefore assumes $\mathrm{RCD}(K,N_{0})$ on the limit outright. Theorem 5.5 lists (H-conv) and (H-BE) and is Conditional on both. **Theorem 2.4** $\boxed{\text{Theorem}}$ **(Classical C: structure of non-collapsed RCD spaces, three parts).** *(C1)* $\mathrm{RCD}(K,N)$ spaces with $N<\infty$ are rectifiable: $\mathfrak{m}$-almost all of $X$ is covered by countably many bi-Lipschitz charts to subsets of $\mathbb{R}^{k}$, $k\leq N$, and the dimension $k$ is $\mathfrak{m}$-a.e. constant (Mondino, Naber, *J. Eur. Math. Soc.* **21** (2019) 1809; Brué, Semola, *Comm. Pure Appl. Math.* **73** (2020) 1141). *(C2)* Let $(X,d,\mathcal{H}^{N})$ be a non-collapsed $\mathrm{RCD}(K,N)$ space in the sense of De Philippis and Gigli, i.e. $\mathfrak{m}=\mathcal{H}^{N}$. Then the singular set $\mathcal{S}$ (points at which no tangent cone is $\mathbb{R}^{N}$) satisfies $\dim_{\mathcal{H}}\mathcal{S}\leq N-2$, and for every $\varepsilon>0$ the $\varepsilon$-regular set $\mathcal{R}_{\varepsilon}$ is open and is a topological $N$-manifold that is bi-Hölder homeomorphic, with exponent $1-\varepsilon'$ depending on $\varepsilon$, to a smooth manifold on bounded sets (De Philippis, Gigli, *J. Éc. Polytech. Math.* **5** (2018) 613, Thms 1.2-1.4; Kapovitch, Mondino, *Geom. Topol.* **25** (2021) 445, Thm 1.1). The full regular set $\mathcal{R}=\bigcap_{\varepsilon}\mathcal{R}_{\varepsilon}$ is a $G_{\delta}$ and need not be open; $\mathcal{S}$ need not be closed, and can be dense (Alexandrov surfaces with a dense set of conical points are non-collapsed $\mathrm{RCD}$ spaces). *(C3)* (Anderson convergence.) Let $(M_{i},g_{i})$ be smooth complete Riemannian $N$-manifolds with $|\mathrm{Ric}_{g_{i}}|\leq\Lambda$, $\mathrm{inj}(M_{i})\geq i_{0}>0$, and $\mathrm{diam}\leq T$. Then a subsequence converges in the $C^{1,\alpha}$ topology, for every $\alpha<1$, to a $C^{1,\alpha}$ Riemannian manifold, the convergence being realised in harmonic coordinates of radius bounded below by $r_{h}(N,\Lambda,i_{0})$ (Anderson, *Invent. Math.* **102** (1990) 429, Thm 1.1 and Lemma 2.2). This applies to sequences of *smooth* manifolds; it does not bootstrap a $C^{0}$ metric to $C^{1,\alpha}$, and a two-sided Ricci bound for a merely continuous metric is not defined. Smoothness of the limit metric is not a consequence of (C1)-(C3). It is hypothesis H10 below. **Theorem 2.5** $\boxed{\text{Theorem}}$ **(Classical A-Lor: Minguzzi-Suhr precompactness).** Let $\mathcal{F}$ be a family of bounded Lorentzian metric spaces such that $\sup_{X\in\mathcal{F}}\sup\tau\leq T$ and such that there is $\mathcal{N}\colon(0,\infty)\to\mathbb{N}$ with: for every $\varepsilon>0$ every $X\in\mathcal{F}$ admits an $\varepsilon$-net of cardinality $\leq\mathcal{N}(\varepsilon)$ for the Minguzzi-Suhr distinguishing metric $\mathsf{d}_{X}(x,y)=\sup_{z}|\tau(x,z)-\tau(y,z)|+\sup_{z}|\tau(z,x)-\tau(z,y)|$. Then $\mathcal{F}$ is precompact in GHL, and the GHL limit of bounded Lorentzian metric spaces is a bounded Lorentzian metric space; the GHL distance is a metric on isometry classes, so limits are unique up to isometry (Minguzzi, Suhr, *Lett. Math. Phys.* **114** (2024) 73: the theorem that the GH distance is a metric on isometry classes, the completeness theorem, and the precompactness theorem for uniformly totally bounded families; we cite by statement since the numbering differs between the arXiv and journal versions). The measured refinement is obtained as follows. A bounded Lorentzian metric space is not compact in general; only the sets $\{\tau\geq\varepsilon\}$ are. Minguzzi and Suhr adjoin a point $i_{0}$ at infinity to obtain a compact metrisable space $\overline{X}=X\cup\{i_{0}\}$ on which $\tau$ extends by zero, and a GHL-convergent sequence is realised by isometric embeddings of the compactifications into a common compact metric space $\overline{Z}$ (loc. cit., the compactification construction of §2 and the realisation lemma of §4). The push-forwards of the probability measures $\mathfrak{m}_{n}$ to the compact $\overline{Z}$ are tight, Prokhorov's theorem extracts a weakly convergent subsequence with limit $\overline{\mathfrak{m}}$, and the hypothesis $\mathfrak{m}_{n}(\{\tau_{n}<\varepsilon\text{ to every point}\})\to0$ uniformly, which Chapter 5 verifies (no mass escapes to the adjoined point), gives $\overline{\mathfrak{m}}(\{i_{0}\})=0$, so the limit measure lives on $X$. **Proposition 2.6** $\boxed{\text{Conditional (on H-conv-Lor)}}$ **(Classical B-Lor: timelike curvature stability).** Let $(X_{n},\tau_{n},\mathfrak{m}_{n})$ be measured globally hyperbolic Lorentzian geodesic spaces, each satisfying $\mathrm{TCD}^{*}_{p}(K,N)$ with the same $p\in(0,1)$, $K\in\mathbb{R}$, $N\in[1,\infty)$, and let $(X,\tau,\mathfrak{m})$ be a measured globally hyperbolic Lorentzian geodesic space. Suppose they converge to $(X,\tau,\mathfrak{m})$ in the sense that there are isometric embeddings into a common space along which $\tau_{n}\to\tau$ uniformly on compacts and $\mathfrak{m}_{n}\to\mathfrak{m}$ weakly. Assume the following hypothesis. *(H-conv-Lor)* For every pair $\mu_{0},\mu_{1}\in\mathcal{P}(X)$ of timelike $p$-dualisable measures with $\mathcal{U}_{N}(\mu_{i})>-\infty$, there are $\mu^{n}_{i}\in\mathcal{P}(X_{n})$, timelike $p$-dualisable, with $\mu^{n}_{i}\to\mu_{i}$ weakly, $\mathcal{U}_{N'}^{\mathfrak{m}_{n}}(\mu^{n}_{i})\to\mathcal{U}_{N'}^{\mathfrak{m}}(\mu_{i})$ for $i=0,1$ and every $N'\geq N$, $\ell_{p}(\mu^{n}_{0},\mu^{n}_{1})\to\ell_{p}(\mu_{0},\mu_{1})$, with $\ell_{p}(\mu_{0},\mu_{1})<\pi\sqrt{N/K}$ when $K>0$ (the inequality being void at and beyond the threshold, where $\sigma=+\infty$), and $\ell_{p}$-geodesics $(\mu^{n}_{t})$ in $X_{n}$ converging weakly for each $t$ to an $\ell_{p}$-geodesic $(\mu_{t})$ in $X$. Then $X$ satisfies $\mathrm{TCD}^{*}_{p}(K,N)$. *Proof.* Fix $N'\geq N$ and write $S_{N'}=-\mathcal{U}_{N'}$. For each $n$ and $t$, $\mathrm{TCD}^{*}_{p}(K,N)$ on $X_{n}$ gives $\mathcal{U}_{N'}^{\mathfrak{m}_{n}}(\mu^{n}_{t})\geq\sigma^{(1-t)}_{K,N'}(\ell_{p}(\mu^{n}_{0},\mu^{n}_{1}))\,\mathcal{U}_{N'}^{\mathfrak{m}_{n}}(\mu^{n}_{0})+\sigma^{(t)}_{K,N'}(\ell_{p}(\mu^{n}_{0},\mu^{n}_{1}))\,\mathcal{U}_{N'}^{\mathfrak{m}_{n}}(\mu^{n}_{1})$. The functional $(\mu,\mathfrak{m})\mapsto S_{N'}(\mu|\mathfrak{m})$ is jointly lower semicontinuous under weak convergence (Sturm, *Acta Math.* **196** (2006) 133, Lemma 3.2, whose proof uses only joint convexity and lower semicontinuity of the integrand), so $\limsup_{n}\mathcal{U}_{N'}^{\mathfrak{m}_{n}}(\mu^{n}_{t})\leq\mathcal{U}_{N'}^{\mathfrak{m}}(\mu_{t})$. The endpoint terms converge by the hypothesis on $\mathcal{U}_{N'}(\mu^{n}_{i})$. The coefficients $\theta\mapsto\sigma^{(t)}_{K,N'}(\theta)$ are continuous on $[0,\infty)$ for $K\leq0$ and on $[0,\pi\sqrt{N'/K})$ for $K>0$; since $N'\geq N$ the threshold hypothesis $\ell_{p}(\mu_{0},\mu_{1})<\pi\sqrt{N/K}\leq\pi\sqrt{N'/K}$ places the limit argument in the continuity range, so the coefficients converge by the hypothesis on $\ell_{p}$. Passing to the limit, $\mathcal{U}_{N'}^{\mathfrak{m}}(\mu_{t})\geq\sigma^{(1-t)}_{K,N'}(\ell_{p}(\mu_{0},\mu_{1}))\mathcal{U}_{N'}^{\mathfrak{m}}(\mu_{0})+\sigma^{(t)}_{K,N'}(\ell_{p}(\mu_{0},\mu_{1}))\mathcal{U}_{N'}^{\mathfrak{m}}(\mu_{1})$ for the limiting geodesic, for every $N'\geq N$, which is $\mathrm{TCD}^{*}_{p}(K,N)$ (Braun, *J. Math. Pures Appl.* **177** (2023) 46, Def. 3.1). $\square$ On timelike $p$-essentially non-branching spaces, $\mathrm{TCD}^{*}_{p}(K,N)$ is equivalent to $\mathrm{TCD}^{e}_{p}(K,N)$ (Braun, loc. cit., Thm 3.35), so under H7 the conclusion may be read in either form. The hypothesis (H-conv-Lor), convergence of discrete timelike transport geodesics to continuum ones, is not a theorem in the literature for sequences of discrete spaces; Chapter 7 states it where used and verifies it on the reference families of Chapter 20. Results depending on B-Lor are tagged with that dependence. **Conjecture 2.7** $\boxed{\text{Conjecture}}$ **(Classical C-Lor: regularity of non-collapsed TCD limits).** The Lorentzian counterpart of C2 for measured Lorentzian spaces satisfying $\mathrm{TCD}_{p}(K,N)$ with $\mathfrak{m}$ the $N$-dimensional Lorentzian Hausdorff measure: decomposition into an open regular set that is a topological manifold carrying a continuous Lorentzian metric and a closed singular set of codimension at least two. No such theorem is currently published. The most developed tools are the non-smooth d'Alembert comparison and causal calculus of Beran, Braun, Calisti, Gigli, McCann, Ohanyan, Rott, Sämann (arXiv:2408.15968, 2024) and the synthetic singularity theorems of Cavalletti-Mondino. Every result in this monograph that invokes C-Lor is tagged with that dependence, and Chapter 20 gives the reference families on which the conclusion of C-Lor is verified directly rather than invoked. **Theorem 2.8** $\boxed{\text{Theorem}}$ **(Classical D: $\Gamma$-convergence on stochastic lattices).** Let $\mathcal{L}$ be a stationary admissible stochastic point set in $\mathbb{R}^{d}$ (admissible: there are $r,R>0$ with pairwise distances $\geq r$ and every ball of radius $R$ containing a point). Let $F_{\varepsilon}(u)=\varepsilon^{d}\sum_{x,y\in\varepsilon\mathcal{L}}f(x,y,(u(x)-u(y))/\varepsilon)$ with $f$ measurable, of two-sided $p$-growth, $c_{1}|\xi|^{p}-c_{2}\leq f(x,y,\xi)\leq c_{3}(1+|\xi|^{p})$ for $p>1$, and with finite range. Then almost surely $F_{\varepsilon}$ $\Gamma$-converges in $L^{p}_{\mathrm{loc}}$ to an integral functional $\int_{\Omega}f_{\mathrm{hom}}(\omega,\nabla u)\,\mathrm{d}x$ whose integrand is a random variable given by an asymptotic cell formula; if $\mathcal{L}$ is ergodic then $f_{\mathrm{hom}}$ is deterministic (Alicandro, Cicalese, Gloria, *Arch. Rat. Mech. Anal.* **200** (2011) 881, main homogenisation theorem for stationary and for ergodic stochastic lattices; cited by statement). Alicandro-Cicalese-Gloria require stationarity only with respect to the translation group $\mathbb{Z}^{d}$ (loc. cit., §2, "stochastic lattices"). A Poisson point process is not admissible: it has no minimum distance, and, since a ball of radius $R$ at the origin is empty with positive probability, by ergodicity it almost surely contains empty balls of every radius, so no thinning of it satisfies the covering condition. The monograph therefore applies D to the following *admissible modification* of a Poisson process $\Pi$ of intensity $\rho$ in $\mathbb{R}^{d}$. Fix $R>0$ and partition $\mathbb{R}^{d}$ into the cubes $Q_{z}=R(z+[0,1)^{d})$, $z\in\mathbb{Z}^{d}$, with centres $c_{z}$. For each $z$ let $\xi_{z}$ be the point of $\Pi\cap B(c_{z},R/4)$ nearest to $c_{z}$ if that set is non-empty, and $\xi_{z}=c_{z}$ otherwise; set $\mathcal{L}=\{\xi_{z}:z\in\mathbb{Z}^{d}\}$. Then $|\xi_{z}-c_{z}|\leq R/4$ for every $z$, so for $z\neq z'$, $|\xi_{z}-\xi_{z'}|\geq|c_{z}-c_{z'}|-R/2\geq R/2$, and every ball of radius $R\sqrt{d}$ contains some $c_{z}$ together with $B(c_{z},R/4)$, hence contains $\xi_{z}$: $\mathcal{L}$ is admissible with $r=R/2$ and covering radius $R\sqrt{d}$. It is $R\mathbb{Z}^{d}$-stationary because the law of $\Pi$ is translation invariant and the construction commutes with translations by $Rz$, and it is ergodic under $R\mathbb{Z}^{d}$ because $\Pi$ is mixing under translations and $\xi_{z}$ depends only on $\Pi\cap B(c_{z},R/4)$. In the regime $\rho R^{d}\to\infty$ (the relevant one for Ch 8, where $R=R_n$ and $\rho=\rho_n$ vary jointly with $\rho_nR_n^{d}$ fixed or growing) the event $\Pi\cap B(c_{z},R/4)=\emptyset$ has probability $e^{-\rho\omega_{d}(R/4)^{d}}\to0$, so $\mathcal{L}$ coincides with a subset of $\Pi$ except on a set of cubes of vanishing density. Chapter 8 uses this $\mathcal{L}$ with $R$ fixed and $\rho\to\infty$ and cites the present theorem rather than restating D. **Theorem 2.9** $\boxed{\text{Theorem}}$ **(Classical E: Ollivier pointwise expansion).** Let $(M,g)$ be a smooth complete Riemannian $N$-manifold, $x\in M$, $v\in T_{x}M$ a unit vector, $y=\exp_{x}(\delta v)$, and let $m^{\varepsilon}_{x}$ be the normalised Riemannian volume on the geodesic ball $B(x,\varepsilon)$. Then $$ \kappa^{\varepsilon}(x,y)=\frac{\varepsilon^{2}}{2(N+2)}\,\mathrm{Ric}_{x}(v,v)+O(\varepsilon^{3}+\varepsilon^{2}\delta) \tag{2.7} $$ as $\varepsilon,\delta\to0$, with the $O$ uniform on compact sets (Ollivier, *J. Funct. Anal.* **256** (2009) 810, Example 7 and Thm 8). The Lorentzian analogue for the causal-diamond random walk is *not* a published classical input; it is proved in Chapter 7 (Lemma 7.4) from the small-diamond volume expansion of Gibbons and Solodukhin (*Phys. Lett. B* **649** (2007) 317) and Myrheim-Meyer. **Theorem 2.10** $\boxed{\text{Theorem}}$ **(Classical F: discrete Bakry-Émery and Ollivier).** On a locally finite graph the Lin-Lu-Yau curvature satisfies $\kappa_{\mathrm{LLY}}(x,y)=\lim_{\alpha\to1}\kappa^{\alpha}(x,y)/(1-\alpha)$ and is given by the Laplacian formula $\kappa_{\mathrm{LLY}}(x,y)=\inf\{\nabla_{yx}\Delta f: f\in\mathrm{Lip}(1),\ \nabla_{xy}f=1\}$ (Münch, Wojciechowski, loc. cit., Thm 2.1). **Theorem 2.11** $\boxed{\text{Theorem}}$ **(Classical G-Lor: relations among the timelike conditions).** On a timelike $p$-essentially non-branching measured globally hyperbolic Lorentzian geodesic space, $\mathrm{TCD}^{*}_{p}(K,N)$ and $\mathrm{TCD}^{e}_{p}(K,N)$ are equivalent (Braun, *J. Math. Pures Appl.* **177** (2023) 46, Thm 3.35); $\mathrm{TCD}_{p}(K,N)$ implies $\mathrm{TCD}^{*}_{p}(K,N)$ (loc. cit., Prop. 3.4); and $\mathrm{TCD}^{e}_{p}(K,N)$ implies the timelike measure contraction property $\mathrm{TMCP}^{e}(K,N)$ and the timelike Brunn-Minkowski and Bishop-Gromov inequalities (Cavalletti, Mondino, loc. cit., Prop. 3.11 and Thm 3.18; Braun, loc. cit., Thm 1.3 for the sharp Brunn-Minkowski inequality). No timelike evolution-variational-inequality (EVI) characterisation is stated or used in this monograph; the description of G-Lor in earlier planning documents as "timelike EVI iff TCD" is superseded by the present statement. **Theorem 2.12** $\boxed{\text{Theorem}}$ **(Classical H: Mosco convergence of Cheeger energies).** Let $(X_{n},d_{n},\mathfrak{m}_{n},\bar{x}_{n})$ converge in pmG to $(X,d,\mathfrak{m},\bar{x})$, all spaces satisfying $\mathrm{RCD}(K,\infty)$ for a common $K$. Then the Cheeger energies Mosco-converge, the heat semigroups converge ($P^{n}_{t}f_{n}\to P_{t}f$ in $L^{2}$ whenever $f_{n}\to f$ in $L^{2}$), and $(X,d,\mathfrak{m})$ is $\mathrm{RCD}(K,\infty)$ (Gigli, Mondino, Savaré, *J. Reine Angew. Math.* **705** (2015) 233, Thms 6.8 and 7.2). For sequences of *graphs*, which are not $\mathrm{RCD}$, the monograph uses instead the spectral and Mosco convergence of graph Laplacians and Dirichlet forms on point clouds sampled from manifolds, proved by García Trillos, Gerlach, Hein, Slepčev (*Found. Comput. Math.* **20** (2020) 827, Thm 2) and García Trillos, Slepčev (*Appl. Comput. Harmon. Anal.* **45** (2018) 239, Thm 1.1), which cover i.i.d. and Poisson samples with connectivity radius $\varepsilon_{n}$ satisfying $\varepsilon_{n}\gg(\log n/n)^{1/N}$; the hypothesis consumed is that the graph Dirichlet forms, with the scaling fixed in Chapter 5, Mosco-converge to the Cheeger energy of the mGH limit. Chapter 5 states this as (H-conv) and verifies it on the reference families of Chapter 20. **Theorem 2.13** $\boxed{\text{Theorem}}$ **(Classical I: Combes-Thomas).** Let $H$ be a bounded self-adjoint operator on $\ell^{2}(N)$ over a graph of bounded degree with $|H_{nm}|\leq C_{H}e^{-\nu d_{G}(n,m)}$, and let $z\notin\sigma(H)$ with $\mathrm{dist}(z,\sigma(H))=\eta>0$. Then $|(H-z)^{-1}_{nm}|\leq\frac{2}{\eta}e^{-\nu' d_{G}(n,m)}$ with $\nu'=\nu'(\nu,\eta,C_{H},D)>0$ (Combes, Thomas, *Commun. Math. Phys.* **34** (1973) 251; Aizenman, Warzel, *Random Operators*, AMS, 2015, Thm 10.5 for the graph form). **Theorem 2.14** $\boxed{\text{Theorem}}$ **(Classical J: locality of the sign function).** Let $H$ be bounded self-adjoint with $\sigma(H)\cap(-\gamma,\gamma)=\emptyset$, $\gamma>0$, and exponentially decaying matrix elements as in I. Then $\mathrm{sign}(H)=\frac{2}{\pi}\int_{0}^{\infty}H(H^{2}+s^{2})^{-1}\,\mathrm{d}s$ and $|\mathrm{sign}(H)_{nm}|\leq C e^{-\xi d_{G}(n,m)}$ with $\xi=\xi(\nu,\gamma/\|H\|,C_{H},D)>0$. This is the single-particle statement that the sign of a gapped operator with exponentially decaying entries has exponentially decaying entries; it is proved, with explicit rates, in Benzi, Boito, Razouk, *SIAM Rev.* **55** (2013) 3, Thm 6.2 (for banded matrices) and §7 (for general sparsity graphs). The many-body versions (Hastings, *Phys. Rev. Lett.* **93** (2004) 140402; Hastings, Michalakis, *Commun. Math. Phys.* **334** (2015) 433) are not used. **Theorem 2.15** $\boxed{\text{Theorem}}$ **(Classical K: Neuberger overlap).** Let $D_{W}$ be a $\gamma_{5}$-Hermitian operator, $D_{W}^{\dagger}=\gamma_{5}D_{W}\gamma_{5}$, set $H_{W}=\gamma_{5}(D_{W}-m_{W})$ with $m_{W}\in\mathbb{R}$ (so that $H_{W}$ is self-adjoint) and $0\notin\sigma(H_{W})$, and $D_{\mathrm{ov}}=\frac{1}{a}(\mathbf{1}+\gamma_{5}\,\mathrm{sign}(H_{W}))$. Then $D_{\mathrm{ov}}$ is $\gamma_{5}$-Hermitian, satisfies the Ginsparg-Wilson relation $\gamma_{5}D_{\mathrm{ov}}+D_{\mathrm{ov}}\gamma_{5}=aD_{\mathrm{ov}}\gamma_{5}D_{\mathrm{ov}}$, and $\sigma(aD_{\mathrm{ov}})$ lies on the circle $|z-1|=1$ (Neuberger, *Phys. Lett. B* **417** (1998) 141; Ginsparg, Wilson, *Phys. Rev. D* **25** (1982) 2649; Lüscher, *Phys. Lett. B* **428** (1998) 342). The statement is algebraic and uses no translation invariance. **Theorem 2.16** $\boxed{\text{Theorem}}$ **(Classical L: Brouwer).** A continuous self-map of a non-empty compact convex subset of $\mathbb{R}^{n}$ has a fixed point (Brouwer, *Math. Ann.* **71** (1912) 97; Hatcher, *Algebraic Topology*, Cambridge, 2002, Cor. 2.15 for $n$ arbitrary). **Theorem 2.17** $\boxed{\text{Theorem}}$ **(Classical M: implicit function theorem).** If $F\colon\mathbb{R}^{n}\times\mathbb{R}^{m}\to\mathbb{R}^{n}$ is $C^{1}$, $F(x_{0},y_{0})=0$, and $D_{x}F(x_{0},y_{0})$ is invertible, then there are neighbourhoods and a $C^{1}$ map $x(y)$ with $F(x(y),y)=0$, $x(y_{0})=x_{0}$, unique in the neighbourhood (Dieudonné, *Foundations of Modern Analysis*, Academic Press, 1969, Thm 10.2.1). **Theorem 2.18** $\boxed{\text{Theorem}}$ **(Classical N: Kazhdan-Lusztig at roots of unity).** Let $\mathfrak{g}$ be complex simple with lacing number $m\in\{1,2,3\}$ and dual Coxeter number $h^{\vee}$, and $q=e^{i\pi/\ell}$. *(N1, $m\mid\ell$.)* If $\ell=m(k+h^{\vee})$ for an integer $k\geq1$, the category of finite-dimensional tilting modules of Lusztig's form of $U_{q}(\mathfrak{g})$ modulo negligible morphisms is a ribbon fusion category braided-equivalent to the category of integrable highest-weight modules of $\widehat{\mathfrak{g}}$ at level $k$ (Kazhdan, Lusztig, *J. Amer. Math. Soc.* **6** (1993) 905, **7** (1994) 335, 383; Finkelberg, *Geom. Funct. Anal.* **6** (1996) 249, with the correction in Finkelberg, *Geom. Funct. Anal.* **23** (2013) 810). *(N2, $\gcd(m,\ell)=1$.)* If $\ell$ is coprime to $m$ and $\ell>h$ (the Coxeter number), the semisimplified tilting category is a ribbon fusion category whose simple objects are indexed by the dominant weights $\lambda$ with $\langle\lambda+\rho,\theta_{s}^{\vee}\rangle<\ell$, $\theta_{s}$ the highest short root (Sawin, *J. Knot Theory Ramif.* **15** (2006) 1245, Thm 1). No braided equivalence with an affine Lie algebra category is asserted in this case. Chapter 16 uses type $B_{2}$ ($m=2$) and must therefore specify the parity of $\ell$; Theorem 2.19 records the consequences of each choice. **Theorem 2.19** $\boxed{\text{Theorem}}$ **(Classical O: Andersen-Paradowski modularity).** Let $\mathfrak{g}$ be of type $B_{2}$, $q=e^{i\pi/\ell}$. *(O1, $\ell$ even, $\ell=2(k+3)$.)* The semisimplified tilting category is a unitary modular tensor category whose simple objects are indexed by the dominant weights of level $\leq k$, with $S$-matrix the Kac-Peterson matrix of $\widehat{\mathfrak{so}}(5)_{k}$ and twists $\theta_{\lambda}=q^{\langle\lambda,\lambda+2\rho\rangle}$; modularity is Andersen-Paradowski (*Commun. Math. Phys.* **169** (1995) 563, Thm 3.1 and Prop. 3.4) together with Turaev's nondegeneracy criterion, and unitarity is Wenzl (*J. Amer. Math. Soc.* **11** (1998) 261, Thm 3.7), both proved for $\ell$ divisible by the lacing number. *(O2, $\ell$ odd.)* The semisimplified tilting category of N2 is a ribbon fusion category; Sawin proves that it is modular for type $B_{n}$ with $\ell$ odd (loc. cit., Thm 2, case $B$), and its unitarity is established by Rowell for the odd-$\ell$ type-$B$ categories via the Hermitian structure on the braid-group representations (Rowell, *Contemp. Math.* **413** (2006) 215, Thm 4.2 and the discussion of type $B$ at odd roots, where the positivity of the form is verified by a case analysis). Chapter 16 states which case it uses and inherits exactly the corresponding status. **Theorem 2.20** $\boxed{\text{Theorem}}$ **(Classical P: Ajtai-Komlós-Tusnády).** Let $X_{1},\dots,X_{n}$ be i.i.d. uniform on $[0,1]^{2}$ and $\mu_{n}$ their empirical measure. Then $\mathbb{E}\,W_{1}(\mu_{n},\mathrm{Leb})\asymp\sqrt{\log n/n}$. In dimension $k\geq3$, $\mathbb{E}\,W_{1}(\mu_{n},\mathrm{Leb})\asymp n^{-1/k}$ (Ajtai, Komlós, Tusnády, *Combinatorica* **4** (1984) 259; Talagrand, *Ann. Probab.* **22** (1994) 919; Fournier, Guillin, *Probab. Theory Relat. Fields* **162** (2015) 707, Thm 1 for the general-dimension rate with explicit constants). For i.i.d. samples from the normalised volume of a compact Riemannian $k$-manifold, $k\geq3$, the rate $\mathbb{E}\,W_{1}(\mu_{n},\mathrm{vol})\leq C n^{-1/k}$ holds with $C$ depending on the manifold (Weed, Bach, *Bernoulli* **25** (2019) 2620, Thm 1 with the upper Wasserstein dimension equal to $k$; García Trillos, Slepčev, *Canad. J. Math.* **67** (2015) 1358, Thm 1.1 for the $\infty$-transport version with a $\log$ factor). For a Poisson process of intensity $\rho$ on the same manifold, conditioning on the number of points $n\sim\mathrm{Poisson}(\rho\,\mathrm{vol})$ gives i.i.d. uniform points, and $\mathbb{E}\,W_{1}\leq C\,\mathbb{E}[n^{-1/k}\mathbf{1}_{n\geq1}]+\mathrm{diam}\cdot\mathbb{P}(n=0)\leq C'(\rho\,\mathrm{vol})^{-1/k}$, the last step by monotonicity, $n^{-1/k}\leq(\rho\,\mathrm{vol}/2)^{-1/k}$ on $\{n\geq\rho\,\mathrm{vol}/2\}$, and the Chernoff bound $\mathbb{P}(n<\rho\,\mathrm{vol}/2)\leq e^{-\rho\,\mathrm{vol}(1-\log2)/2}$ on the complement, where $W_{1}\leq\mathrm{diam}$. **Theorem 2.21** $\boxed{\text{Theorem}}$ **(Classical Q: isoperimetry and Nash-Sobolev on graphs).** *(Q1, infinite graphs.)* On an infinite connected graph of bounded degree $D$ with counting measure, the isoperimetric inequality $|\partial B|\geq C_{\mathrm{iso}}|B|^{(N_{0}-1)/N_{0}}$ for all finite $B$ is equivalent, with constants depending only on $C_{\mathrm{iso}},N_{0},D$, to the Sobolev inequality $\|f\|_{N_{0}/(N_{0}-1)}\leq C\|\nabla f\|_{1}$ for finitely supported $f$, where $\|\nabla f\|_{1}=\sum_{(x,y)\in E}|f(x)-f(y)|$, and implies the Nash inequality $\|f\|_{2}^{2+4/N_{0}}\leq C_{\mathrm{Nash}}\,\mathcal{E}(f)\|f\|_{1}^{4/N_{0}}$ with $\mathcal{E}(f)=\sum_{(x,y)\in E}(f(x)-f(y))^{2}$ and $C_{\mathrm{Nash}}=C_{\mathrm{Nash}}(C_{\mathrm{iso}},N_{0})$ (Coulhon, Saloff-Coste, *Rev. Mat. Iberoam.* **9** (1993) 293, Prop. 1 and Thm 1; Varopoulos, Saloff-Coste, Coulhon, *Analysis and Geometry on Groups*, Cambridge, 1992, §VI.1). *(Q2, finite graphs, the form matching H4'.)* On a finite connected graph of bounded degree $D$ with $|N|=M$, the relative isoperimetric inequality $|\partial B|\geq C_{\mathrm{iso}}|B|^{(N_{0}-1)/N_{0}}$ for all $B$ with $|B|\leq M/2$ is equivalent, with constants depending only on $C_{\mathrm{iso}},N_{0},D$, to the Sobolev inequality $\|f-\mathrm{med}(f)\|_{N_{0}/(N_{0}-1)}\leq C\|\nabla f\|_{1}$ for all $f$, with $\mathrm{med}(f)$ a median, and implies the Nash inequality $\|f-\bar f\|_{2}^{2+4/N_{0}}\leq C_{\mathrm{Nash}}\,\mathcal{E}(f)\|f-\bar f\|_{1}^{4/N_{0}}$ for the mean-zero part, equivalently the relative Faber-Krahn inequality $\lambda_{1}(B)\geq c\,|B|^{-2/N_{0}}$ for the Dirichlet eigenvalue of every $B$ with $|B|\leq M/2$, with $c=c(C_{\mathrm{iso}},N_{0},D)$ (Chung, *Spectral Graph Theory*, AMS, 1997, Thm 2.9 for the median form; Coulhon, Grigor'yan, *Duke Math. J.* **89** (1997) 133, Prop. 2.1 and Thm 1.1 for the Faber-Krahn equivalence on graphs; the constants' dependence on $D$ arises from comparing the counting measure with the degree measure, $1\leq\deg\leq D$). The Dirichlet form $\mathcal{E}$ with counting measure and the generator of the continuous-time simple random walk of Theorem 2.23, which uses the degree measure, have Rayleigh quotients comparable within the factor $D$. **Theorem 2.22** $\boxed{\text{Theorem}}$ **(Classical R: Varopoulos ultracontractivity).** The Nash inequality with exponent $N_{0}$ for a symmetric Markov semigroup $P_{t}$ is equivalent to the on-diagonal bound $\sup_{x}p_{t}(x,x)\leq C t^{-N_{0}/2}$ for all $t>0$, with $C=C(C_{\mathrm{Nash}},N_{0})$; on a graph with counting measure $p_{t}(x,x)\leq1$ for all $t$, so the bound for $t\leq1$ holds with $C\geq1$ and the content is the bound for $t\geq1$ (Varopoulos, *J. Funct. Anal.* **63** (1985) 240; Carlen, Kusuoka, Stroock, *Ann. Inst. H. Poincaré Probab. Statist.* **23** (1987) 245, Thm 2.1). **Theorem 2.23** $\boxed{\text{Theorem}}$ **(Classical S: Davies off-diagonal bound on graphs).** For the continuous-time simple random walk on a graph of bounded degree $D$, the heat kernel satisfies $p_{t}(x,y)\leq\sqrt{p_{t}(x,x)p_{t}(y,y)}\exp\big(-d_{G}(x,y)\,\mathrm{arsinh}(d_{G}(x,y)/t)+\sqrt{t^{2}+d_{G}(x,y)^{2}}-t\big)$ for all $x,y$ and $t>0$ (Davies, *J. Lond. Math. Soc.* **47** (1993) 65, Thm 10; Delmotte, *Rev. Mat. Iberoam.* **15** (1999) 181, for the discrete-time form). **Theorem 2.24** $\boxed{\text{Theorem}}$ **(Classical T: uniqueness of the hyperfinite $\mathrm{II}_{1}$ factor).** Any two hyperfinite factors of type $\mathrm{II}_{1}$ are isomorphic (Murray, von Neumann, *Ann. Math.* **44** (1943) 716, Thm XIV). By definition, an infinite-dimensional factor with a faithful normal tracial state is of type $\mathrm{II}_{1}$; hyperfiniteness plays no role in that statement. Connes (*Ann. Math.* **104** (1976) 73) proves that injective factors are hyperfinite, which is used in Chapter 14 only through the remark that hyperfiniteness of the inductive limit is automatic from its construction and does not require Connes's theorem. **Theorem 2.25** $\boxed{\text{Theorem}}$ **(Classical U: Henneaux-Teitelboim unimodular decomposition).** For the action $S=\frac{1}{16\pi G}\int(R-2\Lambda)\sqrt{-g}\,\mathrm{d}^{d}x+S_{\mathrm{mat}}$ varied subject to $\sqrt{-g}=\omega$ for a fixed density $\omega$, the field equations are the trace-free Einstein equations $R_{\mu\nu}-\frac1d Rg_{\mu\nu}=8\pi G(T_{\mu\nu}-\frac1d Tg_{\mu\nu})$; taking the divergence, the contracted Bianchi identity $\nabla^{\mu}R_{\mu\nu}=\frac12\partial_{\nu}R$ together with $\nabla^{\mu}T_{\mu\nu}=0$ gives $\frac{d-2}{2d}\partial_{\nu}R+\frac{8\pi G}{d}\partial_{\nu}T=0$, so $\frac{d-2}{2}R+8\pi GT=d\Lambda_{0}$ for a constant $\Lambda_{0}$ that is an integration constant (for $d=4$: $R+8\pi GT=4\Lambda_{0}$, which is the trace of $G_{\mu\nu}+\Lambda_{0}g_{\mu\nu}=8\pi GT_{\mu\nu}$), and substituting back yields the full Einstein equations with cosmological constant $\Lambda_{0}$ (Henneaux, Teitelboim, *Phys. Lett. B* **222** (1989) 195; Unruh, *Phys. Rev. D* **40** (1989) 1048). **Theorem 2.26** $\boxed{\text{Theorem}}$ **(Classical V: random tensor networks and the replica formula).** For a random tensor network on a graph $\Gamma$ with Haar-random node tensors of bond dimension $\chi$ and a boundary region $A$, the Rényi-$n$ entropies of $A$ satisfy $S_{n}(A)=|\gamma_{A}|\log\chi+S^{\mathrm{bulk}}_{n}(\gamma_{A})+o(1)$ as $\chi\to\infty$ when the minimal cut $\gamma_{A}$ is unique, where $|\gamma_{A}|$ is the number of edges in the cut and $S^{\mathrm{bulk}}_{n}$ the bulk Rényi entropy across it, the expected $n$-th moment of the reduced density matrix being computed by a classical Ising-type partition function whose leading term is the minimal cut with corrections suppressed by powers of $\chi^{-1}$ (Hayden, Nezami, Qi, Thomas, Walter, Yang, *J. High Energy Phys.* **11** (2016) 009, Thm 1 (Rényi entropies, Eqs. 3.8-3.10) and Thm 2 (bulk corrections)). *Physical input, not a theorem:* the island formula for the entropy of Hawking radiation as a minimisation over quantum extremal surfaces is derived in the physics literature from the replica trick with replica wormholes (Penington, Shenker, Stanford, Yang, *J. High Energy Phys.* **03** (2022) 205; Almheiri, Hartman, Maldacena, Shaghoulian, Tajdini, *Rev. Mod. Phys.* **93** (2021) 035002). **Remark 2.27 (on Classical F).** An Ollivier bound $\kappa_{\mathrm{LLY}}\geq K$ yields the $L^{\infty}$ gradient estimate of B1; it does not in general yield the Bakry-Émery estimate $\Gamma(P_{t}f)\leq e^{-2Kt}P_{t}\Gamma(f)$, the two gradient estimates being distinct on graphs (Münch, Wojciechowski, loc. cit., §1 and §4, where the relation between the $\ell^{\infty}$ and the $\Gamma$-form estimates is discussed). The dimensional discrete condition $\mathrm{CDE}'(K,n)$ of Bauer, Horn, Lin, Lippner, Mangoubi, Yau (*J. Diff. Geom.* **99** (2015) 359) is logically independent of the Ollivier bound and is not assumed anywhere in this monograph. ### 2.10. Dimensions and ledger With $\hbar,c,G$ restored, (2.1) carries the dimension of $d$; (2.2) is dimensionless; (2.4) and (2.5) are dimensionally homogeneous once $K$ carries the dimension of $d^{-2}$ and $\tau$ that of time; (2.6) is an equation between actions divided by $\hbar$; (2.7) is dimensionless with $\mathrm{Ric}$ of dimension $\mathrm{length}^{-2}$ and $\varepsilon$ a length. | Result | Classical inputs | Hypotheses used | Steps | |---|---|---|---| | Remark 2.1 | Bombelli-Henson-Sorkin (cited) | none | boost invariance of the link region | | Theorems 2.2-2.5, 2.8-2.26 | each is itself the cited input | none (statements only) | hypotheses listed in each statement; discharged at the chapter of use | | Prop 2.6 (B-Lor) | Sturm lsc lemma; Braun Def. 3.1, Thm 3.35 | (H-conv-Lor) over all $N'\geq N$, with the $K>0$ threshold; limit assumed a Lorentzian geodesic space | limit passage in the entropy inequality for each $N'$; endpoint convergence and $\ell_{p}$ convergence are the hypothesis | | Thm 2.3 (B), chain to RCD | Münch-Wojciechowski; Kuwada Thm 2.2 ($p=1$, $q=\infty$); AGS Thm 4.17; Savaré | (H-BE), since $W_{1}$ contraction does not give $\mathrm{BE}(K,\infty)$ on a non-smooth limit | B1 on graphs; B2 on the limit gives the Lipschitz bound only; B3 under (H-BE) | | Thm 2.8 (D), admissible modification | ACG $\mathbb{Z}^{d}$-stationary framework; Poisson mixing | none | hard-core and covering bounds from $|\xi_{z}-c_{z}|\leq R/4$; stationarity and ergodicity under $R\mathbb{Z}^{d}$ | | Conj. 2.7 (C-Lor) | none (open) | none | statement only | | Thm 2.12 (H), graph case | García Trillos-Gerlach-Hein-Slepčev; García Trillos-Slepčev | (H-conv), Mosco convergence of graph forms, stated in Chapter 5 | the graph case is a hypothesis, not a consequence of the RCD stability theorem | --- ## Chapter 3. The Axioms of Lach-Space ### 3.1. The six axioms The axioms are stated first in the compact form used throughout, then made precise where the compact form leaves a choice. **Axiom L1 (Discrete causal structure).** The fundamental object is a locally finite directed acyclic graph $\mathsf{Q}=(N,E)$. The transitive closure $\preceq$ of $E$ is the causal order. **Axiom L2 (Edge tension).** Each edge $e\in E$ carries a tension $T(e)\in[0,\infty)$. **Axiom L3 (Ollivier-Ricci curvature).** The discrete Ricci curvature of $e=(n,m)\in E$ is $$ \mathcal{K}(e)=1-\frac{W_{1}(\mu_{n},\mu_{m})}{d_{G}(n,m)}, \tag{3.1} $$ where $\mu_{n}$ is the normalised counting measure on $N^{+}(n)$, $W_{1}$ is taken with respect to $d_{G}$, and $d_{G}(n,m)=1$ for an edge. **Axiom L4 (Metric emergence).** For $n\preceq m$, $$ d_{\mathrm{eff}}(n,m)=\inf_{\gamma\colon n\to m}\sum_{e\in\gamma}T(e)^{-1}, \tag{3.2} $$ the infimum over directed paths. **Axiom L5 (Energy functional and graph dynamics).** The Lach-Space energy is $$ E[\mathsf{Q}]=\sum_{e\in E}T(e)\,\mathcal{K}(e), \tag{3.3} $$ and the graph evolves under the gradient flow $\mathrm{d}\mathsf{Q}/\mathrm{d}s=-\nabla E[\mathsf{Q}]$ in the configuration space of locally finite directed graphs of bounded degree. **Axiom L6 (Proper-time postulate).** Each edge carries a Hilbert space $\mathcal{H}_{e}$ and a self-adjoint edge Hamiltonian $H_{e}$ with discrete spectrum $\lambda_{0}(e)\leq\lambda_{1}(e)\leq\cdots$ and spectral gap $\Delta(e)=\lambda_{1}(e)-\lambda_{0}(e)>0$. The proper time across $e$ is $$ \tau(e)=\frac{2\pi\hbar}{\Delta(e)}. \tag{3.4} $$ ### 3.2. Precise forms Three of the axioms require a precise reading. **Definition 3.3 (Out-measure, curvature at a sink, convention for zero tension).** If $N^{+}(n)=\emptyset$, set $\mu_{n}=\delta_{n}$. In (3.2) the convention $1/0=+\infty$ is used, so that a path through an edge of zero tension has infinite weight and $d_{\mathrm{eff}}$ is defined on the whole causal relation. With these conventions (3.1) is defined for every edge of a locally finite graph, and $\mathcal{K}(e)\in[-2,1]$: for $a\in N^{+}(n)$ and $b\in N^{+}(m)$ the path $a,n,m,b$ shows $d_{G}(a,b)\leq3$, so $W_{1}(\mu_{n},\mu_{m})\leq3$ and $\mathcal{K}(e)\geq-2$; the upper bound because $W_{1}\geq0$. The lower bound is attained asymptotically: if $N^{+}(n)=\{m,a_{1},\dots,a_{k}\}$ and $N^{+}(m)=\{b\}$ with no other edges, then $W_{1}(\mu_{n},\mu_{m})=\frac{1}{k+1}\cdot1+\frac{k}{k+1}\cdot3\to3$ and $\mathcal{K}(e)\to-2$ as $k\to\infty$. If $N$ is infinite the graph distance between two nodes of the same connected component is finite, and the Wasserstein distance in (3.1) involves only finitely many nodes by local finiteness. **Definition 3.4 (Configuration space and dynamics).** Fix $D\in\mathbb{N}$ and $00$. **Remark 3.6 (Scaling of Ollivier curvature).** If $\mathsf{Q}$ is a graph obtained by sampling a smooth $N$-manifold at mesh $\varepsilon$, with $\mu_{n}$ supported on nodes within distance $\sim\varepsilon$, Classical E shows $\mathcal{K}(e)$ is of order $\varepsilon^{2}$. A sequence $(\mathsf{Q}_{n})$ with mesh $\varepsilon_{n}\to0$ therefore has $\mathcal{K}_{n}\to0$ pointwise on any smooth target. Wherever a *uniform lower bound* on curvature is imposed along a scaling sequence in this monograph, it is imposed on the rescaled curvature $$ \mathcal{K}^{(n)}(e):=\varepsilon_{n}^{-2}\,\mathcal{K}_{n}(e), \tag{3.5} $$ with $\varepsilon_{n}$ the scale of the sequence, defined in Chapter 5 as $\varepsilon_{n}=n^{-1}$ when tensions are rescaled by $T_{n}=nT$. ### 3.3. Structural hypotheses The theorems of Parts II through V are proved under subsets of the following hypotheses. Each constrains the class of QSNs to which the result applies. For a sequence $(\mathsf{Q}_{n})$ a hypothesis is said to hold *uniformly* if it holds for every $n$ with constants independent of $n$, with the curvature hypotheses H2 and H6 read for the rescaled curvature (3.5). **(H1) Bounded degree.** $\deg^{+}(n)+\deg^{-}(n)\leq D<\infty$ for all $n$. **(H1') Bilateral tension bound.** H1 and $00$, $C_{\mathrm{vol}}>0$ with $$ \text{(i)}\quad\frac{|\partial B|}{|B|^{(N_{0}-1)/N_{0}}}\geq C_{\mathrm{iso}}\quad\text{for all finite }B\subset N\text{ with }|B|\leq|N|/2, \qquad \text{(ii)}\quad|B(n,r)|\leq C_{\mathrm{vol}}\,r^{N_{0}}\quad\text{for all }n\in N,\ r\geq1. $$ Part (ii) is new in v3. Part (i) alone gives a polynomial *lower* volume bound (Lemma 5.3); bounded degree alone gives only an exponential upper bound, and no theorem in the literature derives a polynomial upper bound from an Ollivier lower bound on general graphs. The two parts together give doubling. **(H5) Spectral non-degeneracy.** $\Delta(e)\geq\Delta_{\min}>0$ for all $e$, and $\Delta(e)\leq\Delta_{\max}<\infty$. **(H6$^{+}$) Uniform timelike Ollivier upper bound.** The proper-time Ollivier curvature $\mathcal{K}_{\tau}(e)$, defined in Chapter 5 (§5.4) by replacing $d_{G}$ and the out-measures in (3.1) by $d_{\tau}$ and the proper-time-weighted out-measures, satisfies $\mathcal{K}_{\tau}(e)\leq K^{+}$ for all $e$. The direction is reversed relative to H2: by the one-sided shell expansion of Lemma 7.4, a timelike Ricci *lower* bound on a smooth target corresponds to an Ollivier *upper* bound for the causal random walk, so an upper bound is the hypothesis relevant to timelike curvature-dimension conditions. The lower-bound form H6 of earlier editions is withdrawn. **(H7) Non-branching timelike geodesics.** In the GHL limit of Theorem 5.6, maximising causal curves with a common initial segment coincide. **(H8) Density of central elements.** (Chapter 14.) For the inductive system $(\mathcal{A}^{(D)}_{\Gamma},\iota_{\Gamma,\Gamma'})$ of edge-mode algebras, the representation cutoffs satisfy $D_{e}\to\infty$ along the directed set, and for every $\Gamma$ and every $z$ in the centre of $\mathcal{A}^{(D)}_{\Gamma}$ with $\tau_{\Gamma}(z)=0$ one has $\inf_{\Gamma'\supseteq\Gamma}\|\,\mathbb{E}_{\Gamma'}(\iota_{\Gamma,\Gamma'}z)\,\|_{2}=0$, where $\mathbb{E}_{\Gamma'}$ is the trace-preserving conditional expectation onto the centre of $\mathcal{A}^{(D)}_{\Gamma'}$. **(H9) Random-tensor-network scrambling.** (Chapter 15.) The physical state on the QSN restricted to the black-hole and radiation regions is, at each time, a random tensor network state in the sense of Classical V with Haar-random node tensors of bond dimension $\chi\geq\chi_{0}$, and the unitary evolution scrambles the region on a timescale $t_{\mathrm{scr}}\leq\beta\log S_{\mathrm{BH}}$. **(H-conv) Mosco convergence.** (Chapter 5.) The graph Dirichlet forms of the sequence, with the scaling of §5.2, Mosco-converge to the Cheeger energy of the mGH limit (Theorem 2.12, graph case). **(H-BE) Curvature-dimension on the limit.** (Chapter 5.) The mGH limit of Theorem 5.5 is an $\mathrm{RCD}(K,N_{0})$ space. This is assumed, not derived: the route B1, B2, B3 after Theorem 2.3 does not close, since no theorem converts a graph Ollivier bound into a graph Bakry-Émery bound and no step upgrades $\mathrm{RCD}(K,\infty)$ to $\mathrm{RCD}(K,N_{0})$. **(H-cf) Causal fraction.** (Chapter 5.) The hypothesis of Lemma 5.4 and Theorem 5.6(a), stated in §5.4, bounding below the fraction of causally related pairs in every graph ball. **(H-TCD) Timelike curvature-dimension on the limit.** (Chapter 7.) The GHL limit of Theorem 5.6 is assumed to satisfy $\mathrm{TCD}^{*}_{p}(K,N_{0})$; this replaces clause (iv) of H-conv-Lor. **(H-conv-Lor) Lorentzian transport convergence.** (Chapter 7.) The hypothesis stated in Proposition 2.6. **(H10) Regularity of the limit.** (Chapters 5, 7, 8.) The continuum limit delivered by Theorem 5.5 or Theorem 5.6 is, on its regular set, a smooth manifold with a $C^{2}$ metric. This is not a consequence of Classical C or C-Lor, which yield continuity only; it is the hypothesis under which curvature tensors of the limit are defined pointwise and under which Theorem 8.1 is stated. ### 3.4. Consistency **Proposition 3.1** $\boxed{\text{Proposition}}$ **(Consistency of L1-L6).** *There exists a model of L1-L6, namely the Hasse diagram of a Poisson sprinkling into a bounded globally hyperbolic region of Minkowski space with tensions equal to the proper times of links and edge Hamiltonians constructed below. In this model L1, L2, L3, L4 hold almost surely, the functional of L5 is finite and the dynamics of Definition 3.4 is well defined, and L6 holds, the constructed Hamiltonians satisfying H5 in addition.* *Proof.* *Probability space.* Let $R\subset\mathbb{R}^{1,d-1}$ be the closed causal diamond $J^{+}(p)\cap J^{-}(q)$ for $p\prec q$; it is compact with $\mathrm{vol}(R)=V<\infty$. Let $(\Omega,\mathcal{F},\mathbb{P}_{\rho})$ be the canonical space of locally finite counting measures on $R$ with the $\sigma$-algebra generated by the counting maps $\omega\mapsto\omega(A)$, $A$ Borel, and $\mathbb{P}_{\rho}$ the Poisson law of intensity measure $\rho\,\mathrm{vol}|_{R}$ (Last, Penrose, *Lectures on the Poisson Process*, Cambridge, 2017, Thm 3.6 for existence and uniqueness of the law). Write $\Pi=\Pi(\omega)$ for the support of $\omega$. Every "almost surely" below is with respect to $\mathbb{P}_{\rho}$ at the fixed $\rho>0$. The number of points $\omega(R)$ is Poisson with mean $\rho V$, hence $\mathbb{P}_{\rho}(\omega(R)<\infty)=1$. The process is simple, i.e. $\mathbb{P}_{\rho}$-almost surely $\omega(\{x\})\leq1$ for every $x$, because the intensity measure is diffuse (loc. cit., Prop. 6.9). Let $A_{\mathrm{null}}$ be the event that two distinct points of $\Pi$ are null-related. It is measurable: $A_{\mathrm{null}}=\bigcap_{j\geq1}\bigcup_{(Q,Q')}\{\omega(Q)\geq1,\ \omega(Q')\geq1\}$, the union over pairs of disjoint rational boxes $Q,Q'\subset R$ of diameter $<1/j$ such that some $x\in Q$, $y\in Q'$ are null-related, each set in the union being in $\mathcal{F}$ and the union countable. It is null: conditional on $\omega(R)=k$, the points are i.i.d. uniform on $R$ (loc. cit., Prop. 3.8 combined with Thm 3.6; the binomial-conditional representation), the set $\{(x,y)\in R^{2}:\eta(y-x,y-x)=0\}$ is closed of Lebesgue measure zero in $R^{2}$ since it is the zero set of a non-constant polynomial, and a countable union over the $\binom{k}{2}$ pairs and over $k$ of null events is null. Let $\Omega_{0}=\{\omega(R)<\infty\}\cap\{\text{simple}\}\cap A_{\mathrm{null}}^{c}$, a set of full measure. *L1.* Fix $\omega\in\Omega_{0}$. Let $N=\Pi$ and let $E$ be the set of links: $(x,y)\in E$ iff $x\prec y$ and $\Pi\cap I(x,y)=\emptyset$, where $I(x,y)$ is the open causal interval. The graph is finite, hence locally finite. It is acyclic because $\prec$ is a strict partial order on Minkowski space and every edge is oriented from past to future, so a directed cycle would give $x\prec x$. The transitive closure of $E$ coincides with the restriction of $\prec$ to $\Pi$: if $x\prec y$ in $\Pi$, the finite set $\Pi\cap[x,y]$ is totally ordered along some maximal chain from $x$ to $y$ whose consecutive elements are links. Thus $\preceq$ on $N$ is the Minkowski causal order, and L1 holds almost surely. *L2.* For a link $(x,y)$ set $T(x,y)=\sqrt{-\eta(y-x,y-x)}$, the Minkowski proper time from $x$ to $y$. This is non-negative, and on $\Omega_{0}$ it is strictly positive for every link because no two points are null-related. L2 holds on $\Omega_{0}$. *L3.* Each $N^{+}(x)$ is finite, $\mu_{x}$ is a probability measure on a finite set, $d_{G}$ is finite on each connected component, and $W_{1}$ is a finite linear programme. Thus (3.1) defines $\mathcal{K}(e)\in[-2,1]$ for every edge, with the convention of Definition 3.3 at the maximal elements of $\Pi$, and L3 holds. *L4.* For $x\preceq y$ there is at least one directed path; each term $T(e)^{-1}$ is finite on $\Omega_{0}$ by L2; there are finitely many paths. Hence $d_{\mathrm{eff}}(x,y)<\infty$, and L4 holds. Note that $d_{\mathrm{eff}}$ is not a metric on $N$ and is not claimed to be one; it is a function on the causal relation. *L5.* $E[\mathsf{Q}]$ is a finite sum of finite terms. For the fixed $\omega\in\Omega_{0}$ let $D=D(\omega)$ be the maximal degree of the sampled graph, a finite random variable; the configuration space $\mathcal{C}_{D(\omega)}$ of Definition 3.4, built with $T_{\min}(\omega),T_{\max}(\omega)$ as below, contains the isomorphism class of $(\mathsf{Q},T)$. The tension flow is a Lipschitz ordinary differential equation on a compact cube with reflection, and the jump process has total rate at most one by construction, so the piecewise-deterministic process is non-explosive and well defined (Definition 3.4). The identification of $T_{\min},T_{\max}$ with the minimum and maximum link proper times of the sample, both positive and finite on $\Omega_{0}$, places the sampled tensions in the admissible cube. *L6.* For every edge choose $\mathcal{H}_{e}=\mathbb{C}^{2}$ and $H_{e}=\mathrm{diag}(0,\Delta(e))$ with $\Delta(e)\in[\Delta_{\min},\Delta_{\max}]$, $0<\Delta_{\min}\leq\Delta_{\max}<\infty$ fixed. This is a bounded self-adjoint operator with simple eigenvalues $0<\Delta(e)$, so (3.4) defines $\tau(e)\in[2\pi/\Delta_{\max},2\pi/\Delta_{\min}]$, and L6 holds; the construction satisfies H5. An infinite-dimensional witness is $\mathcal{H}_{e}=\ell^{2}(\mathbb{N})$ with $H_{e}=\Delta(e)\,\mathsf{N}$, $\mathsf{N}$ the number operator $(\mathsf{N}\psi)(k)=k\psi(k)$ on the domain $\{\psi:\sum k^{2}|\psi(k)|^{2}<\infty\}$; it is self-adjoint (multiplication by a real function on $\ell^{2}$, Reed, Simon, *Methods of Modern Mathematical Physics I*, Academic Press, 1980, Prop. in §VIII.3), has compact resolvent since $(H_{e}+1)^{-1}$ is diagonal with entries $\to0$, and has simple spectrum $\{0,\Delta(e),2\Delta(e),\dots\}$, so $\lambda_{1}-\lambda_{0}=\Delta(e)$ and L6 and H5 hold as before. What is *not* admissible is the Casimir of $\mathrm{Spin}(4,1)$ on $L^{2}(\mathrm{Spin}(4,1))$, or its restriction to finitely many unitary irreducibles: the Casimir is a scalar on each irreducible, the irreducibles are infinite-dimensional, and $L^{2}$ of a non-compact group decomposes as a direct integral over the principal series, so the spectrum is continuous and no spectral gap in the sense of Definition 3.5 exists. (The edge Hilbert spaces of Chapter 4 are larger; Proposition 3.1 asserts consistency of the axioms, for which a two-dimensional witness suffices. A finite direct sum of non-trivial unitary irreducible representations of the non-compact group $\mathrm{Spin}(4,1)$ is infinite-dimensional, so the representation cutoff of Chapter 4 is defined on the compact form $\mathrm{Spin}(5)$, as Chapter 21 makes precise.) No relation between $\tau(e)$ and $T(e)$ is asserted by the axioms; in the reference families of Chapter 20 they are tied in the statement of Theorem 20.1. All conclusions hold on $\Omega_{0}$, which has $\mathbb{P}_{\rho}$-measure one. $\square$ **Remark 3.7 (Dimensions).** With $\hbar,c,G$ restored: $T(e)$ has dimensions of time in the model above, so $d_{\mathrm{eff}}$ in (3.2) has dimensions of inverse time and is a *weighted path functional*, not a length; $\mathcal{K}$ is dimensionless; $E[\mathsf{Q}]$ in (3.3) has the dimensions of $T$; $\Delta(e)$ is an energy and $\tau(e)=2\pi\hbar/\Delta(e)$ a time. The identification of $d_{\mathrm{eff}}$ with a geometric length in Chapter 5 is made after multiplication by the fixed unit $T_{\min}\,\ell$ with $\ell$ a length scale set by the rescaling introduced in §5.2 (so that $d_{\mathrm{eff}}T_{\min}\ell$ has the dimension of length); no equation in this chapter requires $T$ and $\tau$ to share units. **Remark 3.8 (Uniformity in $\rho$).** Proposition 3.1 does not verify H1 with a constant $D$ uniform in $\rho$. For a sprinkling of intensity $\rho$ into a fixed causal diamond in $d$ dimensions, the expected number of links from a point at fixed position grows without bound as $\rho\to\infty$: the expected number of links from the bottom point of an Alexandrov interval of volume $V$ is $\sum_{y}\mathbb{P}(\text{interval}(x,y)\text{ empty})$, which is $\rho\int_{R\cap J^{+}(x)}e^{-\rho\,\mathrm{vol}(I(x,y))}\,\mathrm{d}y$. Write $y=(t,r)$ in coordinates centred at $x$ with $r=|\vec y|$ and $\delta=t-r>0$; then $\mathrm{vol}(I(x,y))=c_{d}(t^{2}-r^{2})^{d/2}=c_{d}(\delta(t+r))^{d/2}$. For $d=2$ the change of variables $u=\rho\,\mathrm{vol}(I)$ gives $\int_{0}^{\rho V}e^{-u}\,\mathrm{d}u/u$ regularised at $u\sim1$, i.e. $\log(\rho V)+O(1)$. For $d>2$ the region $\{\rho\,\mathrm{vol}(I)\lesssim1\}$ is a shell along the cone of thickness $\delta\lesssim\rho^{-2/d}$ (at spatial extent of order the region's size), whose expected occupancy is $\rho\times\rho^{-2/d}\times O(1)=O(\rho^{(d-2)/d})$ (the computation is in Bombelli, Henson, Sorkin, *Mod. Phys. Lett. A* **24** (2009) 2579, §3, done there for the infinite-Minkowski case; restricting to a bounded region truncates the integral at the region's size and produces the finite exponents stated; for the finite-region asymptotics see also Glaser, Surya, *Phys. Rev. D* **88** (2013) 124026, §II). The uniform-degree hypothesis along a scaling sequence therefore excludes raw link graphs of sprinklings into diamonds of fixed size; Chapter 20 builds sequences satisfying it by a thickened-slab construction. Proposition 3.1 also does not assert that the Minkowski proper time is the proper time of L6; it asserts that the two structures can coexist on the same graph. ### 3.5. Partial independence A set of axioms is independent if no axiom follows from the others. For axioms stated in an informal signature this requires a decision about what is held fixed. We hold fixed the signature $(N,E,T,\mathcal{K},d_{\mathrm{eff}},E[\cdot],\mathcal{H}_{e},H_{e},\tau)$ with $\mathcal{K},d_{\mathrm{eff}},E[\cdot],\tau$ treated as primitive functions that the axioms constrain. Under this reading L3, L4, L5, L6 each *fix* a primitive in terms of others, and independence of such an axiom means that the remaining axioms are satisfiable with a different assignment of that primitive. **Proposition 3.2** $\boxed{\text{Proposition}}$ **(Partial independence).** *None of L1, L3, L6 is a consequence of the other five axioms. Specifically:* *(a) there is a structure satisfying L2-L6, with L5 read in its compact form (3.3) and the move constraints of Definition 3.4 relaxed to drop acyclicity, in which the underlying graph is not acyclic;* *(b) there is a structure satisfying L1, L2, L4, L5, L6 in which $\mathcal{K}$ is a function on $E$ different from (3.1);* *(c) there is a structure satisfying L1-L5, with a choice of edge Hilbert spaces and Hamiltonians with positive gaps, in which $\tau$ is a function on $E$ different from (3.4).* *Proof.* *(a)* Let $n\geq3$, $N=\mathbb{Z}/n\mathbb{Z}$, $E=\{(i,i+1):i\in\mathbb{Z}/n\mathbb{Z}\}$. The graph is locally finite of degree two but contains the directed cycle $0\to1\to\cdots\to n-1\to0$, so L1 fails. Set $T\equiv1$ (L2). Each $N^{+}(i)=\{i+1\}$, so $\mu_{i}=\delta_{i+1}$ and $W_{1}(\mu_{i},\mu_{i+1})=d_{G}(i+1,i+2)=1$, giving $\mathcal{K}(e)=0$ for every edge; (3.1) is well defined, so L3 holds. Directed paths exist between any ordered pair, and $d_{\mathrm{eff}}(i,j)$ is the directed distance, finite; L4 holds (the relation $\preceq$ is now total and not antisymmetric, but L4 does not require antisymmetry). $E[\mathsf{Q}]=0$ is finite and the dynamics of Definition 3.4 is well defined on the configuration space with acyclicity dropped from the move constraints; L5 holds in the form stated in (a). Take $\mathcal{H}_{e}=\mathbb{C}^{2}$, $H_{e}=\mathrm{diag}(0,1)$, $\tau(e)=2\pi$; L6 holds. Thus L2-L6 hold and L1 fails. *(b)* Let $\mathsf{Q}$ be any finite directed acyclic graph of bounded degree with at least one edge, $T\equiv1$, and edge Hamiltonians as in (a). Define $$ \widetilde{\mathcal{K}}(e)=c\big(\deg^{+}(n)+\deg^{-}(m)-2\big)\qquad\text{for }e=(n,m), $$ with $c\neq0$ chosen so that $\widetilde{\mathcal{K}}\neq\mathcal{K}$ on at least one edge. Such $c$ exists: write $a(e)=\deg^{+}(n)+\deg^{-}(m)-2\in\mathbb{Z}_{\geq0}$. If $a(e)\neq0$ for some edge $e$, any $c\notin\{\mathcal{K}(e)/a(e)\}$ works. If $a\equiv0$, every node has at most one out-edge and every node at most one in-edge, so the graph is a disjoint union of directed paths; on the last edge $(n,m)$ of a path of length at least one, $N^{+}(m)=\emptyset$ so $\mu_{m}=\delta_{m}$ by Definition 3.3, and $N^{+}(n)=\{m\}$ so $\mu_{n}=\delta_{m}$, giving $W_{1}=0$ and $\mathcal{K}(n,m)=1\neq0=\widetilde{\mathcal{K}}(n,m)$ for every $c$. Replace L3 by the assignment $\mathcal{K}:=\widetilde{\mathcal{K}}$. L1, L2, L6 are untouched. L4 concerns $d_{\mathrm{eff}}$, which depends only on $T$. L5 defines $E$ from $T$ and whichever $\mathcal{K}$ is in force, and the dynamics of Definition 3.4 is well defined for any bounded edge function. Hence L1, L2, L4, L5, L6 hold with a curvature different from (3.1): L3 is not forced by the others. *(c)* Let $\mathsf{Q}$ be any finite directed acyclic graph of bounded degree with at least one edge $e_{0}$, and tensions $T\colon E\to[T_{\min},T_{\max}]$ with $T(e_{0})=2$. Equip each edge with $\mathcal{H}_{e}=\mathbb{C}^{2}$ and $H_{e}=\mathrm{diag}(0,1)$, so $\Delta\equiv1$, and define $\widetilde{\tau}(e)=T(e)^{-1}$. L1-L5 hold: they do not mention $\tau$. L6 requires $\tau(e)=2\pi/\Delta(e)=2\pi$ on every edge, but $\widetilde{\tau}(e_{0})=1/2\neq2\pi$. Thus L1-L5 hold together with a choice of edge Hamiltonians with positive gaps and a proper-time assignment that is not of the form (3.4) for those Hamiltonians: L6 is not forced by the others. (For a different choice of Hamiltonians, namely $\Delta(e)=2\pi T(e)$, the same $\widetilde{\tau}$ would satisfy (3.4); the independence claim is existential in the Hamiltonians, as stated.) $\square$ **Remark 3.9 (Why L2, L4, L5 carry no independence claim).** In the signature adopted above, L4 is the definition of $d_{\mathrm{eff}}$ from $T$ and L5 is the definition of $E$ from $T$ and $\mathcal{K}$ together with a dynamical postulate. A counter-model for L4 would be a structure with L1, L2, L3, L5, L6 and a different $d_{\mathrm{eff}}$; one exists (take $d_{\mathrm{eff}}\equiv1$ on the causal relation), but the fact establishes only that L4 is a definition, not that it is a substantive constraint, and we do not count definitions as independent axioms. The same applies to the functional half of L5. The dynamical half of L5 is a postulate about evolution; its negation (a static graph) is consistent with L1-L4, L6, so in that weak sense it is independent. (The retraction record in Appendix A.5 lists the v2.x counter-models for L2, L4, L5, with the mathematical reason for their failure: the proposed counter-model for L2 set $T\equiv1$ and so satisfied L2, and the proposed counter-model for L5 changed the functional while silently keeping (3.1) in force, so that L3 was reimposed rather than tested.) For L2, any structure with L1, L3-L6 and $T$ taking a negative value fails L2 while the other axioms, which only use $T$ through $T^{-1}$ and through $E$, remain formally satisfiable when $T\neq0$; but L4 with negative tensions produces a function $d_{\mathrm{eff}}$ that is not a pseudo-distance on chains, and we decline to count an axiom system with a sign-indefinite "distance" as a model of L4. The honest statement is that L2, L4, L5 are constitutive definitions and a dynamical postulate rather than constraints whose independence is a meaningful question in the informal signature. A fully rigorous independence analysis would require a formal signature with specified sorts and default values for each primitive; we do not supply one. ### 3.6. Reading guide for the hypotheses The hypothesis schedule is not a list of conjectured properties of nature, and none of the axioms contains a constant that requires dimensional restoration beyond Remark 3.7. H1', H2, H3, H4', H5 are conditions on a *sequence* of QSNs, verified on explicit sequences in Chapter 20. H6 and H7 concern the timelike geometry and are verified on the same sequences. H8 is a definitional constraint on admissible inductive systems and is imposed by construction in Chapter 14. H9 is a modelling hypothesis on the state, and the Page-time result that uses it is conditional on it. H10 is a regularity hypothesis on the limit, satisfied by the reference families of Chapter 20 because their limits are smooth by construction, and not derivable from the synthetic theory for arbitrary sequences. Appendix A gives the full dependency table. ### 3.7. Hypothesis discharge ledger | Result | Classical inputs | Hypotheses used | Steps where consumed | |---|---|---|---| | Prop 3.1 | Poisson process existence and simplicity (Last-Penrose Thm 3.6, Prop. 3.8, Prop. 6.9); Davis PDMP non-explosion | none (H5 is satisfied by the witness, not assumed) | probability space: finiteness, simplicity, measurability and nullity of $A_{\mathrm{null}}$; L1: acyclicity of $\prec$; L2, L4: no null links; L5: Def. 3.4 on a finite graph with $D=D(\omega)$; L6: $\mathcal{H}_{e}=\mathbb{C}^{2}$, $H_{e}=\mathrm{diag}(0,\Delta(e))$ | | Prop 3.2 | none | none | (a) cycle $\mathbb{Z}/n\mathbb{Z}$ with L5 in compact form; (b) degree functional $\widetilde{\mathcal{K}}$, existence of $c$ via the path case; (c) $\widetilde{\tau}=T^{-1}$ with $\Delta\equiv1$, existential in the Hamiltonians | | Defs 3.3-3.5, Remarks 3.6-3.9 | Classical E (for the $\varepsilon^{2}$ scaling in 3.6); Bombelli-Henson-Sorkin and Glaser-Surya (link counts in 3.8) | none | statements only | --- ## Referee response (review1.md, 39 findings) | # | Sev. | Action | |---|---|---| | 1 | MAJOR | Range of $\mathcal{K}$ corrected to $[-2,1]$ with the referee's extremal example written out (Def 3.3, Prop 3.1 step L3). Later chapters must use $-2$. | | 2 | MINOR | Added $0<\mathfrak{m}(B)<\infty$ to the doubling hypothesis in Thm 2.2; Villani citation corrected to Particular Case 5.16. | | 3 | MINOR | B2 split into global form (Kantorovich-Rubinstein, no kernel) and local form (Kuwada Thm 2.2, length space), with the use of each stated. | | 4 | MAJOR | B3 restated via $\mathrm{BE}(K,\infty)$ with the Gaussian volume bound and intrinsic-distance identity; AGS Thm 4.17 and Cor. 4.18 cited; $L^{1}$/$L^{2}$ equivalence attributed to Savaré 2014. | | 5 | MAJOR | "closed" deleted from C2; $G_{\delta}$ nature of $\mathcal{R}$ and possible density of $\mathcal{S}$ stated. | | 6 | MAJOR | C3 restated as Anderson's $C^{1,\alpha}$ convergence theorem for smooth sequences with two-sided Ricci and injectivity bounds; the $C^{0}$ bootstrap claim removed. | | 7 | MAJOR | §2.5 now defines $\mathrm{TCD}^{e}_{p}$ (CM), $\mathrm{TCD}^{*}_{p}$ and $\mathrm{TCD}_{p}$ (Braun) with the correct coefficients and Braun's Prop. 3.4 and Thm 3.35; Thm 2.11 rewritten accordingly. | | 8 | MINOR | Smooth characterisation written as $\mathrm{Ric}(v,v)\geq K|g(v,v)|$ with the sign convention made explicit. | | 9 | MAJOR | Measured refinement of A-Lor now passes through the Minguzzi-Suhr one-point compactification, with the no-mass-escape condition stated as a hypothesis to be verified in Chapter 5; theorem numbers replaced by citation by statement. | | 10 | MAJOR | B-Lor relabelled Proposition 2.6; (H-conv-Lor) strengthened with endpoint entropy convergence and $\ell_{p}$ convergence; proof written out with Sturm's joint lsc lemma. | | 11 | MINOR | D restated: stationary gives a random integrand a.s., ergodic gives deterministic; two-sided $p$-growth made explicit; cited by statement. | | 12 | MINOR | Sentence rewritten; the negative claim moved to Remark 2.27 with reference. | | 13 | MAJOR | H citations replaced by García Trillos-Gerlach-Hein-Slepčev 2020 and García Trillos-Slepčev 2018; redundant phrase removed. | | 14 | MINOR | J now cites Benzi-Boito-Razouk and derives from Thm 2.13; $C_{H}$ added to the dependency list; many-body sources marked unused. | | 15 | MINOR | $m_{W}\in\mathbb{R}$ added to K. | | 16 | MAJOR | N split into N1 ($m\mid\ell$, Finkelberg level $k$) and N2 ($\gcd(m,\ell)=1$, Sawin, no affine equivalence); O split into O1 ($\ell$ even, Andersen-Paradowski and Wenzl) and O2 ($\ell$ odd, Sawin modularity, Rowell unitarity); Chapter 16 must declare its case. | | 17 | MINOR | P: manifold rate cited to Weed-Bach and García Trillos-Slepčev; de-Poissonisation argument written out. | | 18 | MAJOR | Q split into Q1 (infinite graphs) and Q2 (finite graphs, median and relative Faber-Krahn forms matching H4'); constants' dependence on $D$ and the counting/degree measure comparison stated. | | 19 | MINOR | R restated for all $t>0$ with the $t\leq1$ case explained. | | 20 | MINOR | T relabelled; Murray-von Neumann for uniqueness; definitional sentence corrected; Connes's role clarified. | | 21 | MAJOR | U corrected to $\frac{d-2}{2}R+8\pi GT=d\Lambda_{0}$ with the divergence computation shown and the $d=4$ trace check. | | 22 | MINOR | V: HNQTWY statement quoted with Rényi-$n$ form and error structure; island formula labelled as physical input. | | 23 | MINOR | $\mathrm{RCD}^{*}=\mathrm{RCD}$ (Cavalletti-Milman) cited. | | 24 | MINOR | $\widetilde{G}_{k}=k^{d-2}G_{k}$ in general $d$. | | 25 | MINOR | Remark 3.8 rewritten with the correct exponents ($\log\rho$ for $d=2$, $\rho^{(d-2)/d}$ for $d>2$), the derivation sketched, and Glaser-Surya cited for the finite region. | | 26 | MAJOR | Renumbered: Defs 3.3-3.5, Remarks 3.6-3.9; Props 3.1, 3.2 kept as anchors; ledger updated. | | 27 | MINOR | Cross-references to "Definition 5.2", "Definition 5.3", "Definition 20.1" replaced by section references (§5.4, §5.2) and Theorem 20.1. | | 28 | MINOR | Convention $1/0=+\infty$ added (Def 3.3). | | 29 | MAJOR | Def 3.4 now specifies the uniform proposal kernel with total rate $\leq1$; non-explosion via Davis's PDMP theorem; Prop 3.1 step L5 updated. | | 30 | MINOR | Prop 3.1 retagged Proposition; statement reads "L6 holds, the construction satisfying H5". | | 31 | MAJOR | Witness changed to $\mathcal{H}_{e}=\mathbb{C}^{2}$, $H_{e}=\mathrm{diag}(0,\Delta(e))$; the non-compactness of $\mathrm{Spin}(4,1)$ noted with deferral to Chapter 21. | | 32 | MINOR | Measurability of $A_{\mathrm{null}}$ shown via rational boxes; simplicity cited to Last-Penrose Prop. 6.9. | | 33 | MINOR | $D=D(\omega)$ stated. | | 34 | MINOR | Prop 3.2(a) statement now says L5 is read in compact form with acyclicity dropped. | | 35 | MINOR | Existence of $c$ argued via $a(e)$ and the directed-path case. | | 36 | MAJOR | Prop 3.2(c) restated existentially in the Hamiltonians; proof fixed with $\Delta\equiv1$, $T(e_{0})=2$; irrelevant condition deleted. | | 37 | MINOR | Retraction reference in Remark 3.9 now gives the mathematical reasons and points to Appendix A.5. | | 38 | (none) | Global checks re-run after edits: no dashes, no forbidden phrases. | | 39 | MINOR | Chapter 2 ledger now lists Prop 2.6 with (H-conv-Lor) and Thm 2.12 graph case with (H-conv). | ### Pass 2 (review1b.md: 3 MAJOR, 13 MINOR) | # | Sev. | Action | |---|---|---| | B.1 | MAJOR | B2 restated with Kuwada's correct exponent pairing ($p=1\leftrightarrow q=\infty$ gives the Lipschitz-constant bound only). The closing sentence of B3 replaced: the $W_{1}$-to-$\mathrm{BE}(K,\infty)$ step is recorded as the new hypothesis (H-BE), added to §3.3 and to the Chapter 1 summary; von Renesse-Sturm cited for the smooth case. Theorem 5.5 must list (H-BE); Chapter 5 to be re-refereed. | | B.2 | MAJOR | Thm 2.8 now defines the admissible modification $\mathcal{L}=\{\xi_{z}\}$ (nearest Poisson point within $R/4$ of each cube centre, else the centre), proves hard-core radius $R/2$, covering radius $R\sqrt{d}$, $R\mathbb{Z}^{d}$-stationarity and ergodicity, and the vanishing density of centre-substitutions as $\rho\to\infty$; the false thinning claim deleted; Chapter 8 directed to cite Thm 2.8. | | B.3 | MAJOR | (H-conv-Lor) quantified over all $N'\geq N$; proof runs for each $N'$. | | B.4 | MINOR | Limit assumed a measured globally hyperbolic Lorentzian geodesic space; $K>0$ threshold $\ell_{p}<\pi\sqrt{N/K}$ added with the void-inequality convention, and the continuity range of $\sigma$ made explicit in the proof. | | B.5 | MINOR | $\mu,\nu\in\mathcal{P}_{1}$ added to B1. | | B.6 | MINOR | Derivation clause deleted; Benzi-Boito-Razouk cited as source. | | B.7 | MINOR | Wenzl reference corrected to *J. Amer. Math. Soc.* **11** (1998) 261. | | B.8 | MINOR | Jensen replaced by monotonicity plus the explicit Chernoff bound. | | B.9 | MINOR | "Remark 3.5'" corrected to "Remark 3.7". | | B.10 | MINOR | Move-set bound corrected to $|N|^{2}+|N|2^{D}+2|E|$. | | B.11 | MINOR | Remark 3.8 sketch rewritten: $e^{-u}$ regularisation for $d=2$; interval volume $c_{d}(\delta(t+r))^{d/2}$, shell thickness $\rho^{-2/d}$, occupancy $\rho^{(d-2)/d}$. | | B.12 | MINOR | Remark 2.27 moved to the end of §2.9. Chapter 3 order (3.3-3.6, 3.1, 3.2, 3.7-3.9) retained because SPEC anchors Props 3.1-3.2; Appendix C to note it. | | B.13 | MINOR | Boxed tags added to Theorems 2.2-2.26 (Theorem), Conj. 2.7 (Conjecture), Prop. 2.6 (Conditional on H-conv-Lor); (H-conv), (H-BE), (H-conv-Lor) entered in the §3.3 schedule. | | B.14 | MINOR | Sentence added to Thm 2.11 stating that no timelike EVI is used and superseding SPEC's description. | | B.15 | MINOR | Dimensional remark corrected: $d_{\mathrm{eff}}T_{\min}\ell$ is a length. | | B.16 | MINOR | Prop 3.1 step L5 now speaks of the isomorphism class of $(\mathsf{Q},T)$ and of $\mathcal{C}_{D(\omega)}$ built with $T_{\min}(\omega),T_{\max}(\omega)$. | | B.17, B.18 | (none) | Numerical anchor and global checks acknowledged; re-run after edits: no dashes, no forbidden phrases. | # Part II. The Quantum Spacetime Network and Emergent Geometry --- ## Chapter 4. The QSN: Nodes, Edges, Hilbert Space ### 4.1. The underlying graph Throughout Part II, $\mathsf{Q}=(N,E)$ denotes a QSN satisfying Axioms L1 to L6 of Chapter 3. We write $N^+(n)=\{m\in N:(n,m)\in E\}$ for the out-neighbourhood of $n$, $N^-(n)$ for the in-neighbourhood, and $\deg(n)=|N^+(n)|+|N^-(n)|$ for the total valence. The causal order $\preceq$ is the reflexive transitive closure of $E$. Since $\mathsf{Q}$ is acyclic, $\preceq$ is a partial order. The causal diamond of a comparable pair $n\preceq m$ is $J(n,m)=\{p\in N:n\preceq p\preceq m\}$, and the causal future and past are $J^+(n)=\{p:n\preceq p\}$, $J^-(n)=\{p:p\preceq n\}$. **Definition 4.1** (Graph metric and effective metric). The graph metric $d_G(n,m)$ is the minimum number of edges in an undirected path from $n$ to $m$, with $d_G(n,m)=\infty$ if no such path exists. The effective metric $d_{\mathrm{eff}}$ is given by Axiom L4. The graph ball is $B_G(n,r)=\{m:d_G(n,m)\le r\}$ and the effective ball is $B_{\mathrm{eff}}(n,r)=\{m:d_{\mathrm{eff}}(n,m)\le r\}$. The reference measure $\mu$ is the counting measure on $N$; its restriction to a finite set $B$ has mass $\mu(B)=|B|$. The effective metric of L4 is defined on all pairs by allowing undirected paths in the infimum (the formula in L4 is stated for causally related pairs; for the purposes of Chapter 5 we use the symmetrised version, summing $T(e)^{-1}$ along undirected paths). When $\mathsf{Q}$ is the Hasse diagram of a sprinkled Lorentzian manifold, the undirected symmetrisation is the discrete analogue of the Riemannian metric obtained from a Wick-rotated Lorentzian one; the Lorentzian data are carried separately by $d_\tau$ (Definition 5.0 below). ### 4.2. Node structure **Definition 4.2** (Node Hilbert space). Each node $n\in N$ carries $$\mathcal{H}_n=\mathcal{H}_{\mathrm{geo}}(n)\otimes\mathcal{H}_{\mathrm{mat}}(n).\tag{4.1}$$ *Geometric factor.* Let $\mathcal{E}(n)$ be the set of edges incident to $n$. Each $e\in\mathcal{E}(n)$ is labelled by an $\mathrm{SU}(2)$ irreducible representation $j_e\in\tfrac12\mathbb{Z}_{\ge0}$. The geometric factor is the intertwiner space $$\mathcal{H}_{\mathrm{geo}}(n)=\mathrm{Inv}_{\mathrm{SU}(2)}\Bigl(\bigotimes_{e\in\mathcal{E}(n)}V_{j_e}\Bigr),\tag{4.2}$$ the $\mathrm{SU}(2)$-invariant subspace of the tensor product of the representation spaces $V_{j_e}$ (Ashtekar, Lewandowski, *Class. Quant. Grav.* **21** (2004) R53). Under H1 the valence is bounded by $D$, so $\dim\mathcal{H}_{\mathrm{geo}}(n)<\infty$ for each fixed assignment of spins. The Ashtekar-Lewandowski volume operator $\widehat{V}_n$ acts on (4.2) with discrete spectrum $$\widehat{V}_n\,|\iota\rangle=\ell_P^{3}\,v(j_{e_1},\dots,j_{e_{\deg n}};\iota)\,|\iota\rangle,$$ where $v$ is the node-local intertwiner polynomial of Bianchi (*Nucl. Phys. B* **807** (2009) 591). The only properties of $\widehat{V}_n$ used in this monograph are: (i) $\widehat{V}_n\ge0$; (ii) for each fixed assignment of spins with $j_e\le j_{\max}$, its spectrum on the finite-dimensional space (4.2) is a finite set, so the minimum nonzero eigenvalue $v_{\min}\ell_P^3$ over all such assignments exists and $v_{\min}=v_{\min}(D,j_{\max})>0$ (a minimum over finitely many finite-dimensional spectra); no positivity of the infimum over unbounded spins is asserted; (iii) $\widehat{V}_n$ commutes with the $\mathrm{SU}(2)$ action at $n$. All three are established in Ashtekar-Lewandowski (loc. cit., Thm 4.5) and Brunnemann-Thiemann (*Class. Quant. Grav.* **23** (2006) 1289). *Matter factor.* $\mathcal{H}_{\mathrm{mat}}(n)=\mathcal{F}_f(n)\otimes\mathcal{F}_b(n)$ where $\mathcal{F}_f(n)$ is the finite-dimensional fermionic Fock space on the 48 Weyl species of the Standard Model with a right-handed neutrino in each generation (dimension $2^{48}$), and $\mathcal{F}_b(n)$ is the bosonic Fock space of the Higgs doublet, truncated at a node-occupation cutoff $n_b^{\max}$ when a finite-dimensional model is required. Canonical anticommutation relations $\{\psi_I(n),\psi_J^\dagger(n)\}=\delta_{IJ}$ hold on $\mathcal{F}_f(n)$. ### 4.3. Edge structure **Definition 4.3** (Edge Hilbert space). Each directed edge $e=(n,m)$ carries $$\mathcal{H}_e=\mathcal{H}_{\mathrm{grav}}(e)\otimes\mathcal{H}_{\mathrm{gauge}}(e),\qquad \mathcal{H}_{\mathrm{grav}}(e)=L^2(\mathrm{Spin}(4,1)),\quad \mathcal{H}_{\mathrm{gauge}}(e)=L^2(G_{\mathrm{SM}}),\tag{4.3}$$ with $G_{\mathrm{SM}}=\mathrm{SU}(3)_c\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$. Here $L^2$ is taken with respect to Haar measure. Since $\mathrm{Spin}(4,1)$ is non-compact, $L^2(\mathrm{Spin}(4,1))$ decomposes by the Plancherel theorem into a direct integral of principal-series representations; in all constructions below we work in the compact real form $\mathrm{Spin}(5)$ wherever a finite-dimensional truncation is required, and return to $\mathrm{Spin}(4,1)$ by the continuation of Chapter 21. The holonomy operators $\widehat{U}_{\mathrm{grav}}(e)$ and $\widehat{U}_{\mathrm{SM}}(e)$ act by left multiplication. The edge Hamiltonian $H_e$ of Axiom L6 is not the Casimir of $\mathrm{Spin}(4,1)$: the quadratic Casimir on $L^2(\mathrm{Spin}(4,1))$ has continuous spectrum (the principal series contributes the half-line $[\tfrac94,\infty)$ by Plancherel, Harish-Chandra, *Trans. AMS* **75** (1953) 185), so it has no gap and L6 would be ill-posed. We fix instead the following model, which is the minimal structure making L6 and H5 well-posed. **Definition 4.3'** (Edge dynamical space and Hamiltonian). To each edge $e$ attach a separable Hilbert space $\mathcal{H}_e^{\mathrm{dyn}}=\ell^2(\mathbb{Z}_{\ge0})$ with orthonormal basis $(\varepsilon_k)_{k\ge0}$, and the number operator $\mathsf{N}\varepsilon_k=k\varepsilon_k$ with domain $\mathrm{Dom}(\mathsf{N})=\{\sum c_k\varepsilon_k:\sum k^2|c_k|^2<\infty\}$. The edge Hamiltonian is $$H_e=\Delta(e)\,\mathsf{N},\qquad \mathrm{Dom}(H_e)=\mathrm{Dom}(\mathsf{N}),\qquad \Delta(e)>0.\tag{4.3'}$$ $H_e$ is self-adjoint (it is a real diagonal operator on its maximal domain, hence self-adjoint by the multiplication-operator form of the spectral theorem, Reed-Simon I, Thm. VIII.4), has compact resolvent ($(H_e+1)^{-1}$ is diagonal with entries $(\Delta(e)k+1)^{-1}\to0$), discrete spectrum $\{0,\Delta(e),2\Delta(e),\dots\}$, and spectral gap $\lambda_1-\lambda_0=\Delta(e)$. The full edge space is $\mathcal{H}_e\otimes\mathcal{H}_e^{\mathrm{dyn}}$ with $H_e$ acting on the second factor. The number $\Delta(e)$ is edge data on the same footing as $T(e)$; H5 is the requirement $\Delta_{\min}\le\Delta(e)\le\Delta_{\max}$ uniformly, and Chapter 6 records that no relation between $\Delta(e)$ and the holonomy sector is derived. ### 4.4. Total Hilbert space and the gauge-invariant subspace **Definition 4.4** (Kinematical and physical Hilbert spaces). The kinematical Hilbert space is the tensor product $$\mathcal{H}_{\mathsf{Q}}=\Bigl(\bigotimes_{n\in N}\mathcal{H}_n\Bigr)\otimes\Bigl(\bigotimes_{e\in E}\mathcal{H}_e\Bigr),\tag{4.4}$$ understood for infinite $N$ as the incomplete (von Neumann) tensor product with respect to a fixed reference product vector. The local gauge group at a node $n$ is $G_n=\mathrm{Spin}(4,1)\times G_{\mathrm{SM}}$, acting on $\mathcal{H}_e$ by right multiplication for $e$ outgoing from $n$ and by left multiplication for $e$ incoming, and on $\mathcal{H}_{\mathrm{mat}}(n)$ by the Standard Model representation. The physical subspace is $$\mathcal{H}_{\mathsf{Q}}^{\mathrm{phys}}=\overline{\Bigl\{\,\psi\in\mathcal{H}_{\mathsf{Q}}:\ \int_{G_n}g\cdot\psi\,\mathrm{d}g=\psi\ \ \forall n\in N\Bigr\}},\tag{4.5}$$ where the integral is the group average over the compact part of $G_n$ and a rigging-map average (Ashtekar-Lewandowski-Marolf-Mourão-Thiemann, *J. Math. Phys.* **36** (1995) 6456) over the non-compact factor. The physical subspace is non-zero: the product of invariant spin-network states with the Fock vacuum lies in it. **Proposition 4.5** $\boxed{\text{Proposition}}$ (Local finiteness of the physical algebra). *Under H1, for every finite subset $B\subset N$, the algebra $\mathcal{A}_B$ of bounded operators on $\mathcal{H}_{\mathsf{Q}}^{\mathrm{phys}}$ generated by gauge-invariant operators supported on $B$ and its incident edges, with spin labels and bosonic occupations bounded by $j_{\max}$ and $n_b^{\max}$, is finite-dimensional, and $\dim\mathcal{A}_B\le C(D,j_{\max},n_b^{\max})^{|B|}$.* *Proof.* Each node contributes $\dim\mathcal{H}_{\mathrm{geo}}(n)\le(2j_{\max}+1)^{D}$ (the intertwiner space is a subspace of the tensor product of at most $D$ factors of dimension at most $2j_{\max}+1$) times $\dim\mathcal{H}_{\mathrm{mat}}(n)\le 2^{48}(n_b^{\max}+1)^4$. Each incident edge with spin cutoff contributes the finite-dimensional subspace of $L^2(\mathrm{Spin}(5))\otimes L^2(G_{\mathrm{SM}})$ spanned by matrix coefficients of representations of highest weight bounded by $j_{\max}$, which has dimension at most $c(j_{\max})$ by the Peter-Weyl theorem. The number of edges incident to $B$ is at most $D|B|$. The operator algebra on a Hilbert space of dimension $M$ has dimension $M^2$. Multiplying gives the bound with $C=\bigl((2j_{\max}+1)^{D}2^{48}(n_b^{\max}+1)^{4}c(j_{\max})^{D}\bigr)^2$. $\square$ Proposition 4.5 is the structural statement that underlies every "UV finiteness" claim made for the QSN: on bounded regions with bounded representation labels, all observable algebras are finite-dimensional, and no regulator is needed to define entropies, traces, or partition functions. ### 4.5. The graph-local Hamiltonian The microscopic dynamics is generated by $$H_{\mathsf{Q}}=H_{\mathrm{geo}}+H_{\mathrm{ent}}+H_{\mathrm{gauge}}+H_{\mathrm{mat}},\tag{4.6}$$ with $$H_{\mathrm{geo}}=\tfrac12\alpha_{\mathrm{geo}}\sum_{n\in N}\mathrm{tr}\bigl(\widehat{C}^{(2)}_n\bigr)+\sum_{n\in N}V(\widehat{V}_n),\qquad V(x)=\tfrac{\kappa}{2\ell_P^{6}}(x-v_{\min}\ell_P^{3})^{2},$$ where $\widehat{C}^{(2)}_n$ is the quadratic $\mathrm{SU}(2)$ Casimir acting on $\mathcal{H}_{\mathrm{geo}}(n)$ and the stabilising potential has its minimum at the minimum nonzero volume eigenvalue; $$H_{\mathrm{ent}}=-\lambda\sum_{(n,m)\in E}\bigl(\widehat{E}_{nm}+\widehat{E}_{nm}^{\dagger}\bigr),$$ where $\widehat{E}_{nm}$ is the edge creation operator raising the Schmidt rank of the state across the bipartition $\{n\}\,|\,\{m\}$ by one (Hayden, Nezami, Qi, Thomas, Walter, Yang, *JHEP* **11** (2016) 009); $$H_{\mathrm{gauge}}=-\frac{1}{g_s^{2}}\sum_{\square}\mathrm{tr}\bigl(\widehat{U}_\square+\widehat{U}_\square^\dagger\bigr)-t\sum_{e=(n,m)}\sum_{I}\bigl(\psi_I^\dagger(n)\widehat{U}_{\mathrm{SM}}(e)\psi_I(m)+\mathrm{h.c.}\bigr)$$ in Kogut-Susskind form (Kogut, Susskind, *Phys. Rev. D* **11** (1975) 395), the plaquette sum running over minimal undirected cycles of $\mathsf{Q}$; and $H_{\mathrm{mat}}$ a node-local polynomial in the matter creation and annihilation operators containing the Yukawa couplings and the Higgs potential. Each term of (4.6) commutes with the local gauge action, so $H_{\mathsf{Q}}$ preserves $\mathcal{H}_{\mathsf{Q}}^{\mathrm{phys}}$. Under H1 and the representation cutoffs of Proposition 4.5, $H_{\mathsf{Q}}$ restricted to a finite region is a bounded self-adjoint operator on a finite-dimensional space. $H_{\mathsf{Q}}$ is not the object that controls the continuum-limit theorems of Chapter 5; those depend only on the combinatorial data $(N,E,T,\mathcal{K})$ of Axioms L1 to L5. $H_{\mathsf{Q}}$ enters in Chapters 14 and 15 through the states it prepares. ### 4.6. Reference frames as antichain foliations **Definition 4.6** (Antichain foliation). A subset $\Sigma\subset N$ is an antichain if no two distinct elements of $\Sigma$ are comparable under $\preceq$. A reference frame is a map $t:N\to\mathbb{Z}$ such that (i) each level set $\Sigma_k=t^{-1}(k)$ is a maximal antichain in $\{n:t(n)\ge k\}$, and (ii) $t$ is strictly increasing along edges: $(n,m)\in E\Rightarrow t(m)>t(n)$. **Proposition 4.7** $\boxed{\text{Proposition}}$ (Existence of reference frames). *Every locally finite DAG in which every node has finite causal past admits a reference frame. If in addition H3 holds, every reference frame takes finitely many values on any causal diamond.* *Proof.* Define $t(n)$ to be the length of the longest directed path terminating at $n$. Finiteness of the causal past of $n$ together with local finiteness implies that the set of directed paths terminating at $n$ is finite, so $t(n)<\infty$. If $(n,m)\in E$, any longest path to $n$ extends by $e$ to a path to $m$, so $t(m)\ge t(n)+1$. The level set $\Sigma_k$ is an antichain: if $n\preceq m$ with $n\ne m$ and both in $\Sigma_k$, the longest path to $n$ extended by a directed path $n\to m$ gives $t(m)>t(n)$, contradiction. Maximality within $\{t\ge k\}$: any $p$ with $t(p)>k$ has a longest path which passes through some node of level exactly $k$ (levels change by at least one along each edge, and the path starts at level 0), so $p$ is comparable to an element of $\Sigma_k$. Under H3, the diameter bound together with H1' implies that every directed path inside a causal diamond has at most $T\cdot T_{\max}$ edges (each edge has effective length at least $T_{\max}^{-1}$), so $t$ takes at most that many distinct values on the diamond. $\square$ The level function $t$ is the discrete analogue of a global time function on a globally hyperbolic spacetime; under the hypotheses of Theorem 5.6 it converges, after rescaling, to a Cauchy time function on the continuum limit. We shall not need this refinement. **Remark 4.8** (Dimensions). With $\hbar,c,G$ restored, $\widehat{V}_n$ has eigenvalues $\ell_P^3v$ with $\ell_P=\sqrt{\hbar G/c^3}$; $H_e=\Delta(e)\mathsf{N}$ has dimension of energy; the stabilising potential $V(x)=\tfrac{\kappa}{2\ell_P^6}(x-v_{\min}\ell_P^3)^2$ has dimension of energy when $\kappa$ is an energy; the hopping $t$ and $\lambda$ are energies and $g_s$ is dimensionless. **Hypothesis discharge ledger, Chapter 4.** | Result | Classical inputs | Hypotheses used | Where consumed | |---|---|---|---| | Def 4.3' (well-posedness of $H_e$) | Reed-Simon spectral theorem | H5 for uniform bounds | self-adjointness, gap | | Prop 4.5 | Peter-Weyl | H1, representation cutoffs | dimension count | | Prop 4.7 | none | L1, finite pasts; H1', H3 for the finiteness clause | longest-path construction | --- ## Chapter 5. Discrete Curvature and Metric Emergence ### 5.1. The two distance functions **Definition 5.0** (Lorentzian distance on the QSN). For $n\preceq m$, $$d_\tau(n,m)=\sup_{\gamma:n\to m}\sum_{e\in\gamma}\tau(e),\tag{5.1}$$ the supremum over directed paths from $n$ to $m$, with $\tau(e)=2\pi\hbar/\Delta(e)$ the proper time of Axiom L6. For $n\not\preceq m$ set $d_\tau(n,m)=0$. The effective metric $d_{\mathrm{eff}}$ of L4 (symmetrised as in Definition 4.1) plays the role of a Riemannian distance; $d_\tau$ plays the role of a Lorentzian time separation. The two are related only through the underlying graph, not by any formula, and the two continuum-limit theorems of this chapter are logically independent. **Lemma 5.1** $\boxed{\text{Theorem}}$ (Lorentzian pre-length space). *Let $\mathsf{Q}$ satisfy L1 and L6, and let H1 hold. Then $(N,\preceq,d_\tau)$ is a Lorentzian pre-length space in the sense of Kunzinger-Sämann (*Ann. Glob. Anal. Geom.* **54** (2018) 399, Def. 2.8): $N$ is a metric space (with $d_G$), $\preceq$ is a preorder, $d_\tau:N\times N\to[0,\infty]$ is lower semicontinuous, $d_\tau(n,m)>0$ only if $n\preceq m$, $n\ne m$, and the reverse triangle inequality* $$d_\tau(n,m)\ge d_\tau(n,p)+d_\tau(p,m)\qquad(n\preceq p\preceq m)\tag{5.2}$$ *holds. If in addition the causal past of every node is finite, the supremum in (5.1) is attained.* *Proof.* $(N,d_G)$ is a metric space: $d_G$ is symmetric, vanishes exactly on the diagonal, and satisfies the triangle inequality by path concatenation. The topology induced by $d_G$ on $N$ is discrete. Every function on a discrete space is continuous, hence $d_\tau$ is lower semicontinuous. Since $\mathsf{Q}$ is acyclic, $\preceq$ is antisymmetric; it is reflexive and transitive by construction, so it is a preorder (indeed a partial order). By definition $d_\tau(n,m)=0$ unless $n\preceq m$; when $n=m$ the only directed path is the empty path, of length 0, so $d_\tau(n,n)=0$ as required by the pre-length axioms. For (5.2): let $n\preceq p\preceq m$ and let $\gamma_1:n\to p$, $\gamma_2:p\to m$ be directed paths. Their concatenation $\gamma_1\gamma_2$ is a directed path $n\to m$, so $$d_\tau(n,m)\ge\sum_{e\in\gamma_1}\tau(e)+\sum_{e\in\gamma_2}\tau(e).$$ Taking the supremum over $\gamma_1$ and then over $\gamma_2$ gives (5.2). If every causal past is finite, the set of directed paths from $n$ to $m$ is a subset of the set of directed paths terminating at $m$, which is finite (each such path is a chain in the finite poset $J^-(m)$ and the number of chains in a finite poset is finite); the supremum over a finite set is attained. $\square$ ### 5.2. Ollivier curvature on a cylinder: status Recall Axiom L3: for $e=(n,m)\in E$, $$\mathcal{K}(e)=1-\frac{W_1(\mu_n,\mu_m)}{d_G(n,m)},\qquad \mu_n=\frac{1}{|N^+(n)|}\sum_{p\in N^+(n)}\delta_p,$$ where $W_1$ is the $1$-Wasserstein distance with respect to $d_G$. For an edge, $d_G(n,m)=1$. **Conjecture 5.2** (Ollivier curvature on a cylinder; formerly Lemma 5.2, proof withdrawn). *Let $(M,g_M)$ be a compact smooth Riemannian $d$-manifold with $\mathrm{Ric}_M\ge K_Mg_M$, $K_M>0$, and let $\mathsf{Q}_\rho$ be a bounded-degree coarse-graining (in the sense of C1: $k$-nearest-causal-neighbour truncation with $k=D$) of the Hasse diagram of a Poisson sprinkling of density $\rho$ into a slab $[0,T_0]\times M$ of the cylinder $(\mathbb{R}\times M,-\mathrm{d}t^2+g_M)$, equipped with the out-neighbour measures of L3 and $d_G$. Then there exist $K=K(K_M,d,D)>0$ and a scale $\epsilon_\rho\to0$ such that, with $\mathbb{P}_\rho$-probability tending to one, the rescaled curvature $\epsilon_\rho^{-2}\mathcal{K}(e)\ge K$ holds for all but a vanishing fraction of the edges.* *Retraction record.* The v3.0 proof of this statement (five steps: thin-shell identification of out-neighbourhoods, Poisson concentration, comparison with continuum shell measures, transport-rate input P, and the expansion of Lemma 7.4) fails at three independent points, identified in the second referee pass (review2.md, item 36). (a) In a slab of height $T_0$ the Hasse out-degree of a sprinkled point in dimension $d+1\ge3$ is of order $\rho\,\mathrm{vol}\{\tau\le r_\rho,\ t\le T_0\}\asymp\rho^{(d-1)/(d+1)}T_0^{d-1}\to\infty$, not $O(\log\rho)$: the future proper-time shell is unbounded along the light cone and its volume is not $O(\rho^{-1}\log\rho)$; this is C1. (b) The comparison step used a triangle inequality between $W_1^{d_G}$ (an infimum over couplings) and the causal cost $W_1^\tau$ (a supremum over causal couplings); no such inequality holds. (c) Even for the bounded-degree truncation, the rapidity cutoff of the truncated out-neighbourhood is $\chi_0=\tfrac{1}{d+1}\log\rho+O(1)$, so the rapidity factor $\overline{\sinh^2\chi}$ in the expansion of Lemma 7.4 is of order $r_\rho^{-2}$ and the curvature term is $O(1)$, not $O(r_\rho^2)$; this is the scale-invariant curvature fluctuation of Proposition 20.9 (C15). The statement is retained as a conjecture about the bounded-degree truncation, with $\epsilon_\rho$ left unspecified, because no argument in this monograph determines whether a rescaling exists under which a uniform lower bound holds. The transport-rate input P (the $k^{-1/D}$ rate for $D\ge3$ is Talagrand, *Ann. Probab.* **22** (1994) 919, and Dobrić-Yukich, *J. Theor. Probab.* **8** (1995) 97; Ajtai-Komlós-Tusnády is the $D=2$ case) is not at fault and is retained in Appendix A for use in Chapter 20. **Remark 5.2.1.** No result of Chapter 5 depends on Conjecture 5.2. Theorems 5.5 and 5.6 take H2 (where used) as a hypothesis on the sequence; Conjecture 5.2 is the statement that some physically constructed sequence satisfies it, which is Open Problem 22.1. ### 5.3. Volume doubling The analytic backbone of both continuum-limit theorems is a uniform volume-doubling inequality. The proof uses only discrete analysis; no synthetic Ricci theorem is invoked. Throughout this section we fix a QSN satisfying H1', H2, H4', with $N$ finite (for infinite $N$ the argument applies to each finite causal diamond, which is all that Theorems 5.5 and 5.6 require). Let $\Delta_G$ be the normalised graph Laplacian on $\ell^2(N,\mu)$: $$(\Delta_Gf)(n)=\sum_{m\sim n}w_{nm}\bigl(f(n)-f(m)\bigr),\qquad w_{nm}=\frac{T(e_{nm})}{\sum_{p\sim n}T(e_{np})},$$ where $m\sim n$ means $m\in N^+(n)\cup N^-(n)$ and $e_{nm}$ is the edge joining them. The weights $w_{nm}$ define a random walk on $N$ with transition kernel $P(n,m)=w_{nm}$, which is reversible with respect to the measure $\pi(n)=\sum_{p\sim n}T(e_{np})$. Under H1', $T_{\min}\le\pi(n)/\deg(n)\le T_{\max}$, so $\pi$ is comparable to $\mu$ with constants $T_{\min}$ and $DT_{\max}$. We write $P_t=e^{-t\Delta_G}$ for the continuous-time heat semigroup and $p_t(n,m)$ for its kernel with respect to $\pi$. Hypothesis H4' is two-sided (C5): the isoperimetric lower bound $|\partial B|\ge C_{\mathrm{iso}}|B|^{(N_0-1)/N_0}$ for $|B|\le|N|/2$, and the polynomial upper volume bound $|B_G(n,r)|\le C_{\mathrm{vol}}^{G}\,r^{N_0}$ for $r\ge1$. The upper bound is not derivable from H2: an Ollivier bound $\mathcal{K}\ge K>0$ yields only a diameter bound (Ollivier, loc. cit., Prop. 23, the discrete Bonnet-Myers estimate $\mathrm{diam}\le2/K$), and $K=0$ yields nothing about growth. The Ollivier curvature on any graph lies in $[-2,1]$. Lemma 5.3 does not use H2 at all. **Lemma 5.3** $\boxed{\text{Theorem}}$ (QSN volume doubling). *Let $\mathsf{Q}$ be a finite connected QSN satisfying L1, L2, L4 and hypotheses H1', H4' (two-sided). Then there is a constant $C_{\mathrm{dbl}}=C_{\mathrm{dbl}}(D,N_0,T_{\max}/T_{\min},C_{\mathrm{iso}},C^{G}_{\mathrm{vol}})$ such that for every $n\in N$ and every $r\ge1$ with $r\le c_0|N|^{1/N_0}$ (where $c_0=c_0(D,N_0,T_{\max}/T_{\min},C_{\mathrm{iso}},C^G_{\mathrm{vol}})$),* $$|B_{\mathrm{eff}}(n,2r)|\le C_{\mathrm{dbl}}\,|B_{\mathrm{eff}}(n,r)|.\tag{5.5}$$ *Moreover the effective balls satisfy the two-sided polynomial volume bound* $$c_{\mathrm{vol}}\,(rT_{\min})^{N_0}\le|B_{\mathrm{eff}}(n,r)|\le C_{\mathrm{vol}}\,(rT_{\max})^{N_0}\qquad(1\le r\le c_0|N|^{1/N_0}),\tag{5.6}$$ *with $c_{\mathrm{vol}},C_{\mathrm{vol}}$ depending on the same parameters.* The proof is in five steps. *Step A: metric equivalence.* Since every edge has $T(e)^{-1}\in[T_{\max}^{-1},T_{\min}^{-1}]$, every undirected path of $k$ edges has effective length in $[kT_{\max}^{-1},kT_{\min}^{-1}]$. Taking the infimum over paths, $$T_{\max}^{-1}\,d_G(n,m)\le d_{\mathrm{eff}}(n,m)\le T_{\min}^{-1}\,d_G(n,m).\tag{5.7}$$ Consequently $B_G(n,rT_{\min})\subseteq B_{\mathrm{eff}}(n,r)\subseteq B_G(n,rT_{\max})$, and it suffices to prove doubling and (5.6) for graph balls, with the ratio $T_{\max}/T_{\min}$ entering the constants. Under H1, $|B_G(n,r)|\le 2D^r$, so all balls are finite. *Step B: isoperimetry to Nash (Classical Q2, finite-graph form with additive term).* Classical input Q2 (Coulhon, Grigor'yan, *Duke Math. J.* **89** (1997) 133, Prop. 3.1 and Thm. 3.3, finite-graph case; Coulhon, Saloff-Coste, *Rev. Mat. Iberoam.* **9** (1993) 293, Thm. 1; Part I Theorem 2.9). Hypotheses: (i) finite connected weighted graph with edge weights in $[T_{\min},T_{\max}]$ and degree $\le D$; (ii) $|\partial B|\ge C_{\mathrm{iso}}|B|^{(N_0-1)/N_0}$ for all $B$ with $|B|\le|N|/2$; (iii) $N_0\ge3$. Conclusion: the relative Nash inequality $$\|f\|_2^{2+4/N_0}\le C_{\mathrm{N}}\Bigl(\mathcal{E}(f,f)+|N|^{-2/N_0}\|f\|_2^{2}\Bigr)\|f\|_1^{4/N_0}\qquad\text{for all }f:N\to\mathbb{R},\tag{5.8}$$ where $\mathcal{E}(f,f)=\tfrac12\sum_{n\sim m}\pi(n)w_{nm}(f(n)-f(m))^2$ and $C_{\mathrm{N}}=C_{\mathrm{N}}(C_{\mathrm{iso}},N_0,D,T_{\max}/T_{\min})$. The additive term removes the support restriction of (ii): for general $f$ one applies the restricted inequality to $(f-\mathrm{med}f)_\pm$, each supported on at most half the vertices, and the median term is controlled by $|N|^{-2/N_0}\|f\|_2^2$ through $\|f\|_1^{4/N_0}\ge|N|^{-4/N_0}\cdots$; this is Coulhon-Grigor'yan Thm. 3.3. We record the derivation of the restricted inequality because its hypotheses must match H4'. For $f\ge0$ supported on a set of measure at most $|N|/2$, the Maz'ya coarea formula gives $$\sum_{n\sim m}\pi(n)w_{nm}|f(n)-f(m)|=\int_0^\infty|\partial\{f>s\}|_{w}\,\mathrm{d}s\ \ge\ T_{\min}C_{\mathrm{iso}}\int_0^\infty|\{f>s\}|^{(N_0-1)/N_0}\,\mathrm{d}s\ \ge\ T_{\min}C_{\mathrm{iso}}\|f\|_{N_0/(N_0-1)},$$ where $|\partial B|_w=\sum_{n\in B,m\notin B}T(e_{nm})\ge T_{\min}|\partial B|$ and the last inequality is the layer-cake representation with Minkowski's integral inequality. This is the $L^1$-Sobolev inequality $\|f\|_{N_0/(N_0-1)}\le(T_{\min}C_{\mathrm{iso}})^{-1}\|\nabla f\|_{1,w}$. Applying it to $f^2$ and using $\|\nabla f^2\|_{1,w}=\sum\pi w|f(n)^2-f(m)^2|\le\bigl(\sum\pi w(f(n)-f(m))^2\bigr)^{1/2}\bigl(\sum\pi w(f(n)+f(m))^2\bigr)^{1/2}\le2\sqrt2\,\mathcal{E}(f,f)^{1/2}(DT_{\max})^{1/2}\|f\|_2$ gives $$\|f\|^{2}_{2N_0/(N_0-1)}\le2\sqrt2\,(DT_{\max})^{1/2}(T_{\min}C_{\mathrm{iso}})^{-1}\,\mathcal{E}(f,f)^{1/2}\|f\|_2.$$ Hölder interpolation $\|f\|_2\le\|f\|_1^{\theta}\|f\|_{q}^{1-\theta}$ with $q=2N_0/(N_0-1)$ and $\tfrac12=\theta+\tfrac{1-\theta}{q}$, i.e. $\theta=\tfrac{1}{N_0+1}$, combined with the previous display, yields (5.8) without the additive term on the restricted class (the exponent bookkeeping: $\|f\|_2^{2+4/N_0}\le\|f\|_1^{4/N_0}\|f\|_q^{2}$ after raising to the appropriate power, and $\|f\|_q^2\le C\mathcal{E}^{1/2}\|f\|_2$ then gives $\|f\|_2^{2+4/N_0}\le C\mathcal{E}^{1/2}\|f\|_2\|f\|_1^{4/N_0}$, and dividing by $\|f\|_2$ and squaring the resulting inequality for $\|f\|_2^{1+4/N_0}$ against $\mathcal{E}^{1/2}$ gives the displayed form with $C_{\mathrm{N}}=C^2$ after adjusting exponents; the computation is recorded in Part I, proof of Theorem 2.9). *Step C: Nash to ultracontractivity (Classical R, relative form).* Classical input R (Carlen, Kusuoka, Stroock, *Ann. Inst. H. Poincaré Probab. Statist.* **23** (1987) 245, Thm. 2.1; Coulhon, Grigor'yan, loc. cit., Thm. 3.3 for the finite-graph relative version). Hypotheses: a symmetric Markov semigroup $P_t=e^{-t\Delta_G}$ on $\ell^2(N,\pi)$ whose Dirichlet form satisfies (5.8) for all $f$. Conclusion: $$\sup_{n,m}p_t(n,m)\le C_{\mathrm{uc}}\,t^{-N_0/2}\qquad(1\le t\le c_1|N|^{2/N_0}),\tag{5.9}$$ with $C_{\mathrm{uc}},c_1$ depending on $(C_{\mathrm{N}},N_0)$. Proof of the implication: for $u_t=P_tf$ with $\|f\|_1=1$, $\tfrac{\mathrm{d}}{\mathrm{d}t}\|u_t\|_2^2=-2\mathcal{E}(u_t,u_t)\le-2C_{\mathrm{N}}^{-1}\|u_t\|_2^{2+4/N_0}+2|N|^{-2/N_0}\|u_t\|_2^2$ (using $\|u_t\|_1\le1$ by contractivity); for $t\le c_1|N|^{2/N_0}$ the additive term is dominated by the main term whenever $\|u_t\|_2^2\ge(2C_{\mathrm{N}})^{N_0/2}|N|^{-1}$, which holds on the stated time range by comparison with the solution of $\phi'=-C_{\mathrm{N}}^{-1}\phi^{1+2/N_0}$, giving $\|u_t\|_2^2\le(2t/(N_0C_{\mathrm{N}}))^{-N_0/2}$; then $\|P_t\|_{1\to\infty}\le\|P_{t/2}\|_{1\to2}^2$ by symmetry. Symmetry of $P_t$ on $\ell^2(\pi)$ is reversibility of $P$ with respect to $\pi$, which holds because $\pi(n)w_{nm}=T(e_{nm})=\pi(m)w_{mn}$. *Step D: Gaussian off-diagonal bound (Classical S).* Classical input S (Davies, *J. Anal. Math.* **58** (1992) 99, Thm. 1; Delmotte, *Rev. Mat. Iberoam.* **15** (1999) 181, Thm. 1.3 and §2.2 for the graph case). Hypotheses: a weighted graph with weights in $[T_{\min},T_{\max}]$ and degree $\le D$ (bounded geometry), and the on-diagonal bound (5.9) on a time interval. Conclusion: on the same interval and for $t\ge d_G(n,m)$, $$p_t(n,m)\le C_{\mathrm{G}}\,t^{-N_0/2}\exp\Bigl(-\frac{d_G(n,m)^2}{C_{\mathrm{G}}\,t}\Bigr),\tag{5.10}$$ and for $t0$, $\sup_n\sum_mw_{nm}(e^{\alpha(\psi(n)-\psi(m))}-1)^2\le D(T_{\max}/T_{\min})(e^{\alpha}-1)^2$, which bounds the perturbation of the Dirichlet form and yields (5.10) after optimising over $\alpha$. H2 is not used. *Step E: polynomial volume bounds and doubling.* The upper bound in (5.6) is the second half of H4': $|B_G(n,r)|\le C^{G}_{\mathrm{vol}}r^{N_0}$, hence $\pi(B_G(n,r))\le DT_{\max}C^{G}_{\mathrm{vol}}r^{N_0}$. The lower bound in (5.6): fix $r\ge1$ with $r^2\le c_1|N|^{2/N_0}$, set $t=r^2$, and let $C\ge2$ be a constant to be chosen. Decompose the complement of $B_G(n,Cr)$ into dyadic annuli $A_j=B_G(n,2^{j+1}Cr)\setminus B_G(n,2^{j}Cr)$, $j\ge0$. On $A_j$ with $2^jCr\le t=r^2$ (i.e. $2^jC\le r$), (5.10) gives $p_t\le C_{\mathrm{G}}r^{-N_0}e^{-4^jC^2/C_{\mathrm{G}}}$; on $A_j$ with $2^jC>r$ the jump-count bound gives $p_t\le(DT_{\max}/T_{\min})e^{-2^jCr\log(2^jC/(er))}\le(DT_{\max}/T_{\min})e^{-2^jCr}$ once $2^jC\ge e^2r$, and for $r<2^jC0$ such that for every $n\in N$ and every $k\ge1$ with $|B_G(n,k)|\le|N|/2$, $$|\{m\succeq n:\ \ell^+(n,m)\le k\}|\ \ge\ \phi_0\,|B_G(n,k)|,$$ where $\ell^+(n,m)$ is the number of edges of the longest directed path from $n$ to $m$. H-cf says that the causal future of every node occupies a fixed fraction of its undirected neighbourhood at every scale. It is a hypothesis on the QSN; it is not implied by H1', H2, H4', H5. (The v3.0 text claimed to derive it from H2 through the out-walk; the argument produced an out-step probability $(T_{\min}/(DT_{\max}))^{O(\sqrt k)}$, exponentially small in $\sqrt k$, which a polynomial volume bound does not compensate. The claim is withdrawn, review2 item 19.) **Lemma 5.4** $\boxed{\text{Conditional (on H-cf)}}$ (Timelike volume doubling). *Let $\mathsf{Q}$ be a finite connected QSN satisfying L1, L2, L4, L6 and hypotheses H1', H4' (two-sided), H5 (with $\Delta_{\min}\le\Delta(e)\le\Delta_{\max}$), and H-cf. Define the timelike ball $B_\tau^+(n,r)=\{m\succeq n:d_\tau(n,m)\le r\}$. Then there is a constant $C_{\mathrm{dbl}}^\tau=C_{\mathrm{dbl}}^\tau(D,N_0,T_{\max}/T_{\min},C_{\mathrm{iso}},C^{G}_{\mathrm{vol}},\tau_{\max}/\tau_{\min},\phi_0)$, with $\tau_{\min}=2\pi\hbar/\Delta_{\max}$ and $\tau_{\max}=2\pi\hbar/\Delta_{\min}$, such that for every $n$ and every $r\ge\tau_{\max}$ with $r/\tau_{\min}\le c_0|N|^{1/N_0}$,* $$|B_\tau^+(n,2r)|\le C_{\mathrm{dbl}}^\tau\,|B_\tau^+(n,r)|,\tag{5.11}$$ *and $c\,(r/\tau_{\max})^{N_0}\le|B_\tau^+(n,r)|\le C\,(r/\tau_{\min})^{N_0}$ on the same range, with $c=\phi_0c_{\mathrm{vol}}T_{\min}^{N_0}$ and $C=C^{G}_{\mathrm{vol}}$.* *Proof.* By H5, $\tau_{\min}\le\tau(e)\le\tau_{\max}$ for every edge, and for every directed path $\gamma$ of $k$ edges, $k\tau_{\min}\le\sum_{e\in\gamma}\tau(e)\le k\tau_{\max}$. Taking the supremum over directed paths from $n$ to $m$, $$\tau_{\min}\,\ell^+(n,m)\le d_\tau(n,m)\le\tau_{\max}\,\ell^+(n,m).\tag{5.12}$$ Let $B_\ell^+(n,k)=\{m\succeq n:\ell^+(n,m)\le k\}$. By (5.12), $B_\ell^+(n,\lfloor r/\tau_{\max}\rfloor)\subseteq B_\tau^+(n,r)\subseteq B_\ell^+(n,\lceil r/\tau_{\min}\rceil)$. *Upper bound.* A directed path of $k$ edges is an undirected path of $k$ edges, so $B_\ell^+(n,k)\subseteq B_G(n,k)$ and $|B_\ell^+(n,k)|\le C^{G}_{\mathrm{vol}}k^{N_0}$ by H4'. *Lower bound.* By H-cf and the lower bound in (5.6) (Lemma 5.3, which uses only H1', H4'), $|B_\ell^+(n,k)|\ge\phi_0|B_G(n,k)|\ge\phi_0c_{\mathrm{vol}}T_{\min}^{N_0}k^{N_0}$ for $1\le k\le c_0|N|^{1/N_0}$ (the restriction $|B_G(n,k)|\le|N|/2$ of H-cf holds on this range by the upper bound in (5.6) after shrinking $c_0$). *Doubling.* $|B_\tau^+(n,2r)|\le|B_\ell^+(n,\lceil2r/\tau_{\min}\rceil)|\le C^{G}_{\mathrm{vol}}(2r/\tau_{\min}+1)^{N_0}\le C^{G}_{\mathrm{vol}}(3r/\tau_{\min})^{N_0}$ for $r\ge\tau_{\max}\ge\tau_{\min}$, and $|B_\tau^+(n,r)|\ge|B_\ell^+(n,\lfloor r/\tau_{\max}\rfloor)|\ge\phi_0c_{\mathrm{vol}}T_{\min}^{N_0}(r/(2\tau_{\max}))^{N_0}$ for $r\ge\tau_{\max}$ (so that $\lfloor r/\tau_{\max}\rfloor\ge r/(2\tau_{\max})$). The ratio gives (5.11) with $C^\tau_{\mathrm{dbl}}=6^{N_0}C^{G}_{\mathrm{vol}}(\tau_{\max}/\tau_{\min})^{N_0}/(\phi_0c_{\mathrm{vol}}T_{\min}^{N_0})$, and the stated $c,C$ follow from the same two displays. $\square$ **Remark 5.4.1.** Lemma 5.4 uses no curvature hypothesis. The timelike Ollivier bound H6 enters only in Chapter 7. H-cf is the only place where the causal structure interacts with volume growth; for the bounded-degree sprinklings of Chapter 20 it is open whether H-cf holds with $\phi_0$ independent of $\rho$ (Open Problem 22.1). ### 5.4. The Riemannian continuum limit Let $\mathrm{Mfld}_{K,N_0,T}$ be the class of compact Riemannian manifolds of dimension $N_0$ with $\mathrm{Ric}\ge Kg$ and diameter $\le T$. Throughout Section 5.4 we use the rescaled graph metric $$d_n:=\epsilon_n\,d_G,\qquad\epsilon_n:=n^{-1}T_{\min}^{-1},\tag{5.13a}$$ rather than $d_{\mathrm{eff},n}$. By (5.7), $d_n\le d_{\mathrm{eff},n}\le(T_{\max}/T_{\min})\,d_n$, so the identity map is bi-Lipschitz between the two metrics with constant $T_{\max}/T_{\min}$, and any mGH limit in one metric is bi-Lipschitz homeomorphic (with the same constant) to an mGH limit in the other. The reason for working with $d_n$ is that Ollivier's contraction (Classical B1) is exact in $d_G$, hence in $d_n$, while in $d_{\mathrm{eff},n}$ it acquires a multiplicative prefactor $T_{\max}/T_{\min}\ge1$ which destroys the contraction property (review2 item 21). **Hypothesis H2-ker** (Ollivier bound for the Laplacian kernel). The Ollivier curvature, computed with the undirected tension-weighted kernel $P(n,m)=w_{nm}$ of Section 5.3 and the metric $d_G$, satisfies $\mathcal{K}_P(n,m)\ge K\epsilon_n^{2}$ for all adjacent $n,m$, with $K\ge0$ fixed along the sequence. H2-ker is a different hypothesis from H2 (which is stated for the uniform out-neighbour measure of L3); neither implies the other. H2-ker is the form consumed in Theorem 5.5. (C14, review2 item 13.) **Hypothesis H-conv** (Mosco convergence of Dirichlet forms, Riemannian). Along the mGH-convergent subsequence of Theorem 5.5, the Dirichlet forms $\mathcal{E}_n$ of the rescaled graph Laplacians $\epsilon_n^{-2}\Delta_{G,n}$ on $L^2(N_n,\mu_n)$ Mosco-converge, in the sense of Kuwae-Shioya (*Comm. Anal. Geom.* **11** (2003) 599, Def. 2.11, convergence of forms on varying Hilbert spaces), to the Cheeger energy $\mathrm{Ch}$ of the limit $(X,d_X,\mathfrak{m})$, and $\mathrm{Ch}$ is a quadratic form (so $X$ is infinitesimally Hilbertian). H-conv is a hypothesis on the sequence, not a theorem; it is known to hold for sprinklings into smooth manifolds at the diameter scale (García Trillos, Slepčev, *Arch. Rat. Mech. Anal.* **220** (2016) 193, Thm. 1.4, under the connectivity-radius condition $\epsilon_n\gg(\log|N_n|/|N_n|)^{1/N_0}$) and is open for general QSN sequences. C4 records why it is needed. **Hypothesis H-BE** (curvature-dimension condition on the limit). The limit $(X,d_X,\mathfrak{m})$ satisfies $\mathrm{RCD}(K,N_0)$. Equivalently, under H-conv: $(X,d_X,\mathfrak{m})$ has the Sobolev-to-Lipschitz property (every $f\in W^{1,2}(X)$ with $|\nabla f|\le1$ $\mathfrak{m}$-a.e. has a $1$-Lipschitz representative) and its heat flow satisfies the dimensional Bakry-Émery estimate $\mathrm{BE}(K,N_0)$ (Erbar, Kuwada, Sturm, *Invent. Math.* **201** (2015) 993, Thm. 7 for $\mathrm{BE}(K,N)\Leftrightarrow\mathrm{RCD}^*(K,N)$ under infinitesimal Hilbertianity and Sobolev-to-Lipschitz; Ambrosio, Gigli, Savaré, *Ann. Probab.* **43** (2015) 339, Thm. 4.17 for the $N=\infty$ equivalence; Cavalletti, Milman, *Invent. Math.* **226** (2021) 1, Thm. 1.1 for $\mathrm{RCD}^*(K,N)\Leftrightarrow\mathrm{RCD}(K,N)$). H-BE is stated directly as a condition on the limit because no result in this monograph derives it from the discrete data: the Ollivier bound H2-ker yields the $W_1$ heat contraction on each graph (B1) but (a) there is no theorem converting an Ollivier bound on a graph into a graph Bakry-Émery estimate (the two notions are incomparable on general graphs; Münch, Wojciechowski, *Adv. Math.* **356** (2019) 106759, establish the Laplacian characterisation of Ollivier curvature, not such an implication), (b) even a graph $L^2$ gradient estimate would not pass to the limit under Mosco convergence of the forms alone, since Mosco convergence controls the integrated energies and not the carrés du champ, and (c) no finite-dimensional condition $\mathrm{CD}(K,N)$ with $N<\infty$ is obtainable from Ollivier curvature, and $\mathrm{RCD}(K,\infty)$ together with $N_0$-Ahlfors regularity does not imply $\mathrm{RCD}(K,N_0)$ (a smooth weighted manifold $(M^{N_0},g,e^{-V}\mathrm{vol})$ with $\mathrm{Ric}+\mathrm{Hess}V\ge K$ and non-constant $V$ is a counterexample). H-BE is the curvature conclusion assumed, not derived; review2 items 22, 24, 25, C17. It holds when the limit is a smooth Riemannian manifold with $\mathrm{Ric}\ge K$ (Bakry, Émery, *Séminaire de Probabilités XIX*, Springer 1985, and Sturm, *Acta Math.* **196** (2006) 133, Thm. 1.7), i.e. under H10 with the pointwise bound. **Theorem 5.5** $\boxed{\text{Conditional (on H-conv, H-BE) modulo Classical(A, C)}}$ (Riemannian continuum limit). *Let $(\mathsf{Q}_n)_{n\ge1}$ be a sequence of finite connected QSNs satisfying L1 to L5, and H1' (with fixed $D$ and fixed ratio $T_{\max}/T_{\min}$), H3 with fixed $T$ after rescaling, and H4' (two-sided) with fixed $N_0\ge3$, $C_{\mathrm{iso}}$, $C^{G}_{\mathrm{vol}}$, and $|N_n|\to\infty$. Let $d_n=\epsilon_nd_G$ be the rescaled graph metric (5.13a) and $\mu_n=|N_n|^{-1}\mu$ the normalised counting measure. Suppose H-conv and H-BE hold. Then there exist a compact metric measure space $(X,d_X,\mathfrak{m})$ and a subsequence $(n_k)$ such that* $$(N_{n_k},d_{n_k},\mu_{n_k})\xrightarrow{\ \mathrm{mGH}\ }(X,d_X,\mathfrak{m}),\tag{5.13}$$ *$(X,d_X,\mathfrak{m})$ is an $\mathrm{RCD}(K,N_0)$ space with $\mathfrak{m}=c\,\mathcal{H}^{N_0}$ for a constant $c=c(N_0,C^{G}_{\mathrm{vol}},c_{\mathrm{vol}},D,T_{\max}/T_{\min})>0$, and $X$ decomposes as $X=\mathcal{R}\sqcup\mathcal{S}$ where $\mathcal{R}$ is an open dense subset that is a topological manifold of dimension $N_0$, every point of $\mathcal{R}$ has a neighbourhood bi-Hölder homeomorphic to a Euclidean ball (with a Hölder exponent $\alpha(N_0,K,T)\in(0,1)$), and $\mathcal{S}$ is closed with $\dim_{\mathcal{H}}\mathcal{S}\le N_0-2$. The distance $d_X|_\mathcal{R}$ is the length distance of a continuous Riemannian metric $g$ on $\mathcal{R}$ in the bi-Hölder charts. The mGH limit of $(N_{n_k},d_{\mathrm{eff},n_k},\mu_{n_k})$ along the same subsequence exists and is bi-Lipschitz homeomorphic to $X$ with constant $T_{\max}/T_{\min}$. H2-ker is not needed for the conclusion; its role is recorded in Remark 5.5.3.* *Classical input A (Gromov precompactness).* Hypotheses: a family of compact metric spaces with diameter $\le T$ and metric doubling with a common constant $C_{\mathrm{dbl}}$ (every ball of radius $2r$ is covered by at most $C_{\mathrm{dbl}}$ balls of radius $r$). Conclusion: the family is precompact in the Gromov-Hausdorff topology (Gromov, *Metric Structures for Riemannian and Non-Riemannian Spaces*, Birkhäuser 2007, Prop. 5.2; Burago, Burago, Ivanov, *A Course in Metric Geometry*, AMS 2001, Thm. 7.4.15). Measured refinement: if the spaces carry Borel probability measures that are measure-doubling with constant $C_{\mathrm{dbl}}$, the family is precompact in the measured GH topology, and the limit measure is doubling with the same constant and is a probability measure (Fukaya, *Invent. Math.* **87** (1987) 517, Prop. 2.6; Cheeger, Colding, *J. Diff. Geom.* **46** (1997) 406, §1). No-mass-escape: all spaces are compact with uniformly bounded diameter and the measures are probability measures; a probability measure on a compact space is tight, so the push-forwards through the GH approximations are tight and every weak limit is a probability measure. *Classical input C (regularity of non-collapsed $\mathrm{RCD}$ spaces).* Hypotheses: $(X,d_X,\mathfrak{m})$ is $\mathrm{RCD}(K,N_0)$ and weakly non-collapsed, $\mathfrak{m}\ll\mathcal{H}^{N_0}$ with $\mathfrak{m}$ and $\mathcal{H}^{N_0}$ mutually absolutely continuous and comparable. Conclusion: (C-i) $\mathfrak{m}=c\,\mathcal{H}^{N_0}$ for a constant $c>0$ (Brena, Gigli, Honda, Zhu, *J. Reine Angew. Math.* **794** (2023) 215, Thm. 1.3: weakly non-collapsed $\mathrm{RCD}(K,N)$ spaces are non-collapsed up to a constant factor); (C-ii) $X=\mathcal{R}\sqcup\mathcal{S}$ with $\mathcal{R}$ open dense, every point of $\mathcal{R}$ having a neighbourhood bi-Hölder homeomorphic to a Euclidean ball with exponent $\alpha(N_0,K,\mathrm{diam})\in(0,1)$, and $\mathcal{S}$ closed with $\dim_{\mathcal{H}}\mathcal{S}\le N_0-2$ (De Philippis, Gigli, *J. Éc. Polytech. Math.* **5** (2018) 613, Thm. 1.3 and Thm. 1.4; Kapovitch, Mondino, *Geom. Topol.* **25** (2021) 445, Thm. 1.3, bi-Hölder charts on the regular set of non-collapsed $\mathrm{RCD}$ spaces); (C-iii) on $\mathcal{R}$ the distance is the length distance of a continuous Riemannian metric in the bi-Hölder charts (Cheeger, Colding, loc. cit., Thm. 5.9, for Ricci limits; for non-collapsed $\mathrm{RCD}$ spaces this is Kapovitch-Mondino, loc. cit., Remark 1.5, via the Reifenberg-flatness of the regular set and the Cheeger-Colding structure theory which applies to $\mathrm{RCD}$ spaces by De Philippis-Gigli). The metric is not smooth in general: the $C^{1,\alpha}$ conclusion of Anderson (*Invent. Math.* **102** (1990) 429, Thm. 1.1) requires two-sided Ricci bounds and a lower injectivity-radius bound on smooth approximants, and Cheeger-Naber (*Ann. Math.* **182** (2015) 1093, Thm. 1.6) takes two-sided bounds as hypothesis. Smoothness of $g$ on $\mathcal{R}$ is the separate hypothesis H10 of Part I (C3). *Proof of Theorem 5.5.* *Discharge of A.* By Lemma 5.3 applied to $\mathsf{Q}_n$ (which uses only H1', H4'), the measured spaces $(N_n,d_n,\mu_n)$ are measure-doubling with the constant $C_{\mathrm{dbl}}$ of Lemma 5.3 at all scales $\epsilon_n\le r\le\epsilon_nc_0|N_n|^{1/N_0}$; by H3 and (5.6) the diameter in $d_n$ is at most $T$ and $|N_n|\ge c\,\epsilon_n^{-N_0}$ (the ball of radius $\mathrm{diam}$ is all of $N_n$), so the doubling range covers $[\epsilon_n,T]$ after adjusting $c_0$, and for $r<\epsilon_n$ the balls are singletons and doubling is trivial. Measure doubling implies metric doubling with constant $C_{\mathrm{dbl}}^{5}$: a ball $B(x,2r)$ is covered by the balls $B(y,r)$ for $y$ in a maximal $r$-separated subset $Y$ of $B(x,2r)$; the balls $B(y,r/2)$, $y\in Y$, are pairwise disjoint and contained in $B(x,4r)$; hence $|Y|\min_{y\in Y}\mu(B(y,r/2))\le\mu(B(x,4r))$, and for any $y\in Y$, $\mu(B(x,4r))\le C_{\mathrm{dbl}}^{3}\mu(B(x,r/2))\le C_{\mathrm{dbl}}^{3}\mu(B(y,3r))\le C_{\mathrm{dbl}}^{5}\mu(B(y,3r/4))\le C_{\mathrm{dbl}}^{5}\mu(B(y,r/2))\cdot C_{\mathrm{dbl}}$, giving $|Y|\le C_{\mathrm{dbl}}^{6}$; we use this constant. Classical A yields a subsequence converging in mGH to a compact $(X,d_X,\mathfrak{m})$ with $\mathfrak{m}$ a doubling probability measure. *Volume bounds on the limit.* The two-sided bound (5.6), after normalisation by $|N_n|$ and in the metric $d_n$, reads $c\,r^{N_0}\le\mu_n(B_n(x,r))\le C\,r^{N_0}$ for $\epsilon_n\le r\le T$, with $c=c_{\mathrm{vol}}T_{\min}^{N_0}/(C^G_{\mathrm{vol}}T^{N_0})$ and $C=C^G_{\mathrm{vol}}/(c_{\mathrm{vol}}T_{\min}^{N_0}T^{N_0})$ up to the factor relating $|N_n|$ to $\epsilon_n^{-N_0}T^{N_0}$ (both bounds from (5.6) at $r=T/\epsilon_n$), independent of $n$. These pass to the limit along the GH approximations by the Portmanteau theorem applied to open and closed balls of all but countably many radii (those radii whose spheres have positive $\mathfrak{m}$-measure are countable because the spheres are disjoint and $\mathfrak{m}$ is finite): $c\,r^{N_0}\le\mathfrak{m}(B(x,r))\le C\,r^{N_0}$ for $00$, a uniform bound $N(\epsilon)$ on the cardinality of maximal $\epsilon$-nets (with respect to the Minguzzi-Suhr Noldus-type distance $\mathrm{d}_\tau(x,y)=\sup_z|\tau(x,z)-\tau(y,z)|+|\tau(z,x)-\tau(z,y)|$). Conclusion: the family is precompact in the GHL topology, and GHL limits are bounded Lorentzian metric spaces. Reconciliation with Kunzinger-Sämann: a bounded Lorentzian metric space with continuous $\tau$ and compact topology is a Lorentzian pre-length space (with the metric $\mathrm{d}_\tau$ as the background metric), and it is globally hyperbolic in the Kunzinger-Sämann sense when causal diamonds are compact, which holds by compactness of $X$ and continuity of $\tau$ (closed subsets of a compact space); conversely the discrete $(N,\preceq,d_\tau)$ with the discrete topology is a bounded Lorentzian metric space. Neither "timelike doubling" nor "global hyperbolicity" is among Minguzzi-Suhr's hypotheses; the net bound is. Measured refinement and no-mass-escape: if the spaces carry Borel probability measures and the GHL approximating correspondences $R_n\subset N_n\times X$ are used to push $\mu_n$ forward through a measurable selection $\sigma_n$ of $R_n$ (which exists by the Kuratowski-Ryll-Nardzewski theorem since $R_n$ is closed with compact fibres), the push-forwards $(\sigma_n)_\#\mu_n$ are probability measures on the compact space $X$, hence tight, hence relatively compact by Prokhorov; any two selections differ pointwise by at most the distortion $\delta_n$ in $\mathrm{d}_\tau$, so their push-forwards have bounded-Lipschitz distance at most $\delta_n$ and all limit points coincide. Every limit point is a probability measure, so no mass escapes. *Proof of Theorem 5.6.* *Structure.* By Lemma 5.1, each $(N_n,\preceq,d_{\tau,n})$ is a Lorentzian pre-length space; with the discrete topology it is compact, $d_{\tau,n}$ is continuous, and $\sup d_{\tau,n}\le T$ by H3 and (5.7), (5.12) (a directed path of $k$ edges has $d_\tau\le k\tau_{\max}$ and $d_{\mathrm{eff}}\ge kT_{\max}^{-1}$, so $\sup d_\tau\le\tau_{\max}T_{\max}\,\mathrm{diam}_{d_{\mathrm{eff}}}$). So each is a bounded Lorentzian metric space. *Discharge of A-Lor.* The net bound: the Minguzzi-Suhr distance $\mathrm{d}_\tau$ on $N_n$ is bounded below by $\tfrac{1}{2}$ times the symmetrised $d_{\tau,n}$-separation to the future and past on comparable pairs (take $z=x$ or $z=y$ in the supremum), so a maximal $\epsilon$-net for $\mathrm{d}_\tau$ is contained in a maximal $\epsilon/2$-separated set for the timelike distance, whose cardinality in a timelike ball of radius $T$ is at most $|B^+_\tau(\cdot,T+\epsilon)|/\min|B^+_\tau(\cdot,\epsilon/4)|\le C(T/\epsilon)^{N_0}$ by the two-sided timelike volume bound of Lemma 5.4 (Conditional on H-cf), uniformly in $n$ after normalisation; this gives $N(\epsilon)=C(T/\epsilon)^{N_0}$. Minguzzi-Suhr Thm. 4.5 yields the GHL-convergent subsequence and the limit bounded Lorentzian metric space $(X,\preceq_X,\tau_X)$ with continuous $\tau_X$. The measured refinement yields $\mathfrak{m}$. The timelike volume bounds of Lemma 5.4 pass to the limit by the Portmanteau theorem (timelike balls of all but countably many radii have $\mathfrak{m}$-null boundary, as the spheres are disjoint). Global hyperbolicity is compactness of causal diamonds, from compactness of $X$ and continuity of $\tau_X$. This proves (a). *Discharge of H-TCD and C-Lor for (b).* H-TCD supplies $\mathrm{TCD}_1(K_\tau,N_0)$ and H7 on $X$; the non-collapsing hypothesis of C-Lor is the two-sided timelike volume bound of (a), which is the McCann-Sämann comparability (loc. cit., Prop. 3.6). C-Lor (a conjecture) yields (b). $\square$ **Remark 5.6.1** (Which sequences lie in the hypothesis class). Theorem 5.6 is a statement about sequences satisfying its hypothesis list. The bounded-degree window sprinklings of Chapter 20 are not known to be in this class: Proposition 20.9 shows that their Ollivier curvatures have scale-invariant fluctuations of order one, and Theorem 20.8 shows that at fixed degree bound $D$ their $d_{\tau,n}$ converges to a Lorentz-Finsler time separation $d_{g,D}$ strictly smaller than $d_g$, with $d_g$ recovered only as $D\to\infty$ (C15). Whether H-cf holds for them with $\phi_0$ independent of $\rho$ is also open. Whether any physically constructed sequence lies in the hypothesis class is Open Problem 22.1 of Chapter 22. **Corollary 5.7** $\boxed{\text{Conditional (on H-cf) modulo Classical(A-Lor)}}$ (Uniqueness of the Lorentzian continuum limit). *The GHL limit $(X,\preceq_X,\tau_X)$ of any convergent subsequence in Theorem 5.6(a) is unique up to isometry of bounded Lorentzian metric spaces (bijections preserving $\tau$). Under the hypotheses of (b), on the regular set the isometry preserves the continuous metric $g^{\mathrm{L}}$ as a continuous tensor field. The same holds for the mGH limit of Theorem 5.5 with isometry of metric measure spaces.* *Proof.* Minguzzi-Suhr (loc. cit., Thm. 4.9) prove that the GHL distance is a metric on isometry classes of bounded Lorentzian metric spaces, so limits are unique up to isometry. Under (b), an isometry $\phi$ preserves $\tau_X$, hence the chronological relation $\tau_X>0$ on $\mathcal{M}$, hence the null cones of $g^{\mathrm{L}}$ in the charts; two continuous Lorentzian metrics on an open set with the same null cones are conformal, $g_2=\Omega^2g_1$ with $\Omega>0$ continuous (at each point the two quadratic forms have the same null cone, and a Lorentzian quadratic form is determined by its null cone up to a positive scalar); the time separation of $\Omega^2g$ between $p$ and $q=\exp$-close to $p$ along a timelike direction $v$ is $\Omega(p)\tau_{g}(p,q)+o(\tau_g(p,q))$ by continuity of $\Omega$, so equality of time separations for all pairs forces $\Omega\equiv1$. The mGH statement is Gromov's (Burago-Burago-Ivanov, loc. cit., Thm. 7.3.30). $\square$ **Proposition 5.8** $\boxed{\text{Conditional (on H10, H-cf, H-TCD, C-Lor) modulo Classical(A-Lor)}}$ (Emergent metric formula). *Under the hypotheses of Theorem 5.6(b) and H10 in the form that $g^{\mathrm{L}}$ is $C^{2}$ on $\mathcal{M}$, let $x\in\mathcal{M}$, let $U$ be a $C^2$ chart around $x$ in which $I^+(x)\cap U$ is the chronological future of $x$, and let $x_0(n)\in N_n$ be nodes with $x_0(n)\to x$ under the GHL approximating correspondences. Then, as distributions on $I^+(x)\cap U$,* $$g^{\mathrm{L}}_{\mu\nu}=-\tfrac12\,\lim_{n\to\infty}\partial_\mu\partial_\nu\bigl[d_{\tau,n}(x_0(n),\cdot)^{2}\bigr]\quad\text{on }I^+(x)\cap U,\tag{5.15}$$ *where $d_{\tau,n}(x_0(n),\cdot)$ is transported to $U$ by the approximating correspondences, the limit is in $\mathcal{D}'(I^+(x)\cap U)$, and $g^{\mathrm{L}}_{\mu\nu}$ on the left is the continuous function $g^{\mathrm{L}}_{\mu\nu}(q)$ for $q\in I^+(x)\cap U$ evaluated in the limit $q\to x$ in the sense that $-\tfrac12\partial_\mu\partial_\nu\tau_X(x,\cdot)^2$ is a continuous function on $I^+(x)\cap U$ extending continuously to $x$ with value $g^{\mathrm{L}}_{\mu\nu}(x)$.* *Proof.* In a $C^{2}$ globally hyperbolic Lorentzian manifold, for $q$ in the chronological future of $x$ within a convex normal neighbourhood, the maximising causal curve from $x$ to $q$ is the unique timelike geodesic in that neighbourhood (O'Neill, *Semi-Riemannian Geometry*, Academic Press 1983, Prop. 5.34 and Lem. 5.33), so $\tau_X(x,q)^2=-g_x(\exp_x^{-1}q,\exp_x^{-1}q)$, which is a $C^{1}$ function of $q$ on $I^+(x)\cap U$ (the inverse exponential map of a $C^2$ metric is $C^1$) with $-\tfrac12\partial_\mu\partial_\nu\tau_X(x,\cdot)^2(q)\to g^{\mathrm{L}}_{\mu\nu}(x)$ as $q\to x$ inside $I^+(x)$, by the expansion $\exp_x^{-1}q=(q-x)+O(|q-x|^2)$ with the $O$ controlled by the $C^0$ norm of the Christoffel symbols. The second derivatives exist in the distributional sense on $I^+(x)\cap U$ and are continuous functions there. GHL convergence gives uniform convergence $d_{\tau,n}(x_0(n),\cdot)\to\tau_X(x,\cdot)$ on compact subsets of $I^+(x)\cap U$ through the correspondences (Minguzzi-Suhr, loc. cit., Def. 4.1), hence uniform convergence of the squares, hence convergence in $\mathcal{D}'(I^+(x)\cap U)$ of the second derivatives: for $\varphi\in C_c^\infty(I^+(x)\cap U)$, $\int d_{\tau,n}^2\,\partial_\mu\partial_\nu\varphi\to\int\tau_X^2\,\partial_\mu\partial_\nu\varphi$ by dominated convergence with the uniform bound $\sup|d_{\tau,n}^2|\le T^2$ on $\mathrm{supp}\,\varphi$ as dominating function. No statement is made across the light cone of $x$, where $\tau_X(x,\cdot)^2$ extended by zero is Lipschitz but not $C^1$ and its distributional Hessian carries a singular part supported on the cone (review2 item 35). $\square$ **Remark 5.8.1** (Dimensions). With $\hbar,c,G$ restored: $\tau(e)=2\pi\hbar/\Delta(e)$ has dimension of time when $\Delta(e)$ is an energy; $d_\tau$ is a time; $g^{\mathrm{L}}_{\mu\nu}$ in (5.15) is dimensionless when coordinates carry dimension of length and $\tau^2$ is multiplied by $c^2$; $T(e)$ has dimension of inverse length so that $d_{\mathrm{eff}}$ is a length; $\mathcal{K}$ is dimensionless and $\epsilon_n^{-2}\mathcal{K}$ has dimension of inverse length squared, matching $\mathrm{Ric}$. **Hypothesis discharge ledger, Chapter 5.** | Result | Classical inputs | Hypotheses used | Where consumed | |---|---|---|---| | Lem 5.1 | none | L1, L6, H1 | preorder (L1), $\tau>0$ (L6), finiteness of path sets (H1) | | Conj 5.2 | (withdrawn proof) | | | | Lem 5.3 | Q2, R, S | H1' (Steps A, B, D), H4' lower (Step B), H4' upper (Step E); no curvature hypothesis | as listed | | Lem 5.4 | Lem 5.3 | H1', H4', H5 ((5.12)), H-cf (lower bound) | as listed | | Thm 5.5 | A, C (Brena-Gigli-Honda-Zhu, De Philippis-Gigli, Kapovitch-Mondino) | H1', H3, H4' (Lem 5.3); H-conv, H-BE (assumed on the limit) | A (doubling, H3); C (H-BE, Ahlfors regularity) | | Rem 5.5.3 / Conj 5.9 | B1 (Ollivier), Kuwada | H2-ker | graph-level contraction only | | Thm 5.6 (a) | A-Lor (Minguzzi-Suhr) | H1', H3, H4', H5, H-cf (Lem 5.4 for net bounds) | A-Lor net bound | | Thm 5.6 (b) | C-Lor (conjecture) | H-TCD (incl. H7) | regular set | | Cor 5.7 | Minguzzi-Suhr Thm 4.9 | those of Thm 5.6(a) [(b) for the metric clause] | inherited | | Prop 5.8 | O'Neill convex neighbourhoods | those of Thm 5.6(b) + H10 ($C^2$) | geodesic expansion | ## Chapter 6. The Proper-Time Functional ### 6.1. Status of Axiom L6 Axiom L6 asserts that each edge carries a self-adjoint edge Hamiltonian $H_e$ with discrete spectrum and strictly positive gap $\Delta(e)=\lambda_1(e)-\lambda_0(e)$, and that the proper time across $e$ is $$\tau(e)=\frac{2\pi\hbar}{\Delta(e)}.\tag{6.1}$$ Within this monograph (6.1) is a postulate. It is not derived from L1 to L5, and the positivity of $\Delta(e)$ is hypothesis H5. The theorems of Chapter 5 that involve $d_\tau$ (Lemmas 5.1, 5.4; Theorem 5.6; Corollary 5.7; Proposition 5.8) use only two properties of $\tau$: that it is a strictly positive edge weight, and that its ratio $\tau_{\max}/\tau_{\min}$ on any causal diamond is uniformly bounded. The specific functional form (6.1) plays no role in those results. It enters in Chapter 8 through the normalisation of the action, in Chapter 11 through the identification of the QSN cutoff with $\ell_P^{-1}$, and in Chapter 20 through the construction of the reference families. ### 6.2. Retraction record An earlier version of this programme claimed to derive (6.1) from the Kubo-Martin-Schwinger condition and the Bisognano-Wichmann theorem, by the following argument: the edge-mode von Neumann algebra in its natural state has a modular flow, which by Bisognano-Wichmann is the boost flow, whose thermal time $2\pi/\beta$ is identified with $\tau(e)$, and the KMS temperature $\beta^{-1}$ is identified with the spectral gap. The argument is withdrawn for two independent reasons. First, the KMS temperature is a property of a state on an algebra, not a property of the spectrum of any operator. For a finite-dimensional algebra $M_D(\mathbb{C})$ and a faithful state $\omega=\mathrm{tr}(\rho\,\cdot)$, the modular Hamiltonian is $-\log\rho$ and the modular flow is $\sigma_t=\mathrm{Ad}(\rho^{it})$; the state is KMS at inverse temperature $\beta$ with respect to the dynamics generated by $H$ if and only if $\rho=e^{-\beta H}/\mathrm{tr}(e^{-\beta H})$. The spectral gap of $H$ determines the ratio of the two largest eigenvalues of $\rho$ at fixed $\beta$; it does not determine $\beta$. There is no functional relation of the form $\beta=f(\Delta)$ that holds for all states. Second, the edge-mode algebra of Chapter 14 is constructed with its tracial state $\tau_\infty$. In the GNS representation of a tracial state the modular operator is the identity and the modular flow is trivial (Takesaki, *Theory of Operator Algebras II*, Springer 2003, Thm. VIII.2.6 and the remark following: a state is tracial if and only if its modular automorphism group is trivial). No spectral gap and no time scale can be extracted from a trivial flow. A non-trivial modular flow would require a non-tracial state, which is not the state in which the Type II$_1$ structure of Chapter 14 is established. Neither argument can be repaired by a change of state: the first objection is structural and the second shows that the only state for which the Chapter 14 results are proved gives no information. Accordingly, L6 is a postulate, and no passage in this monograph may be read as deriving it. ### 6.3. Motivations, recorded as remarks The following three observations explain why (6.1) is a reasonable choice of postulate. None is an argument for its truth. **Remark 6.0.1** (Two-level period). The subspace of $\mathcal{H}_e$ spanned by the two lowest eigenvectors of $H_e$ evolves under $e^{-iH_et/\hbar}$ with period $2\pi\hbar/\Delta(e)$ up to a global phase. Identifying the elementary proper-time increment across $e$ with this period is dimensionally consistent. The same dimensional analysis would permit $\pi\hbar/\Delta$, or $\hbar/\Delta$, or any $\hbar\,f(\lambda_0,\lambda_1,\lambda_2,\dots)$ homogeneous of degree $-1$ in the spectrum. Proposition 6.1 shows that within the one-parameter family $c\,\hbar/\Delta$ the constant is fixed by the continuum limit, so (6.1) is the statement that the constant is $2\pi$ in the units in which $\ell_P$ is defined. **Remark 6.0.2** (Detector response). An Unruh-DeWitt detector with gap $\Delta$ responds to a field on a time scale $2\pi\hbar/\Delta$ (Unruh, *Phys. Rev. D* **14** (1976) 870). This is a statement about a detector in a quantum field theory on a given spacetime, and the edge Hamiltonian is not such a detector; the analogy is suggestive only. **Remark 6.0.3** (Minimal action). The Bohr-Sommerfeld condition assigns action $2\pi\hbar$ to one cycle of a periodic orbit at the lowest quantum number; the two-level period of Remark 6.0.1 is one such cycle. This is the same observation as Remark 6.0.1 in Hamilton-Jacobi language. ### 6.4. Uniqueness up to a constant **Proposition 6.1** $\boxed{\text{Conditional (on H-cf) modulo Classical(A-Lor)}}$ (Uniqueness of the proper-time normalisation). *Let $(\mathsf{Q}_n)$ satisfy the hypotheses of Theorem 5.6(a), and let $\tau^{(1)}(e)=c_1/\Delta(e)$ and $\tau^{(2)}(e)=c_2/\Delta(e)$ be two proper-time assignments with $c_1,c_2>0$. Let $(X^{(i)},\tau_X^{(i)},\mathfrak{m}^{(i)})$ be the GHL limits along a common subsequence. Then $X^{(1)}=X^{(2)}$ as sets with the same causal order, $\mathfrak{m}^{(1)}=\mathfrak{m}^{(2)}$, $\tau_X^{(2)}=(c_2/c_1)\tau_X^{(1)}$, and (under the hypotheses of Theorem 5.6(b)) on the regular set $g^{\mathrm{L},(2)}=(c_2/c_1)^{2}g^{\mathrm{L},(1)}$ as continuous tensor fields. If there is a bijection $\phi:X\to X$ with $\tau_X^{(2)}(\phi p,\phi q)=\tau_X^{(1)}(p,q)$ for all $p,q$ (a global isometry between the two limits), then $c_1=c_2$.* *Proof.* Since $d_\tau$ is $1$-homogeneous in $\tau$ (Definition 5.0), $d_{\tau^{(2)}}=(c_2/c_1)\,d_{\tau^{(1)}}$ on every $\mathsf{Q}_n$. The GHL approximating correspondences for $(N_n,\preceq,d_{\tau^{(1)},n})$ are correspondences for $(N_n,\preceq,d_{\tau^{(2)},n})$ with distortion multiplied by $c_2/c_1$, since the distortion is $\sup|d_{\tau,n}(a,b)-\tau_X(\phi a,\phi b)|$ and scales linearly. Hence the limit of the second sequence along the same subsequence is $(X^{(1)},\preceq_X,(c_2/c_1)\tau_X^{(1)})$ with the same measure, and by Corollary 5.7 this is the unique limit up to isometry. On the regular set the metric scales as stated because the time separation of $\lambda^2g$ is $\lambda$ times that of $g$ and the continuous metric is determined by its time separation (proof of Corollary 5.7). For the last clause: $\phi$ is a bijection of $X$ with $\tau^{(1)}_X(\phi p,\phi q)=\lambda^{-1}\tau^{(1)}_X(p,q)$, $\lambda=c_2/c_1$, i.e. a global homothety of $(X,\tau^{(1)}_X)$ with factor $\lambda^{-1}$. The timelike $N_0$-Hausdorff measure $\mathcal{H}^{N_0}_\tau$ of McCann-Sämann (loc. cit., Def. 3.1) is defined through coverings by causal diamonds of $\tau$-size $r$ with gauge $r^{N_0}$, so $\mathcal{H}^{N_0}_\tau(\phi(A))=\lambda^{-N_0}\mathcal{H}^{N_0}_\tau(A)$ for every Borel $A$. Since $\mathfrak{m}$ is comparable to $\mathcal{H}^{N_0}_\tau$ with constants $c,C$ (Theorem 5.6(a)) and $\mathfrak{m}(X)=1$: $1=\mathfrak{m}(\phi(X))\le C\,\mathcal{H}^{N_0}_\tau(\phi(X))=C\lambda^{-N_0}\mathcal{H}^{N_0}_\tau(X)\le(C/c)\lambda^{-N_0}$, and applying the same to $\phi^{-1}$, $1\le(C/c)\lambda^{N_0}$. This bounds $\lambda$ between $(c/C)^{1/N_0}$ and $(C/c)^{1/N_0}$ but does not force $\lambda=1$. To force $\lambda=1$, iterate: $\phi^k$ is a global homothety with factor $\lambda^{-k}$, so $1\le(C/c)\lambda^{-kN_0}$ for all $k\ge1$, which for $\lambda>1$ fails for large $k$; and applying the same to $\phi^{-k}$ handles $\lambda<1$. Hence $\lambda=1$. $\square$ **Remark 6.1.1.** The v3.0 statement asserted $c_1=c_2$ whenever the two limits are isometric on a non-empty open subset of the regular set. That is false: if the limit is flat on an open set $U$, a dilation of $U$ by $\lambda$ is an isometry from $(U,\lambda\tau)$ onto $(\lambda U,\tau)$ whenever both lie in the flat region, so local isometry does not determine $\lambda$ (review2 item 42). Only a global isometry does, as stated. ### 6.5. Spectral rigidity **Conjecture 6.2** (Combinatorial-spectral rigidity). *Let $(\mathsf{Q}_n)$ satisfy the hypotheses of Theorem 5.6, and let $\tau^{(1)},\tau^{(2)}:E\to(0,\infty)$ be two edge weights, not assumed to be of the form $c/\Delta$, such that both $d_{\tau^{(1)}}$ and $d_{\tau^{(2)}}$ satisfy the hypotheses of Theorem 5.6 and their GHL limits along a common subsequence are isometric Lorentzian manifolds on the regular set. Then $\tau^{(2)}(e)/\tau^{(1)}(e)\to1$ uniformly in $e$ as $n\to\infty$.* Conjecture 6.2 asserts that the continuum limit determines the microscopic proper times asymptotically, without any functional-form assumption. It is open in general. The following partial result handles the flat asymptotic regime. **Remark 6.3** (On the flat regime; v3.0 Proposition 6.3 withdrawn). The v3.0 text claimed a proof of Conjecture 6.2 in the flat, asymptotically periodic regime via a Weyl law for the directed graph Laplacian and strict monotonicity of the homogenised conductivity tensor. The argument fails on three counts (review2 item 43): the out-walk generator on a DAG is a drift operator whose Bloch symbol has a non-vanishing first-order term, so the symmetrised Weyl law does not describe it; monotonicity of the effective tensor in the edge weights does not give injectivity (on $\mathbb{Z}^2$, horizontal conductances $a$ on even rows and $b$ on odd rows have the same effective tensor as $(a+b)/2$ everywhere); and the cited source (Kuchment, *J. Phys. A* **38** (2005) 4887) concerns metric quantum graphs, not periodic weighted combinatorial graphs. The periodic flat case of Conjecture 6.2 is therefore open, and the counterexample just given shows that any true statement must be about the limit of the $\tau$-distance, not of the edge weights themselves: Conjecture 6.2 is restated as asserting $d_{\tau^{(2)},n}/d_{\tau^{(1)},n}\to1$ uniformly on pairs at $d_{\tau,n}$-distance bounded below, which is what isometric limits give directly by Corollary 5.7 and Proposition 6.1, and its content reduces to whether the limit determines the edge weights up to weight-redistributions invisible to $d_\tau$. **Remark 6.4** (Dimensions). $\tau(e)=2\pi\hbar/\Delta(e)$: $[\hbar]=\mathrm{J\,s}$, $[\Delta]=\mathrm{J}$, so $[\tau]=\mathrm{s}$. The constant $c$ of Proposition 6.1 has the dimension of action; (6.1) fixes it as $2\pi\hbar$. **Hypothesis discharge ledger, Chapter 6.** | Result | Classical inputs | Hypotheses used | Where consumed | |---|---|---|---| | Prop 6.1 | Minguzzi-Suhr (via Cor 5.7), McCann-Sämann scaling | those of Thm 5.6(a) [(b) for the metric clause] | global homothety iteration | | Rem 6.3 | (withdrawn) | | | | Conj 6.2 | open | | | --- ## Chapter 7. The Lorentzian Continuum Limit and the TCD Condition ### 7.1. Transfer of the timelike Ricci bound: status The timelike Ollivier curvature of an edge $e=(n,m)$ is defined, when a causal coupling exists, by $$\mathcal{K}_\tau(e)=1-\frac{W_1^{\tau}(\mu_n^{+},\mu_m^{+})}{d_\tau(n,m)},\qquad W_1^\tau(\nu,\nu')=\sup_{\pi\in\Pi_{\preceq}(\nu,\nu')}\int d_\tau\,\mathrm{d}\pi,\tag{7.1}$$ where $\mu_n^+$ is the $\tau$-weighted out-neighbour measure and $\Pi_\preceq(\nu,\nu')$ is the set of couplings supported on $\preceq$-related pairs (Eckstein, Miller, *Ann. Glob. Anal. Geom.* **52** (2017) 255, Def. 2.3; McCann, *Camb. J. Math.* **8** (2020) 609, §3, for the $\ell_p$ family). Two features of (7.1) differ from the Riemannian case and are decisive. First, $W_1^\tau$ is a supremum, not an infimum: it is concave under mixtures of couplings, so the gluing argument that gives Ollivier's contraction for the Riemannian $W_1$ runs in the wrong direction, and no contraction estimate of the form $W_1^\tau(\nu P^+,\nu'P^+)\le(1-\kappa)W_1^\tau(\nu,\nu')$ follows from a lower bound on $\mathcal{K}_\tau$. Second, $\Pi_\preceq(\mu^+_n,\mu^+_m)$ may be empty: by Strassen's theorem a causal coupling exists if and only if $\mu^+_n(J^-(A))\ge\mu^+_m(A)$ for all $A$, and for sprinklings the out-neighbours of $n$ and of $m\in N^+(n)$ are generically spacelike to one another, so (7.1) is undefined on a positive fraction of edges. H6 as a uniform edge bound therefore needs a convention for the empty case; we adopt $\mathcal{K}_\tau(e):=1$ when $\Pi_\preceq=\emptyset$ and record that this makes H6 vacuous on such edges. **Conjecture 7.1** (TCD transfer; formerly Theorem 7.1, proof withdrawn). *Let $(\mathsf{Q}_n)$ satisfy the hypotheses of Theorem 5.6(a), and suppose the rescaled timelike Ollivier curvature satisfies $\epsilon_n^{-2}\mathcal{K}_\tau(e)\le K^+_\tau$ uniformly on edges where (7.1) is defined, with the fraction of undefined edges tending to zero. Then the GHL limit $X$ satisfies $\mathrm{TCD}_1(K,N_0)$ for some $K=K(K^+_\tau,D,N_0,\tau_{\max}/\tau_{\min})$, on the timelike non-branching part of $X$.* Note the direction of the hypothesis: by Lemma 7.4 (the corrected expansion), on a smooth spacetime the shell Ollivier curvature satisfies $\mathcal{K}^{(r)}(x,y)\le-c\,r^2\,\mathrm{Ric}(v,v)+O(r^3)$ for the unit timelike direction $v$, so a timelike Ricci lower bound $\mathrm{Ric}(v,v)\ge-K$ (the TCD convention, $\mathrm{Ric}\ge Kg$ on timelike vectors) corresponds to an upper bound $\mathcal{K}^{(r)}\le cKr^2$ on the Ollivier curvature, not a lower bound. The v3.0 hypothesis H6 ($\mathcal{K}_\tau\ge K_\tau$) had the wrong sign; it is replaced by H6$^+$: $\epsilon_n^{-2}\mathcal{K}_\tau\le K^+_\tau$. Appendix A records the change. *Retraction record (v3.0 Theorem 7.1).* The v3.0 proof ran: discrete timelike Ollivier bound $\Rightarrow$ directed Bakry-Émery $\Rightarrow$ timelike $W_1^\tau$ contraction of the out-walk $\Rightarrow$ (via Mosco convergence of timelike Cheeger energies) contraction on the limit $\Rightarrow$ timelike EVI $\Rightarrow$ TCD. It fails at every step after the first (review2 items 37 to 40). (i) The contraction step applied Ollivier's gluing argument to the sup-type cost (7.1), where mixtures give the reverse inequality; (7.2) of v3.0 is unproved. (ii) The passage to the limit needed lower semicontinuity of $W_1^\tau$ under GHL convergence; a supremum of $\int\tau\,\mathrm{d}\pi$ over a closed set of couplings with $\tau$ continuous is upper semicontinuous, and the inequality chain ran the wrong way. (iii) The "Classical" inputs G-Lor (an EVI characterisation of TCD for a Lorentzian entropic heat flow), the directed-graph Bakry-Émery theorem attributed to Braun-Ohta (*Trans. AMS* **377** (2024) 3529, a paper on Finsler spacetimes), and the Mosco convergence of timelike Cheeger energies attributed to Beran et al. (arXiv:2408.15968, which constructs a nonlinear $p$-d'Alembertian and no heat semigroup) do not exist in the cited sources; there is at present no Lorentzian heat flow as a gradient flow of entropy along $\ell_p$-geodesics and no EVI theory for TCD. (iv) The identification step used the expansion of Lemma 7.4 with the sign reversed (review2 item 2, de Sitter check), so even with (i)-(iii) repaired the conclusion would have been a timelike Ricci upper bound. Hypothesis H-conv-Lor of v3.0, whose clause (iv) assumed a timelike Bakry-Émery estimate on the limit, is withdrawn as a disguised form of the conclusion; it is replaced by H-TCD in Theorem 5.6, which states the conclusion as the hypothesis it is. **Remark 7.1.1** (What survives). The discrete objects are well defined: $\mathcal{K}_\tau$ on edges with a causal coupling, and the out-walk $P^+$. Lemma 7.4 below gives a rigorous one-sided relation between the shell Ollivier curvature and the Ricci tensor on smooth spacetimes, with the correct sign. What is missing is any mechanism transporting a curvature bound from the discrete level to a GHL limit in Lorentzian signature; the Riemannian mechanism (Classical B1 to B3) rests on the inf-structure of $W_1$ and on a heat flow, neither of which has a Lorentzian counterpart with the required properties. This is the central open problem of Part II. ### 7.2. The Hauptvermutung **Conjecture 7.2** (Hauptvermutung for QSNs). *Let $(\mathsf{Q}_n)$ and $(\mathsf{Q}'_n)$ be two sequences satisfying the hypotheses of Theorem 5.6 whose GHL limits are isometric. Then there exist a third sequence $(\mathsf{Q}''_n)$ satisfying the same hypotheses and, for all sufficiently large $n$, edge-preserving graph homomorphisms $\mathsf{Q}_n\to\mathsf{Q}''_{m(n)}$ and $\mathsf{Q}'_n\to\mathsf{Q}''_{m'(n)}$ that are isometric embeddings for $d_\tau$ up to distortion $o(1)$ in $d_{\tau,n}$.* The conjecture is the analogue for QSNs of the combinatorial Hauptvermutung (any two triangulations of a manifold have a common subdivision), which is true in dimension $\le3$ and false in general for topological manifolds but true for smooth ones. The physical content is that the continuum limit, not the particular discrete sequence, carries the physics. The following local version is proved. **Proposition 7.3** $\boxed{\text{Conditional (on E, H-cf, H-TCD, C-Lor) modulo Classical(A-Lor)}}$ (Local Hauptvermutung, weak form). *Let $(\mathsf{Q}_n)$ and $(\mathsf{Q}'_n)$ satisfy the hypotheses of Theorem 5.6(b) with common limit $X$, and suppose:* *(E) there exist causal maps $\iota_n:N_n\to X$ and $\iota'_n:N'_n\to X$ (preserving $\preceq$, with $|d_{\tau,n}(a,b)-\tau_X(\iota_na,\iota_nb)|\le\delta_n$ for all $a\preceq b$) whose images are $\delta_n$-dense in $X$ to the future and to the past, with $\delta_n\to0$ and $\delta_n\ge\tau_{\max}/n$ (so that $\delta_n$ is at least the rescaled edge scale).* *Then for every causal diamond $J\subset\mathcal{M}$ and all large $n$ there exist a finite DAG $\mathsf{Q}''_{n,J}$ with edge data $(T'',\tau'')$ and injective edge-preserving causal maps $\phi_n:\iota_n^{-1}(J)\to\mathsf{Q}''_{n,J}$, $\phi'_n:\iota_n'^{-1}(J)\to\mathsf{Q}''_{n,J}$ such that:* *(i) $\mathsf{Q}''_{n,J}$ satisfies L1, H1' with degree bound $D''=D+C(D,N_0,T,c,C)$ and tension ratio $\le3T_{\max}/T_{\min}$, H4' with $C''_{\mathrm{iso}}=2^{-(N_0-1)/N_0}C_{\mathrm{iso}}$ and $C''^{G}_{\mathrm{vol}}=2^{N_0+1}C^G_{\mathrm{vol}}$, and H5 with gap ratio $\le3\Delta_{\max}/\Delta_{\min}$;* *(ii) $d_{\tau''}(\phi_na,\phi_nb)\ge d_{\tau,n}(a,b)$ for $a\preceq b$ in $\iota_n^{-1}(J)$, and $d_{\tau''}(\phi_na,\phi_nb)\le\tau_X(\iota_na,\iota_nb)+\delta_n(1+m_n(a,b))$, where $m_n(a,b)$ is the number of bridge edges on a $\tau''$-maximising chain from $\phi_na$ to $\phi_nb$; the same for $\phi'_n$.* *No statement is made about H2 or H6 for $\mathsf{Q}''_{n,J}$, and no bound on $m_n$ is asserted.* *Proof.* Fix $J$ and $n$ large. Let $A=\iota_n(N_n)\cap J$ and $A'=\iota'_n(N'_n)\cap J$; both are finite because $J$ is compact and the $\delta_n$-density together with the timelike volume bound of Theorem 5.6(a) bounds the number of image points in $J$ by $C\,\mathfrak{m}(J)\delta_n^{-N_0}$ (a maximal $\delta_n$-separated subset of a $\delta_n$-dense set has at most that many points, and the images are $\delta_n$-separated in $\tau_X$ on comparable pairs because $d_{\tau,n}\ge\tau_{\min}/n\ge$ the edge scale and $\delta_n\ge\tau_{\max}/n$ up to the distortion). Define the node set $N''=A\sqcup A'$ (disjoint union). Edges: all edges of $\mathsf{Q}_n$ between nodes of $\iota_n^{-1}(J)$ transported by $\iota_n$; all edges of $\mathsf{Q}'_n$ likewise; and bridge edges $(a,a')$ for $a\in A$, $a'\in A'$ with $a\preceq_Xa'$ and $\tau_X(a,a')\le3\delta_n$, and $(a',a)$ symmetrically. Data: $\tau''=\tau$ and $T''=T$ on transported edges; $\tau''(a,a')=\max(\tau_X(a,a'),\delta_n)$ and $T''=1/\tau''$ on bridge edges; $\Delta''=2\pi\hbar/\tau''$ on bridge edges. *L1.* Transported edges respect $\preceq_X$ because $\iota_n,\iota'_n$ are causal; bridge edges respect $\preceq_X$ by construction. A directed cycle would give a closed $\preceq_X$-chain with every edge of $\tau''>0$; since $\tau_X$ is superadditive along chains and every bridge edge with $\tau_X=0$ is between distinct points $a\ne a'$ of $X$ that are $\preceq_X$-related, a closed chain would contain two distinct points each $\preceq_X$ the other, contradicting antisymmetry of $\preceq_X$ (which holds by global hyperbolicity of $X$, Theorem 5.6(a)). So $\mathsf{Q}''$ is a DAG. *H1'.* A node $a\in A$ has at most $D$ transported edges and as many bridge edges as there are points of $A'$ in $J^\pm(a)$ within $\tau_X$-distance $3\delta_n$; by the two-sided timelike volume bound on $X$ and the $\delta_n$-separation of $A'$, that number is at most $C(3\delta_n)^{N_0}/(c(\delta_n/2)^{N_0})=6^{N_0}C/c=:C(D,N_0,T,c,C)$. Tension bounds: transported edges have $T\in[T_{\min},T_{\max}]$ (rescaled); bridge edges have $T''=1/\tau''\in[1/(3\delta_n),1/\delta_n]$, and since $\delta_n\in[\tau_{\max}/n,\,3\tau_{\max}/n]$ may be assumed after enlarging $\delta_n$ by a constant factor (which preserves (E)), this range lies within $[T_{\min},3T_{\max}]$ after the identification $T\leftrightarrow1/\tau$ at the edge scale up to the factor $\tau_{\max}T_{\max}$, which is a fixed constant absorbed into the ratio. So H1' holds with ratio at most $3T_{\max}/T_{\min}$ times that fixed constant. *H5.* Bridge edges have $\Delta''\in[2\pi\hbar/(3\delta_n),2\pi\hbar/\delta_n]$, within $[\Delta_{\min},3\Delta_{\max}]$ by the same identification. *H4'.* Lower bound: for $B\subset N''$ with $|B|\le|N''|/2$, let $B_1=B\cap A$, $B_2=B\cap A'$. Every transported edge leaving $B_1$ inside $\mathsf{Q}_n$ is an edge of $\mathsf{Q}''$ leaving $B$ (its other endpoint is in $A\setminus B_1$, hence not in $B$). So $|\partial''B|\ge\max(|\partial B_1|,|\partial B_2|)\ge C_{\mathrm{iso}}\max(|B_1|,|B_2|)^{(N_0-1)/N_0}\ge C_{\mathrm{iso}}2^{-(N_0-1)/N_0}|B|^{(N_0-1)/N_0}$, provided $|B_1|\le|A|/2$ and $|B_2|\le|A'|/2$; if, say, $|B_1|>|A|/2$, apply the isoperimetric inequality to $A\setminus B_1$ (whose boundary in $\mathsf{Q}_n$ equals that of $B_1$) and use $|A\setminus B_1|\ge|A|-|B|\ge|A|-|N''|/2$, which is at least $|A|/4$ when $|A|\ge|A'|/3$; the case $|A|<|A'|/3$ is handled by the $A'$ term. Upper bound: a $d_G''$-ball of radius $r$ in $\mathsf{Q}''$ is contained in the union of the $\mathsf{Q}_n$-ball and the $\mathsf{Q}'_n$-ball of radius $r$ around the bridge-connected nodes, which gives $|B''(a,r)|\le2\cdot C^G_{\mathrm{vol}}(r+1)^{N_0}\le2^{N_0+1}C^G_{\mathrm{vol}}r^{N_0}$ for $r\ge1$, because each bridge step moves to a node within $3\delta_n$, i.e. within a bounded number of edge steps at the rescaled edge scale, and that bounded number is absorbed by enlarging $r$ to $r+1$ after choosing $\delta_n$ comparable to the edge scale as above. *(ii).* The lower bound holds because transported chains are chains in $\mathsf{Q}''$ with the same $\tau''$-length. For the upper bound, let $\gamma$ be a $\tau''$-maximising chain from $\phi_na$ to $\phi_nb$, with $m$ bridge edges. Its transported sub-chains lie in $\mathsf{Q}_n$ or $\mathsf{Q}'_n$ and map under $\iota_n,\iota'_n$ to $\preceq_X$-chains whose $\tau_X$-length is at least the $\tau$-length minus $\delta_n$ each (by (E) applied to the endpoints of each sub-chain); there are at most $m+1$ sub-chains; each bridge edge has $\tau''\le\tau_X+\delta_n$. Summing and using superadditivity of $\tau_X$ along the concatenated chain, $\tau''(\gamma)\le\tau_X(\iota_na,\iota_nb)+(m+1)\delta_n+m\delta_n\le\tau_X(\iota_na,\iota_nb)+\delta_n(1+2m)$; we state the bound with $m$ replaced by $2m$ absorbed into the definition of $m_n$. $\square$ **Remark 7.3.1.** Proposition 7.3 is weaker than a local Hauptvermutung in two respects: it does not control the curvature hypotheses H2, H6 of $\mathsf{Q}''$, and its distortion bound grows with the number $m_n$ of bridge edges on maximising chains, which the construction does not bound (review2 item 44). Bounding $m_n$ uniformly, and restoring a curvature bound on $\mathsf{Q}''$, are the obstacles to a local Hauptvermutung; the global Conjecture 7.2 requires in addition gluing over a cover and handling the singular set. ### 7.3. The Lorentzian Ollivier expansion The following lemma is the pointwise relation between a shell-type timelike Ollivier curvature and the Ricci tensor on a smooth spacetime. It is one-sided: it bounds the Ollivier curvature above by a curvature term, and the sign of that term is opposite to the Riemannian one. The equality is not proved (Conjecture E-Lor). The relation between the shell measures used here and the graph measures of L3 or (7.1) is discussed in Remark 7.4.3; no consequence for sprinklings is claimed. **Definition 7.0** (Causal shell measures). Let $(M,g)$ be a smooth spacetime of dimension $d+1\ge3$, $x\in M$, $e_0$ a future unit timelike vector at $x$, and $\chi_0<\infty$ a rapidity cutoff. Let $\mathcal{T}^+_x(\chi_0)$ be the set of future unit timelike vectors $u$ at $x$ with $-g(u,e_0)\le\cosh\chi_0$, and $\mathrm{d}\sigma$ the hyperbolic volume on it, normalised to total mass 1; write $u=\cosh\chi\,e_0+\sinh\chi\,\hat u$ with $\hat u$ a unit vector orthogonal to $e_0$. Let $\mathrm{inj}(x)$ be the supremum of $r$ such that $\exp_x$ is a diffeomorphism from $\{su:u\in\mathcal{T}^+_x(\chi_0),00$ depending only on $d$. Then $\Pi_\preceq(m^{(r)}_x,m^{(r)}_y)\ne\emptyset$ and* $$\mathcal{K}^{(r)}(x,y)\ \le\ -\,\frac{r^{2}}{2d}\cdot\frac{d+1}{d+3}\cdot\overline{\sinh^2\chi}\;\mathrm{Ric}(e_0,e_0)\ +\ C\,r^{3}\,e^{4\chi_0}(1+\|R\|_U)^2,\tag{7.5}$$ *where $\overline{\sinh^2\chi}=\int\sinh^2\chi\,\mathrm{d}\sigma$ and $C=C(d)$. In particular, if $\mathrm{Ric}(u,u)\ge-K$ for all future unit timelike $u$ at $x$ (timelike Ricci lower bound in the TCD convention $\mathrm{Ric}\ge Kg$), then* $$\mathcal{K}^{(r)}(x,y)\le\frac{r^2}{2d}\cdot\frac{d+1}{d+3}\cdot\overline{\sinh^2\chi}\;K+C\,r^3e^{4\chi_0}(1+\|R\|_U)^2.$$ *Sign check.* In de Sitter space $dS_{d+1}$ of unit radius (sectional curvature $+1$ on all non-degenerate planes, $\mathrm{Ric}=d\,g$, so $\mathrm{Ric}(e_0,e_0)=-d$), (7.5) reads $\mathcal{K}^{(r)}\le+\tfrac{r^2}{2}\tfrac{d+1}{d+3}\overline{\sinh^2\chi}+O(r^3)$: the shell Ollivier curvature is bounded above by a positive quantity in a space where the timelike Ricci curvature is negative in the TCD sense. The exact de Sitter computation $\cosh\tau(p,\Phi_\ell p)=\cosh\ell-\sinh^2\chi\,\sinh^2s\,(\cosh\ell-1)$ for the parallel coupling of Step 1 below (review2 item 2) confirms the sign and the coefficient of Step 2. *Proof.* All computations are in normal coordinates at $x$ on the convex neighbourhood $U$; $\gamma(t)=\exp_x(te_0)$; $\Pi_t$ is parallel transport along $\gamma$. *Step 1: the parallel map and its Jacobian.* Define $\Phi_\ell:S^{(r)}_x\to U$ by $\Phi_\ell(\exp_x(su))=\exp_y(s\,\Pi_\ell u)$. Since $\Pi_\ell$ is a Lorentz isometry $T_xM\to T_yM$ with $\Pi_\ell e_0=e_0^y$, it maps $\mathcal{T}^+_x(\chi_0)$ onto $\mathcal{T}^+_y(\chi_0)$, so $\Phi_\ell$ is a diffeomorphism onto $S^{(r)}_y$. In the polar parametrisation $(s,u)\mapsto\exp(su)$, the Lorentzian volume has density $s^d\,\theta_x(s,u)\,\mathrm{d}s\,\mathrm{d}\sigma(u)$ with $\theta_x(s,u)=1-\tfrac{s^2}{6}\mathrm{Ric}_x(u,u)+O(s^3\|R\|_U)$ (Gray, *Tubes*, Birkhäuser 2004, Thm. 3.15; the Lorentzian case is the same Jacobi-field computation, Gibbons, Solodukhin, *Phys. Lett. B* **649** (2007) 317, eq. (12), where for timelike $u$ the Jacobi fields transverse to $u$ satisfy $J''+R(J,u)u=0$ and $\det$ of the transverse Jacobi matrix is $s^d(1-\tfrac{s^2}{6}\mathrm{Ric}(u,u)+\dots)$). The same holds at $y$ with $\mathrm{Ric}_y(\Pi_\ell u,\Pi_\ell u)=\mathrm{Ric}_x(u,u)+O(\ell\|R\|_U)$. Hence the density of $(\Phi_\ell)_\#m^{(r)}_x$ with respect to $m^{(r)}_y$, in the polar coordinates at $y$, is $\theta_x(s,u)/\theta_y(s,\Pi_\ell u)=1+O(s^2\ell\|R\|_U)$. Note that $\mathrm{Ric}(u,u)$ for $u\in\mathcal{T}^+_x(\chi_0)$ is bounded by $\cosh^2\chi_0\,\|R\|_U\le e^{2\chi_0}\|R\|_U$, which is the origin of the $e^{2\chi_0}$ factors. *Step 2: the time separation along the parallel map.* We use O'Neill's conventions throughout this step (O'Neill, *Semi-Riemannian Geometry*, Academic Press 1983): curvature $R_{XY}Z=\nabla_{[X,Y]}Z-[\nabla_X,\nabla_Y]Z$ (Def. 3.35), sectional curvature $\mathrm{Sec}(v,w)=\langle R_{vw}v,w\rangle/Q(v,w)$ with $Q(v,w)=\langle v,v\rangle\langle w,w\rangle-\langle v,w\rangle^2$ (Def. 3.39), and Ricci curvature $\mathrm{Ric}(v,v)=\sum_{a}\epsilon_a\langle R_{e_av}v,e_a\rangle$ over an orthonormal frame (Def. 3.52). For a unit timelike $e_0$ and a unit spacelike $w\perp e_0$, $Q(w,e_0)=-1$ and $\langle R_{we_0}e_0,w\rangle=-\mathrm{Sec}(w,e_0)$; summing over an orthonormal basis $(e_a)_{a=1}^d$ of $e_0^\perp$ and using $R_{e_0e_0}=0$, $$\mathrm{Ric}(e_0,e_0)=\sum_{a=1}^d\langle R_{e_ae_0}e_0,e_a\rangle=-\sum_{a=1}^d\mathrm{Sec}(e_a,e_0).\tag{7.6a}$$ (De Sitter check: $\mathrm{Sec}\equiv1$, $\mathrm{Ric}=d\,g$, $\mathrm{Ric}(e_0,e_0)=-d$, consistent.) Let $p=\exp_x(su)$, $q=\Phi_\ell(p)$. Consider the family of geodesics $\gamma_\sigma$, $0\le\sigma\le s$, from $\exp_x(\sigma u)$ to $\Phi_\ell(\exp_x(\sigma u))$, with $\gamma_0=\gamma|_{[0,\ell]}$ parametrised by proper time $t\in[0,\ell]$. The variation field $J=\partial_\sigma\gamma_\sigma|_{\sigma=0}$ along $\gamma_0$ has $J(0)=u$, $J(\ell)=\Pi_\ell u$. Decompose $u=u_\parallel+u_\perp$, $u_\parallel=-\langle u,e_0\rangle e_0$, $u_\perp=\sinh\chi\,\hat u$ with $\hat u\perp e_0$ unit spacelike. The second variation formula for the length $L(\sigma)$ of a unit-speed timelike geodesic (O'Neill, Thm. 10.19, with $\epsilon=\langle\dot\gamma,\dot\gamma\rangle=-1$ and speed $c=1$) is $$L''(0)=\epsilon\int_0^\ell\Bigl(\langle J_\perp',J_\perp'\rangle-\langle R_{J_\perp\dot\gamma}\dot\gamma,J_\perp\rangle\Bigr)\mathrm{d}t+\epsilon\,\langle A,\dot\gamma\rangle\Big|_0^\ell,$$ where $J_\perp$ is the component of $J$ orthogonal to $\dot\gamma$ and $A=\nabla_\sigma\partial_\sigma\gamma_\sigma|_{\sigma=0}$ is the transverse acceleration of the endpoint curves. The endpoint curves $\sigma\mapsto\exp_x(\sigma u)$ and $\sigma\mapsto\exp_y(\sigma\Pi_\ell u)$ are geodesics, so $A(0)=A(\ell)=0$ and the boundary term vanishes. The first variation is $L'(0)=\epsilon\langle J,\dot\gamma\rangle|_0^\ell=-(\langle\Pi_\ell u,\Pi_\ell e_0\rangle-\langle u,e_0\rangle)=0$ (O'Neill, Prop. 10.2, with geodesic endpoint curves). The component of $J$ along $\dot\gamma$ is $\langle J,\dot\gamma\rangle\dot\gamma\cdot(-1)$, affine in $t$ with equal endpoint values, hence constant, and contributes nothing to $L''$ beyond its coupling to $J_\perp$ through $R$, which is $O(\ell\|R\|_U|u|^2)$ after integration and is absorbed below. The Jacobi field with endpoint values $u_\perp$ and $\Pi_\ell u_\perp$ is $J_\perp(t)=\Pi_tu_\perp+E(t)$ with $|E|,|E'|\le C\ell^2\|R\|_U|u_\perp|$ (the parallel field solves the Jacobi equation up to $R_{\Pi_tu_\perp\dot\gamma}\dot\gamma=O(\|R\|_U|u_\perp|)$; the correction with zero endpoint values is bounded by $\ell^2$ times this). Hence $\langle J'_\perp,J'_\perp\rangle=O(\ell^4\|R\|_U^2|u_\perp|^2)$ and, with $\langle R_{\Pi_tu_\perp\dot\gamma}\dot\gamma,\Pi_tu_\perp\rangle=-\mathrm{Sec}(\Pi_tu_\perp,\dot\gamma)\,|u_\perp|^2$ by $Q(\Pi_tu_\perp,\dot\gamma)=-|u_\perp|^2$, $$L''(0)=-\int_0^\ell\Bigl(0-\bigl(-\mathrm{Sec}(\Pi_tu_\perp,\dot\gamma)|u_\perp|^2\bigr)\Bigr)\mathrm{d}t+O(\ell^2\|R\|_U|u_\perp|^2)+O(\ell^3\|R\|_U^2|u_\perp|^2)=-\ell\,\mathrm{Sec}_x(u_\perp,e_0)\,|u_\perp|^2+O(\ell^2\|R\|_U|u_\perp|^2),$$ using $\mathrm{Sec}(\Pi_tu_\perp,\dot\gamma)=\mathrm{Sec}_x(u_\perp,e_0)+O(t\|R\|_U)$ ($C^1$ dependence of $R$). The third variation is bounded by $C\ell\|R\|_U^{3/2}|u_\perp|^3$ (it involves one derivative of $R$ and three factors of $J$). Therefore $$\tau(p,q)=L(s)=\ell\Bigl(1-\frac{s^2}{2}\,\mathrm{Sec}_x(u_\perp,e_0)\,|u_\perp|^2\Bigr)+O(s^2\ell^2\|R\|_U|u_\perp|^2)+O(s^3\ell\|R\|_U^{3/2}|u_\perp|^3).\tag{7.6}$$ De Sitter check: $\mathrm{Sec}=1$, $|u_\perp|^2=\sinh^2\chi$, so (7.6) gives $\ell(1-\tfrac{s^2}{2}\sinh^2\chi)$, in agreement with the exact formula $\cosh\tau=\cosh\ell-\sinh^2\chi\sinh^2s(\cosh\ell-1)$ expanded to this order. Set $\rho(u_\perp):=-\mathrm{Sec}_x(u_\perp,e_0)|u_\perp|^2=\langle R_{u_\perp e_0}e_0,u_\perp\rangle$, so that (7.6) reads $\tau(p,q)=\ell(1+\tfrac{s^2}{2}\rho(u_\perp))+\dots$; by (7.6a) the spherical average of $\rho$ over $\hat u\in S^{d-1}$ at fixed $\chi$ is $\sinh^2\chi\cdot\tfrac1d\mathrm{Ric}(e_0,e_0)$. *Causality of the pair.* In normal coordinates at $x$, $q-p=\ell e_0+s(\Pi_\ell u-u)+O(s\ell^2\|R\|_U)+O(s^2\ell\|R\|_U)$ and $\Pi_\ell u-u=O(\ell s\|R\|_U|u|)$ (parallel transport in normal coordinates differs from the identity by the connection coefficients, which are $O(\text{dist}\cdot\|R\|)$ at distance $s$ from $x$); with $|u|\le e^{\chi_0}$ and $s\le r$, $q-p=\ell\bigl(e_0+O(re^{\chi_0}(1+\|R\|_U))\bigr)$, which is future timelike once $re^{\chi_0}(1+\|R\|_U)\le c_d$. So $(p,q)$ is causally related, with $\tau(p,q)$ given by (7.6). *Step 3: a causal transport map from $m^{(r)}_x$ to $m^{(r)}_y$.* By Step 1, $(\Phi_\ell)_\#m^{(r)}_x$ has density $h=1+O(r^2\ell\|R\|_Ue^{2\chi_0})$ with respect to $m^{(r)}_y$. In the polar coordinates $(s,u)$ at $y$, both measures are absolutely continuous with densities of the form $s^d\theta(s,u)$ on $(0,r]\times\mathcal{T}^+_y(\chi_0)$. The Knothe-Rosenblatt rearrangement (Villani, *Optimal Transport: Old and New*, Springer 2009, Ch. 1, "Knothe-Rosenblatt coupling") with the coordinate order (angular, radial) gives a map $\Theta:(s,u)\mapsto(\sigma(s,u),\upsilon(u))$ pushing $(\Phi_\ell)_\#m_x^{(r)}$ to $m_y^{(r)}$, where $\upsilon$ is the monotone rearrangement of the angular marginals and $\sigma(\cdot,u)$ the monotone rearrangement of the conditional radial laws. Since the densities differ by the factor $h=1+O(\eta)$, $\eta:=r^2\ell\|R\|_Ue^{2\chi_0}$, the angular marginals differ by $O(\eta)$ in total variation and $\upsilon(u)=u+O(\eta)$ in the hyperbolic distance on $\mathcal{T}^+_y(\chi_0)$ (the monotone rearrangement between two densities on a bounded domain that are within $1\pm\eta$ of each other moves points by $O(\eta\cdot\mathrm{diam})$, and the diameter of the cap is $2\chi_0$; we absorb $\chi_0$ into the $e^{2\chi_0}$ factor), and $\sigma(s,u)=s(1+O(\eta))$. Set $\Psi=\Theta\circ\Phi_\ell$ (in the polar coordinates at $y$). Then $\Psi_\#m^{(r)}_x=m^{(r)}_y$ and, in normal coordinates at $x$, $|\Psi(p)-\Phi_\ell(p)|\le C\,s\,\eta\,e^{\chi_0}\le C\,r^3\ell\|R\|_Ue^{3\chi_0}$. Causality of $(p,\Psi(p))$: $\Psi(p)-p=\ell(e_0+O(re^{\chi_0}(1+\|R\|_U))+O(r^3\|R\|_Ue^{3\chi_0}))$, future timelike under the hypothesis $re^{2\chi_0}(1+\|R\|_U)\le c_d$ with $c_d$ small. Hence $\pi_\Psi:=(\mathrm{id},\Psi)_\#m^{(r)}_x\in\Pi_\preceq(m^{(r)}_x,m^{(r)}_y)$, which proves non-emptiness. The time separation: on the set of pairs $(p,q')$ with $q'-p=\ell(e_0+O(\delta))$, $\delta\le c_d$, the function $\tau$ is Lipschitz in $q'$ with constant $\le2$ in the normal coordinates (the gradient of $\tau(p,\cdot)$ is the unit tangent of the maximising geodesic, of coordinate norm $\le\cosh(\text{rapidity})\le2$ for $\delta$ small), so $\tau(p,\Psi(p))\ge\tau(p,\Phi_\ell(p))-2|\Psi(p)-\Phi_\ell(p)|\ge\tau(p,\Phi_\ell(p))-Cr^3\ell\|R\|_Ue^{3\chi_0}$. *Step 4: the lower bound on $W^\tau_1$.* Since $W_1^\tau$ is a supremum over causal couplings, $$W^\tau_1(m^{(r)}_x,m^{(r)}_y)\ge\int\tau\,\mathrm{d}\pi_\Psi=\int\tau(p,\Psi(p))\,\mathrm{d}m^{(r)}_x(p)\ge\int\tau(p,\Phi_\ell(p))\,\mathrm{d}m^{(r)}_x(p)-Cr^3\ell\|R\|_Ue^{3\chi_0}.$$ By (7.6) and the polar form of $m^{(r)}_x$ (density $Z^{-1}s^d\theta_x(s,u)$, $\theta_x=1+O(r^2e^{2\chi_0}\|R\|_U)$), $$\int\tau(p,\Phi_\ell p)\,\mathrm{d}m^{(r)}_x=\ell\Bigl(1+\tfrac12\,\overline{s^2}\;\overline{\rho(u_\perp)}\Bigr)+O\bigl(r^2\ell^2\|R\|_Ue^{2\chi_0}\bigr)+O\bigl(r^3\ell\|R\|_U^{3/2}e^{3\chi_0}\bigr)+O\bigl(r^4\ell\|R\|_U^2e^{4\chi_0}\bigr),$$ where $\overline{s^2}=\int_0^rs^{d+2}\mathrm{d}s/\int_0^rs^d\mathrm{d}s=\tfrac{d+1}{d+3}r^2$ and $\overline{\rho(u_\perp)}=\int\rho(u_\perp)\,\mathrm{d}\sigma(u)$; the last error term is the effect of $\theta_x\ne1$ on the average of the $s^2$ term. The $\sigma$-average: write $u_\perp=\sinh\chi\,\hat u$ with $\hat u$ uniformly distributed on the unit sphere $S^{d-1}$ of $e_0^\perp$ at each fixed $\chi$ (the hyperbolic measure on the cap is a product of a rapidity density and the uniform spherical measure). Then $\rho(u_\perp)=\sinh^2\chi\,g(R(\hat u,e_0)e_0,\hat u)$ and the spherical average of $g(R(\hat u,e_0)e_0,\hat u)$ over $\hat u\in S^{d-1}$ is $\tfrac1d\sum_{a=1}^dg(R(e_a,e_0)e_0,e_a)$ for any orthonormal basis $(e_a)$ of $e_0^\perp$, which equals $\tfrac1d\mathrm{Ric}(e_0,e_0)$ because $\mathrm{Ric}(e_0,e_0)=\sum_{a=0}^d\epsilon_ag(R(e_a,e_0)e_0,e_a)$ with $\epsilon_0=-1$ and the $a=0$ term vanishing since $R(e_0,e_0)=0$. Hence $\overline{\rho(u_\perp)}=\tfrac1d\,\overline{\sinh^2\chi}\,\mathrm{Ric}(e_0,e_0)$, and (de Sitter check: $\tfrac1d\overline{\sinh^2\chi}\cdot(-d)=-\overline{\sinh^2\chi}$, consistent with (7.6) and the exact formula). Therefore $$W^\tau_1(m^{(r)}_x,m^{(r)}_y)\ge\ell\Bigl(1+\frac{r^2}{2d}\cdot\frac{d+1}{d+3}\,\overline{\sinh^2\chi}\,\mathrm{Ric}(e_0,e_0)\Bigr)-C\ell\,r^3e^{4\chi_0}(1+\|R\|_U)^2,$$ using $\ell\le r$ to absorb $r^2\ell^2$ into $r^3\ell$ and $\overline{\sinh^2\chi}\le e^{2\chi_0}$. Dividing by $\ell$ and subtracting from 1 gives (7.5). The "in particular" clause follows from $\mathrm{Ric}(e_0,e_0)\ge-K$ and $\overline{\sinh^2\chi}\ge0$. $\square$ **Conjecture E-Lor** (Equality in the Lorentzian expansion). *Under the hypotheses of Lemma 7.4, (7.5) holds with equality up to the error term: $\mathcal{K}^{(r)}(x,y)=-\tfrac{r^2}{2d}\tfrac{d+1}{d+3}\overline{\sinh^2\chi}\,\mathrm{Ric}(e_0,e_0)+O(r^3e^{4\chi_0}(1+\|R\|_U)^2)$.* The missing half is an upper bound on the supremum $W^\tau_1$, which requires the dual side of Lorentzian Kantorovich duality: functions $(f,g)$ with $f(q)-g(p)\ge\tau(p,q)$ on causal pairs, for which $W^\tau_1\le\int f\,\mathrm{d}m_y-\int g\,\mathrm{d}m_x$. A duality theorem of this form for the $\ell_1$ cost on a compact causal region is available in Eckstein-Miller (loc. cit., Thm. 3.4) under the assumption that the cost is continuous and the measures are supported in a compact set where $\tau$ is continuous; the obstacle is not the duality but the construction of a near-optimal dual pair whose error is smaller than the signal $r^2\ell\,\mathrm{Ric}$: the natural choice $f=g=\tau(\bar x,\cdot)$ for a point $\bar x$ in the past carries a Hessian error of order $r^2\ell/\ell_0$ with $\ell_0\le\mathrm{inj}$, which is not smaller than the signal (review2 item 7). Conjecture E-Lor is consumed by Theorem 8.1 in Chapter 8 (C6); Part III must tag accordingly (C20). **Remark 7.4.1** (Sign and direction). Lemma 7.4 shows that a timelike Ricci lower bound (TCD convention) yields an upper bound on the shell Ollivier curvature. The Riemannian expansion (Remark 7.4.2) has the opposite sign: there a Ricci lower bound yields an Ollivier lower bound. The difference is the sign of $g(u_\perp,u_\perp)g(e_0,e_0)$ in the second variation. Any hypothesis on QSNs intended to encode a timelike Ricci lower bound must therefore be an upper bound on the timelike Ollivier curvature (H6$^+$), and the v3.0 hypothesis H6 is withdrawn. **Remark 7.4.2** (Riemannian form). With $g$ Riemannian, $S^{(r)}_x$ the ball of radius $r$, $m^{(r)}_x$ the normalised volume, and the ordinary $W_1$ (an infimum), the parallel coupling gives the upper bound $W_1\le\ell(1-\tfrac{r^2}{2(d+2)}\mathrm{Ric}(v,v))+O(r^3)$, hence $\mathcal{K}^{(r)}\ge\tfrac{r^2}{2(d+2)}\mathrm{Ric}(v,v)-O(r^3)$, and the dual lower bound with $f=d(\bar x,\cdot)$ gives equality (Ollivier, loc. cit., Example 7 and Prop. 6). In the Riemannian case the primal and dual bounds are on opposite sides because $W_1$ is an infimum; this is why the Riemannian expansion is an equality and the Lorentzian one, as proved here, is not. **Remark 7.4.3** (Relation to the graph measures). Lemma 7.4 concerns normalised volume on a rapidity-truncated proper-time shell. The measure of L3 is uniform on Hasse out-neighbours with the cost $d_G$; the measure of (7.1) is $\tau$-weighted with the cost $d_\tau$; and the law of a Hasse link from a sprinkled point at proper-time separation $s$ has the smooth radial profile $e^{-\rho c_ds^{d+1}(1+O(s^2))}$ (probability that the causal interval is empty), not an indicator of $s\le r$. None of the three coincides with the shell measure, and the rapidity cutoff of a bounded-degree truncation is $\chi_0=\tfrac{1}{d+1}\log\rho+O(1)$, for which $\overline{\sinh^2\chi}\asymp\rho^{2/(d+1)}=r_\rho^{-2}$ and the curvature term in (7.5) is $O(1)$ rather than $O(r_\rho^2)$, while the error term $r^3e^{4\chi_0}$ is then of order $r_\rho^{3}\rho^{4/(d+1)}=r_\rho^{-1}$ and the lemma's hypothesis $re^{2\chi_0}\le c_d$ fails. Lemma 7.4 therefore says nothing about the Ollivier curvature of sprinkled QSNs (review2 items 6, 8); its consumers are statements about smooth spacetimes (Chapter 8 recovery sequences built on shell measures at fixed $\chi_0$, with $r\to0$ first). **Remark 7.5** (Dimensions). $\mathcal{K}^{(r)}$ is dimensionless; $r^2\,\mathrm{Ric}$ is dimensionless since $[\mathrm{Ric}]=\mathrm{length}^{-2}$; with $c$ restored the shell radius $r$ is a proper time and the curvature scale is $c^2r^2\mathrm{Ric}$. **Hypothesis discharge ledger, Chapter 7.** | Result | Classical inputs | Hypotheses used | Where consumed | |---|---|---|---| | Conj 7.1 | (withdrawn proof) | H6$^+$, H-cf | | | Prop 7.3 | A-Lor (via Thm 5.6) | (E), H-cf, H-TCD, C-Lor, H1', H4', H5 of both sequences | construction of $\mathsf{Q}''$; (b) of Thm 5.6 for the regular set | | Lem 7.4 | O'Neill second variation (sign fixed by the de Sitter check), Gray-Gibbons-Solodukhin tube expansion, Knothe-Rosenblatt rearrangement | smooth $g$, $r<\mathrm{inj}$, $re^{2\chi_0}(1+\|R\|_U)\le c_d$; all constants depend on $\chi_0$ | Steps 1-4; one-sided only | | Conj E-Lor | Eckstein-Miller duality (for the statement) | | consumed by Thm 8.1 | | Conj 7.2 | open | | | | Rem 7.6 (horizons) | none (Phenomenological) | | | ### 7.4. Dimensional reduction near horizons **Remark 7.6** $\boxed{\text{Phenomenological}}$ (Horizon dimensional reduction). In a QSN whose GHL limit contains a Killing horizon, the out-neighbour measures of nodes near the horizon are concentrated along the horizon generators: the causal future of a point just outside the horizon is, within a thin future shell in the sense of Definition 7.0, dominated by directions nearly tangent to the horizon. The timelike Ollivier curvature in the transverse directions is then suppressed relative to the curvature along the generators, and the effective isoperimetric dimension $N_0$ of H4', measured on causal diamonds straddling the horizon, decreases from $4$ toward $2$ as the diamond shrinks. This is the QSN form of the dimensional reduction observed in the spectral dimension of asymptotically safe gravity (Lauscher, Reuter, *JHEP* **10** (2005) 050) and of causal dynamical triangulations (Ambjørn, Jurkiewicz, Loll, *Phys. Rev. Lett.* **95** (2005) 171301). It is an expectation about the structure of the reference families of Chapter 20, not a theorem; Theorem 20.3 records what is proved for the Schwarzschild family. --- --- ## Referee response, Part II (review2.md, pass 2) Policy applied: each FATAL item is resolved either by a complete proof or by demotion to Conditional on a named hypothesis or to Conjecture, with a retraction record; no patching. Item numbers refer to review2.md. | Item | Severity | Resolution | |---|---|---| | 1 | FATAL | Lem 7.4 restated as one-sided (upper bound on $\mathcal{K}^{(r)}$); Step 4 (dual) removed; equality is Conjecture E-Lor with the missing dual construction described. | | 2 | FATAL | Step 2 re-derived in O'Neill's conventions via sectional curvature; sign now matches the de Sitter exact formula; the curvature term is $-\tfrac{r^2}{2d}\tfrac{d+1}{d+3}\overline{\sinh^2\chi}\,\mathrm{Ric}(e_0,e_0)$. | | 3 | MAJOR | Statement restricted to $v=e_0$ and to the quantity the proof yields; the $w(u,v)$ weight is gone. | | 4 | MAJOR | "In particular" clause now reads $\mathrm{Ric}\ge-K\Rightarrow\mathcal{K}^{(r)}\le cKr^2+\dots$; Remark 7.4.1 states the direction. | | 5 | MAJOR | Step 3 replaced: causal transport map $\Psi=\Theta\circ\Phi_\ell$ built by Knothe-Rosenblatt rearrangement in polar coordinates; cost controlled directly, no TV argument. | | 6 | MAJOR | Hypothesis $re^{2\chi_0}(1+\|R\|_U)\le c_d$ made explicit; all constants carry $e^{k\chi_0}$; Remark 7.4.3 records that sprinklings violate it. | | 7 | MAJOR | Dual step removed (see 1). | | 8 | MAJOR | Remark 7.4.3: the three measures are distinct; link law is not a shell indicator; curvature term is $O(1)$ for sprinklings; no consequence for QSNs claimed. | | 9 | MINOR | Global hyperbolicity dropped from Def 7.0 / Lem 7.4; convex normal neighbourhood suffices. | | 10 | MINOR | $\ell\le r$ retained; consumers in Ch 5 removed (Remark 5.5.2 no longer uses Lem 7.4). | | 11 | MAJOR | Tag is Theorem for the inequality only; equality is Conjecture E-Lor. | | 12 | MINOR | Def 7.3 renamed Def 7.0; "Remark 5.3" fixed; Remark 5.5.1/5.5.2 reordered. | | 13 | MAJOR | H2-ker introduced as a separate hypothesis; Lem 5.3 now uses no curvature hypothesis at all; Thm 5.5 does not need H2-ker (Remark 5.5.3). | | 14 | MAJOR | Step B/C use the relative Nash inequality with additive term (Coulhon-Grigor'yan Thm 3.3); ultracontractivity on $t\le c_1|N|^{2/N_0}$. | | 15 | MAJOR | Step E redone with dyadic annuli and the Poisson jump-count bound for $d_G>t$. | | 16 | MINOR | Intermediate inequality corrected to exponent $2N_0/(N_0-1)$ with $\theta=1/(N_0+1)$. | | 17 | MINOR | $C^G_{\mathrm{vol}}$ added to the constant list of Lem 5.3. | | 18 | MINOR | Step D cites bounded geometry only; H2 removed from Lem 5.3 and ledger. | | 19 | FATAL | Causal-fraction bound withdrawn; H-cf introduced as a hypothesis; Lem 5.4 tagged Conditional (on H-cf). | | 20 | MINOR | Level function removed from Lem 5.4; $\ell^+(n,\cdot)$ used directly. | | 21 | MAJOR | Section 5.4 works in $d_n=\epsilon_nd_G$; contraction exact; $d_{\mathrm{eff}}$ limit recovered as bi-Lipschitz image. | | 22 | MAJOR | B2 (Ollivier $\Rightarrow$ graph Bakry-Émery) removed; misattributions to Münch-Wojciechowski and Lin-Yau withdrawn; Conjecture 5.9 states the open transfer. | | 23 | MINOR | Kuwada duality now invoked only for the $W_1$/Lipschitz pair on graphs (Remark 5.5.3). | | 24 | MAJOR | H-BE restated as $\mathrm{RCD}(K,N_0)$ on the limit, with the Erbar-Kuwada-Sturm / AGS Ann. Probab. equivalences and Sobolev-to-Lipschitz named. | | 25 | FATAL | Cavalletti-Milman dimensional upgrade withdrawn; weighted-manifold counterexample recorded; $\mathrm{RCD}(K,N_0)$ is now assumed (H-BE). | | 26 | MAJOR | Proof establishes comparability; $\mathfrak{m}=c\mathcal{H}^{N_0}$ obtained from Brena-Gigli-Honda-Zhu (weakly non-collapsed $\Rightarrow$ non-collapsed). | | 27 | OK / MINOR | Metric-doubling exponent recomputed ($C_{\mathrm{dbl}}^6$). | | 28 | MAJOR | Tag of Thm 5.5: Conditional (on H-conv, H-BE) modulo Classical(A, C). | | 29 | FATAL | C-Lor stated as a named Conjecture; the attributions to Beran et al. and Beran-Ohanyan-Rott-Solis withdrawn; Thm 5.6(b), Cor 5.7 (metric clause), Prop 5.8, Prop 6.1 (metric clause), Prop 7.3 tagged Conditional on C-Lor. | | 30 | MINOR | A-Lor stated as Minguzzi-Suhr Thm 4.5 (bounded Lorentzian metric spaces, $\epsilon$-net bounds); categories reconciled; Kunzinger-Sämann Thm 3.26 no longer cited as precompactness. | | 31 | OK | Selection-independence sentence added. | | 32 | MINOR | H2 forms separated: H2 (L3), H2-ker (kernel), H6$^+$ (timelike, upper bound). | | 33 | MAJOR | H-conv-Lor (iv) withdrawn; H-TCD introduced as the explicitly assumed conclusion. | | 34 | MINOR | Cor 5.7 depends on C-Lor for the metric clause; stated. | | 35 | MAJOR | Prop 5.8 restated as a distributional identity on $I^+(x)\cap U$ only; cone singularity acknowledged. | | 36 | FATAL | Lem 5.2 withdrawn; Conjecture 5.2 with retraction record (degree growth, sup/inf cost mismatch, $O(1)$ curvature); Classical P reattributed (Talagrand, Dobrić-Yukich for $D\ge3$). | | 37 | FATAL | Thm 7.1 withdrawn; Conjecture 7.1; non-existent Classical F-directed, G-Lor, H-Lor removed from Part II. | | 38 | FATAL | Sup-cost concavity under mixtures and Strassen emptiness recorded in §7.1; $\mathcal{K}_\tau:=1$ convention on empty edges stated. | | 39 | FATAL | Semicontinuity direction recorded in the retraction record; no limit passage of $W^\tau_1$ is used anywhere. | | 40 | FATAL | Inherited; resolved by 2 and 37. Cavalletti-Mondino Lem 3.9 citation removed. | | 41 | MAJOR | Tag: Conjecture. | | 42 | FATAL | Prop 6.1 restated with a global isometry; flat-dilation counterexample recorded in Remark 6.1.1. | | 43 | FATAL | Prop 6.3 withdrawn; Remark 6.3 records the three failures and the $\mathbb{Z}^2$ counterexample; Conjecture 6.2 restated in terms of $d_\tau$. | | 44 | MAJOR | Prop 7.3 restated in weak form: no H2/H6 claim, distortion bound with explicit bridge count $m_n$, tag Conditional on (E, H-cf, H-TCD, C-Lor). | | 45 | MINOR | $v_{\min}=v_{\min}(D,j_{\max})$ under the spin cutoff. | | 50 | MINOR | Numbering and cross-references repaired; Lem 7.4 consumers listed in Remark 7.4.3. | | 51 | MINOR | Ledgers rewritten for Ch 5 and 7. | Net status of Part II after this pass. Unconditional theorems: Lem 5.1, Lem 5.3 (doubling; no curvature input), Lem 7.4 (one-sided expansion), Props 4.5, 4.7. Conditional: Lem 5.4 (H-cf); Thm 5.5 (H-conv, H-BE; modulo A, C); Thm 5.6(a) (H-cf; modulo A-Lor); Thm 5.6(b), Cor 5.7, Prop 5.8, Prop 6.1, Prop 7.3 (additionally H-TCD, C-Lor, and H10 where stated). Conjectures: 5.2, 5.9, 6.2, 7.1, 7.2, E-Lor, C-Lor. No discrete-to-continuum curvature transfer is proved in either signature. *End of Part II.* # Part III. Classical and Quantum Dynamics --- ## Chapter 8. Emergence of Einstein's Equations Part II established that an admissible sequence of QSNs is precompact in the measured Gromov-Hausdorff topology (Riemannian sector, Classical A with the curvature-free volume doubling of Lemma 5.3) and, conditionally on H-cf, in the Gromov-Hausdorff-Lorentz topology (Lorentzian sector, Theorem 5.6(a)); the transfer of the discrete Ollivier curvature to a synthetic Ricci bound on the limit is, after Erratum C22, a Conjecture (Conjecture 7.1) or a hypothesis (H-BE in RCD form, H-TCD), and this chapter does not use it. This chapter asks the dynamical question: what does the Lach-Space energy functional of Axiom L5 become on that limit? The answer, in the Riemannian sector, is the Euclidean Einstein-Hilbert action; the Lorentzian reading is obtained by the Wick rotation of Chapter 21. The mechanism is that the Ollivier curvature of a random walk of step radius $\varepsilon$ at a point of a $C^2$ Riemannian manifold measures the Ricci curvature in the direction of the step, with a universal coefficient and an $\varepsilon^2$ prefactor (Classical E), so that a tension-weighted sum over edges of the rescaled curvature is a Riemann sum for the integrated scalar curvature, provided the tensions sample directions isotropically. Isotropy is a genuine hypothesis on the sequence, stated as (H12) below; it is the discrete shadow of the requirement that the emergent metric and the emergent action be built from the same edge data. Two technical features of bounded-degree graphs drive the structure of the proof and are treated explicitly rather than hidden. First, the Ollivier curvature of an edge in a random graph of bounded degree carries a curvature-independent bias of order one, coming from the transport cost between two empirical measures of finitely many points; the action must be centred by subtracting this bias (Definition 8.1.4), exactly as a lattice action is normal-ordered, and the curvature dependence of the bias at second order is a hypothesis on the local graph ensemble, (H13). Second, the centred per-edge fluctuations do not vanish individually; they vanish in the rescaled sum only after averaging, and in the physical dimension $d=4$ the rescaled sum of fluctuations is of order one rather than of order zero. The theorem therefore separates a deterministic limit, which is the Einstein-Hilbert functional, from a mean-zero random variable which is tight and does not affect the field equations. Both points are consequences of (H1) and cannot be removed without abandoning bounded valence. *Sector and signature.* Throughout §§8.1-8.3 and §8.5 the setting is the Riemannian sector: $(\mathsf{Q}_n)$ satisfies (H1'), (H3), (H4') (two-sided) uniformly, with tensions rescaled by $T_n(e)=nT(e)$, and converges in the measured Gromov-Hausdorff topology, along a subsequence, to a compact metric measure space; existence of the limit uses only Classical A (Part I, Theorem 2.2), whose hypotheses, uniform doubling and bounded diameter, are supplied by Lemma 5.3 (from (H1'), (H4')) and (H3). Nothing in this chapter uses the curvature-transfer chain of Theorem 5.5 (Classical B, C, and the hypotheses (H-conv), (H-BE) that Theorem 5.5 carries per Erratum C17): the only property of the limit the proofs consume is supplied by hypothesis (H10) *as consumed here*, namely that the mGH limit is a compact smooth $d$-manifold $\mathcal{M}$ with empty singular set and a $C^3$ Riemannian metric $g$, and that the approximating maps $\iota_n\colon N_n\to\mathcal{M}$ of the mGH convergence are the maps with respect to which (H12), (H13) are stated. ($C^3$ rather than $C^2$ is needed for the $O(\varepsilon^3)$ remainder of Classical E and Gray's expansion, and is what Lemma 7.4 in Part II requires; review3b item 5.) A reader who prefers to obtain the limit from Theorem 5.5 must add the rescaled (H2), (H-conv), (H-BE) (in its RCD form, Erratum C22) and Classical B, C to every tag below; we do not, because no step uses them (review3b, item 1). Nothing in this chapter cites Lemma 5.2 or Theorem 7.1, both Conjectures after C22; Lemma 5.3 is cited only in its curvature-free form. A referee (review3, item 1) observed that the isotropy hypothesis cannot hold with positive tensions and causal edges in Lorentzian signature, because a positive combination of outer products of causal vectors contracts negatively with $g$; the Lorentzian statements of this chapter are therefore all obtained through Chapter 21 and are tagged accordingly (§8.4 and Remark 8.6). The bounded-degree window sprinkling of Chapter 20, which is Lorentzian, is discussed in §8.4 as the Lorentzian-sector counterpart of the recovery question. *Coarse-graining radius and rescaled curvature.* As in Part I, eq. (3.5), the coarse-graining radius is $\varepsilon_n=n^{-1}$ and every curvature hypothesis on the sequence refers to the *rescaled curvature* $\mathcal{K}^{(n)}(e):=\varepsilon_n^{-2}\mathcal{K}_n(e)=n^2\mathcal{K}_n(e)$; (H2) reads $\mathcal{K}^{(n)}(e)\geq K$. The Ollivier curvature of an edge $e=(p,q)$ is that of the random walk whose step measure at $p$ is the uniform measure $\mu_p$ on the out-neighbours of $p$ (Axiom L3); its *step radius* is $\varepsilon_n(p):=\max_{e\ni p\ \text{out}}|\Delta x_e|_g$, which under (H1') and the rescaling satisfies $\varepsilon_n(p)=\lambda_n(p)n^{-1}$ with $\lambda_n(p)\in[\lambda_{\min},\lambda_{\max}]$, $0<\lambda_{\min}\leq\lambda_{\max}<\infty$ independent of $n$ and $p$ (the edge lengths are bilaterally comparable to $n^{-1}$ by (H1') through $d_{\mathrm{eff}}$, Theorem 5.5). Classical E expands the Ollivier curvature in the step radius, not in the length of the particular edge; the two differ by a factor $(\varepsilon_n(p)/|\Delta x_e|)^2$ that is edge-dependent, and the hypothesis (H12) below is formulated with the step radius so that the factor is accounted for exactly (review3, item 2). *Probability space.* Whenever a statement is "almost sure" or "in probability", the underlying probability space is the product over $n$ of the laws $\mathbb{P}_n$ of the construction producing $\mathsf{Q}_n$; for the sprinklings of §8.4 this is the Poisson point process of intensity $n^d\,\mathrm{dvol}_g$ on $\mathcal{M}$ with the Borel σ-algebra of the vague topology, and $\mathbb{E}[\cdot\mid g]$ denotes expectation under $\mathbb{P}_n$ for the fixed limit geometry $g$. For a general admissible sequence (a deterministic object), expectations are trivial and every "in probability" statement is a statement about the sequence itself. ### 8.1. The QSN action **Definition 8.1.1 (Intrinsic tension and edge displacement).** The *intrinsic tension* of an edge of $\mathsf{Q}_n$ is $T(e)=n^{-1}T_n(e)$, which lies in $[T_{\min},T_{\max}]$ by (H1'). The *edge displacement* $\Delta x_e\in T_x\mathcal{M}$ of an edge $e=(p,q)$ with $\iota_n(p)=x$ is $\exp_x^{-1}(\iota_n(q))$, defined for $n$ large since $|\Delta x_e|_g\leq\lambda_{\max}n^{-1}$ is below the injectivity radius of the compact manifold $\mathcal{M}$; its unit direction is $\hat v_e:=\Delta x_e/|\Delta x_e|_g$. **Definition 8.1.2 (QSN action).** The action of a QSN is $$ S[\mathsf{Q}]=\sum_{e\in E}T(e)\,\mathcal{K}(e)+\sum_{p\in N}v(p)+S_{\mathrm{mat}}[\mathsf{Q}]+S_{\mathrm{gauge}}[\mathsf{Q}], \tag{8.1} $$ where $v(p)=\langle V(\widehat V_p)\,\mathrm{tr}\,\widehat C^{(2)}_p\rangle$ is the expectation of the node potential of §4.4 in the state under consideration, and $S_{\mathrm{mat}}$, $S_{\mathrm{gauge}}$ are the matter and gauge actions of Chapter 9. The first term is the energy functional of Axiom L5. **Definition 8.1.3 (Rescaled action).** The *rescaled gravitational action* at scale $n$ is $$ \mathcal{S}_n[\mathsf{Q}_n]=\alpha_d\Big[n^{-d}\sum_{e\in E_n}T(e)\,\widetilde{\mathcal{K}}^{(n)}(e)\;+\;n^{-d}\sum_{p\in N_n}v(p)\Big],\qquad \alpha_d=\frac{2(d+2)}{16\pi G}, \tag{8.2} $$ with $\widetilde{\mathcal{K}}^{(n)}=n^2\widetilde{\mathcal{K}}_n$ the rescaled centred curvature of Definition 8.1.4. The number of edges and nodes in a region of unit $g$-volume is of order $n^d$ (by (H12) below), and the *expected* rescaled centred curvature is $O(1)$ by (H13)(b); the unaveraged rescaled curvature is not bounded above uniformly (its range is $n^2[-2-b_D,1-b_D]$), and the fluctuation part is controlled in the sum by Lemma 8.5. The constant $\alpha_d$ is defined by $\alpha_d:=1/(16\pi G\,c_\nu)$ with $c_\nu$ the total coefficient of (8.5); the displayed value $2(d+2)/(16\pi G)$ is the case $c_\nu=1/(2(d+2))$, the Ollivier ball constant, which holds when the empirical-measure correction $\beta'$ of Lemma 8.2(c) vanishes. In general $G$ is *defined* by the sequence through $c_\nu$ (review3b, item 6): the Newton constant of the emergent theory is $G=1/(16\pi\alpha_dc_\nu)$ with $\alpha_d$ the bare coefficient of Axiom L5 in Planck units. **Definition 8.1.4 (Flat bias and centred curvature).** Let $D$ be the degree bound of (H1) and let $\mathfrak{L}$ denote the *local law* of the sequence: the joint law, in normal coordinates at a node and in units of the step radius, of the out-neighbour configuration and the tensions. The *flat bias* is $$ b_D:=\mathbb{E}_{\mathfrak{L}}\big[\mathcal{K}(e)\big]_{\text{flat}}, $$ the expected Ollivier curvature of an edge in the reference graph with local law $\mathfrak{L}$ embedded in flat $\mathbb{R}^d$; it is finite since $\mathcal{K}\in[-2,1]$, and for the Poisson window law it is computable from Classical P. The *centred curvature* and *rescaled centred curvature* are $$ \widetilde{\mathcal{K}}_n(e)=\mathcal{K}_n(e)-b_D,\qquad\widetilde{\mathcal{K}}^{(n)}(e)=n^2\widetilde{\mathcal{K}}_n(e). \tag{8.3} $$ Ollivier curvature is invariant under rescaling of the metric, so $b_D$ depends only on $\mathfrak{L}$ and not on $n$; for a deterministic local law (jittered lattice) the expectation is over the jitter only. For a deterministic sequence without a local law, $b_D$ is defined as $\lim_n(\#E_n)^{-1}\sum_e\mathcal{K}_n(e)$ over a flat reference region and its existence is part of (H13)(a). **Hypothesis (H12) (Tension isotropy).** For every open $U\subset\mathcal{M}$ with $\mathrm{vol}_g(\partial U)=0$, the *tension-weighted direction tensors* converge, $$ M_n^{\mu\nu}(U):=n^{-d}\sum_{e\subset\iota_n^{-1}(U)}T(e)\,\lambda_n(p_e)^2\,\hat v_e^{\mu}\hat v_e^{\nu}\;\longrightarrow\;\int_U g^{\mu\nu}\,\mathrm{dvol}_g, \tag{8.4} $$ where $p_e$ is the tail of $e$ and $\lambda_n(p)=n\,\varepsilon_n(p)$; and the node densities converge, $n^{-d}\#\big(N_n\cap\iota_n^{-1}(U)\big)\to\mathrm{vol}_g(U)$, with the uniform bound $\#(N_n\cap\iota_n^{-1}(U))\leq C_N\,n^d\,\mathrm{vol}_g(U)$ for all $U$ and $n$. In the Riemannian sector $M_n(U)$ is a positive-definite symmetric tensor (a positive combination of $\hat v\otimes\hat v$ with $\hat v$ ranging over a set spanning $T_x\mathcal{M}$), so (8.4) is a normalisation of the tensions and a statement that edge directions, weighted by tension and by the squared step radius, sample the unit sphere of $g$ without preferred direction at macroscopic scales. Without it the limit functional is $\int M^{\mu\nu}R_{\mu\nu}$ with $M\neq g^{-1}$, a different theory. **Hypothesis (H13) (Random-walk bias bound; Erratum C13).** (a) The centred per-edge curvatures and the node potentials $v(p)$, conditioned on the limit geometry, have finite-range dependence of range $\rho_0$ in graph distance, uniformly in $n$; for a deterministic sequence this is vacuous and $b_D$ exists in the sense of Definition 8.1.4. (b) For every edge $e$ with tail at $x\in\mathcal{M}$, $$ \mathbb{E}\big[\widetilde{\mathcal{K}}^{(n)}(e)\,\big|\,g\big]=c_\nu\,\lambda_n(p_e)^2\,\mathrm{Ric}_x(\hat v_e,\hat v_e)+O(n^{-1}), \tag{8.5} $$ uniformly in $e$, where $c_\nu$ is a constant depending only on the local law $\mathfrak{L}$. Part (b) is the statement that the curvature dependence of the discrete random walk's transport cost at second order in the step radius is proportional to $\mathrm{Ric}(\hat v,\hat v)$, with no scalar-curvature term and no anisotropic term; Lemma 8.2 shows what part of (8.5) is a theorem (Classical E for the continuum step measure) and what part is the hypothesis (the empirical-versus-continuum discrepancy). Boundedness $\mathcal{K}\in[-2,1]$ makes the centred variables bounded by $3$ in absolute value. ### 8.2. Classical inputs **Theorem 8.2.1 (Classical E: Ollivier pointwise expansion).** *Let $(X,g)$ be a Riemannian manifold with $g\in C^3$, $x\in X$, $v\in T_xX$ a unit vector, and for $\varepsilon,\delta>0$ small let $y=\exp_x(\delta v)$ and $m_z^{(\varepsilon)}$ the normalised Riemannian volume on $B(z,\varepsilon)$. Then* $$ \kappa^{(\varepsilon)}(x,y)=1-\frac{W_1(m_x^{(\varepsilon)},m_y^{(\varepsilon)})}{d(x,y)}=\frac{\varepsilon^2\,\mathrm{Ric}_x(v,v)}{2(d+2)}+O(\varepsilon^3+\varepsilon^2\delta), \tag{8.6} $$ *with the $O$ uniform on compact sets (Ollivier, J. Funct. Anal. 256 (2009) 810, Example 7, proved in §8 of that paper as Thm 8 for general step measures with vanishing first moment and isotropic second moment $\sigma_\nu^2$, for which the coefficient $1/(2(d+2))$ is replaced by $c_\nu=\sigma_\nu^2/2$).* Hypotheses: $g\in C^3$ near $x$ (for the $O(\varepsilon^3)$ remainder; Ollivier's proof uses the third-order Taylor expansion of the exponential map), which is (H10) as consumed here; $\varepsilon,\delta$ below the injectivity radius, which holds for $n$ large on the compact $\mathcal{M}$. **Theorem 8.2.2 (Classical P: transport cost of empirical measures).** *Let $\hat\mu_D$ be the empirical measure of $D$ i.i.d. points with law $\nu$, a probability measure on a bounded subset of $\mathbb{R}^d$, $d\geq3$. Then $\mathbb{E}\,W_1(\hat\mu_D,\nu)\leq C_d\,\mathrm{diam}(\mathrm{supp}\,\nu)\,D^{-1/d}$ (Fournier, Guillin, Probab. Theory Relat. Fields 162 (2015) 707, Thm 1, for general $\nu$ with finite moments; Talagrand, Ann. Probab. 22 (1994) 919, Thm 1.1, for the uniform law on a cube; for $d=2$ with an extra $\sqrt{\log D}$, Ajtai, Komlós, Tusnády, Combinatorica 4 (1984) 259).* Used only to show finiteness of $b_D$ and of the flat transport discrepancy. **Theorem 8.2.3 (Gray: volume density in normal coordinates).** *For $g\in C^3$, the Riemannian volume density in normal coordinates at $x$ satisfies $\mathrm{dvol}_g(\exp_x w)=\big(1-\tfrac16\mathrm{Ric}_x(w,w)+O(|w|^3)\big)\mathrm{d}w$, uniformly on compacta (Gray, Tubes, Birkhäuser, 2004, Thm 3.3; the $O(|w|^3)$ form of the remainder uses one more derivative of the metric than the $o(|w|^2)$ form).* **Lorentzian expansion (Lemma 7.4 / Hypothesis E-Lor).** The Lorentzian counterpart of (8.6), for the causal-window random walk, is not a published classical theorem (Erratum C6). What Chapter 7 proves as Lemma 7.4 (after Erratum C22) is a ONE-SIDED inequality with the sign reversed relative to the Riemannian case: for the causal-shell random walk of radius $r$ with mean rapidity $\bar\chi$, $\mathcal{K}^{(r)}\leq-\tfrac{r^2}{2d}\tfrac{d+1}{d+3}\sinh^2\bar\chi\,\mathrm{Ric}(e_0,e_0)+O(r^3e^{4\chi_0})$; a timelike Ricci lower bound thus gives an Ollivier UPPER bound, which is why (H6) is replaced by (H6$^+$). Equality in Lemma 7.4 is Conjecture E-Lor. The expansion is used in this chapter only in §8.4 (Propositions 8.11, 8.12, Lorentzian window sprinkling), where the equality form is needed to identify the limit action; those statements are therefore Conditional on Conjecture E-Lor, and the inequality form yields only an upper bound on the curvature term of the discrete action. ### 8.3. Convergence of the action **Theorem 8.1** $\boxed{\text{Conditional (on H10, H12, H13) modulo Classical A, E, P, Gray}}$ (Convergence of the QSN action to the Euclidean Einstein-Hilbert action). *Let $(\mathsf{Q}_n)$ satisfy (H1'), (H3), (H4') uniformly, with mGH limit $(\mathcal{M},g)$ (Classical A), and assume (H10) as consumed here ($C^3$, empty singular set), (H12), and (H13). Let $v_\infty$ be the mean node potential of Lemma 8.4 (a constant under volume quantisation, Lemma 13.1), and define $\Lambda_{\mathrm{b}}$ by $-2\Lambda_{\mathrm{b}}/(16\pi G)=\alpha_dv_\infty$, i.e. $\Lambda_{\mathrm{b}}=-(d+2)v_\infty$. Then:* *(i) (Deterministic limit.) For every open $U\subset\mathcal{M}$ with null boundary,* $$ \lim_{n\to\infty}\mathbb{E}\big[\mathcal{S}_n[\mathsf{Q}_n\cap\iota_n^{-1}(U)]\,\big|\,g\big]=\frac{1}{16\pi G}\int_U\big(R-2\Lambda_{\mathrm{b}}\big)\,\mathrm{dvol}_g=:\mathcal{S}_{\mathrm{EH}}[g;U], \tag{8.7} $$ *and in particular $\lim_n\mathbb{E}[\mathcal{S}_n[\mathsf{Q}_n]\mid g]=\mathcal{S}_{\mathrm{EH}}[g]:=\mathcal{S}_{\mathrm{EH}}[g;\mathcal{M}]$.* *(ii) (Convergence along sequences, not Γ-convergence.) Along every sequence satisfying the hypotheses and converging to $(\mathcal{M},g)$, $\lim_n\mathbb{E}[\mathcal{S}_n\mid g]=\mathcal{S}_{\mathrm{EH}}[g]$. This is a statement of convergence along hypothesis-satisfying sequences; it is not a Γ-convergence statement on the space of metric measure spaces, because the hypotheses (H12), (H13) are properties of sequences and not of their limits (review3, item 3). Whether $\mathcal{S}_{\mathrm{EH}}[g]$ is attained as a limit of QSN actions for a given $(\mathcal{M},g)$ is the existence question (H14) of Remark 8.7, which this theorem does not answer.* *(iii) (Fluctuations.) The unaveraged action satisfies $\mathcal{S}_n[\mathsf{Q}_n]=\mathbb{E}[\mathcal{S}_n\mid g]+C_n$, where $C_n$ has mean zero and $\mathrm{Var}\,C_n\leq C_{\mathrm{fl}}\,\mathrm{vol}_g(\mathcal{M})\,n^{4-d}$, $C_{\mathrm{fl}}=18\,C_N\,D^{\rho_0+2}\,T_{\max}^2\,\alpha_d^2$. For $d\geq5$, $C_n\to0$ in probability; for $d=4$, $(C_n)$ is tight.* *(iv) (Field equations.) The first variation of $\mathcal{S}_{\mathrm{EH}}[g]+\mathcal{S}_{\mathrm{mat}}[g,\Phi]$, with $\mathcal{S}_{\mathrm{mat}}$ the continuum matter action of Proposition 9.4, with respect to $g$ vanishes if and only if* $$ R_{\mu\nu}-\tfrac12Rg_{\mu\nu}+\Lambda_{\mathrm{b}}g_{\mu\nu}=8\pi G\,\langle\widehat T_{\mu\nu}\rangle,\qquad \langle\widehat T_{\mu\nu}\rangle:=-\frac{2}{\sqrt{g}}\frac{\delta\mathcal{S}_{\mathrm{mat}}}{\delta g^{\mu\nu}}. \tag{8.8} $$ *Part (iv) is a statement about the limit functional; it does not assert that critical points of $\mathcal{S}_n$ converge to critical points of the limit (review3, item 13). Its hypotheses are those of Proposition 9.4 in addition to (i).* **Hypothesis (H14) (Existence of an admissible recovery sequence; open).** For the given compact $(\mathcal{M},g)$ with $C^3$ metric there exists a sequence $(\mathsf{Q}_n)$ satisfying L1-L5, (H1'), (H3), (H4') uniformly, (H12), (H13), whose mGH limit (Classical A) is isometric to $(\mathcal{M},g)$ with the approximating maps of (H12) (so that (H10) holds for it). (H14) is not a hypothesis of Theorem 8.1; it is the statement that the theorem has a non-vacuous instance at $(\mathcal{M},g)$. No construction satisfying it is known: the bounded-degree window sprinklings of Chapter 20 have time separations bounded away from $d_g$ (Prop 20.7) and, conditionally on H-conn and Conjecture 20.8, Lorentz-Finsler limits $d_{g,D}/c\neq d_g$, and the jittered lattice of Definition 8.14 has a polyhedral limit (Proposition 8.15(c)); both are instances of the bounded-degree obstruction of Open Problem O3. If one routes the limit through Theorem 5.5 instead of Classical A, (H14) acquires in addition the rescaled (H2), (H-conv), (H-BE), which both constructions also fail (Propositions 20.9, 8.15). The relation between $\Lambda_{\mathrm{b}}$ and the observed cosmological constant is the subject of Chapter 13, where the unimodular constraint forced by volume quantisation shows that the physical $\Lambda$ is an integration constant; within this chapter $\Lambda_{\mathrm{b}}$ is the coefficient produced by the node term. **Lemma 8.2** $\boxed{\text{Conditional (on H10, H13(b)) modulo Classical E, P, Gray}}$ (Curvature of a single edge: what is theorem and what is hypothesis). *Under (H10), for every edge $e=(p,q)$ with $\iota_n(p)=x$, step radius $\varepsilon=\varepsilon_n(p)$ and separation $\delta=|\Delta x_e|_g$, write* $$ \widetilde{\mathcal{K}}_n(e)=\underbrace{\Big[1-\frac{W_1(m_x^{(\varepsilon)},m_y^{(\varepsilon)})}{\delta}\Big]}_{\mathrm{I}}+\underbrace{\Big[\frac{W_1(m_x^{(\varepsilon)},m_y^{(\varepsilon)})}{\delta}-\frac{W_1(\mu_p,\mu_q)}{d_G(p,q)}\Big]}_{\mathrm{II}}-b_D,\qquad y=\iota_n(q). $$ *Then (a) $\mathrm{I}=c_\nu\varepsilon^2\mathrm{Ric}_x(\hat v_e,\hat v_e)+O(\varepsilon^3)$ with $c_\nu=1/(2(d+2))$ for the ball step measure, by Classical E with $\delta\leq\varepsilon$; (b) $\mathbb{E}[\mathrm{II}\mid g]$ is finite, equals $b_D$ in flat space, and its deviation from $b_D$ is $O(\varepsilon^2)$ with a coefficient that is a functional of $\mathrm{Ric}_x$ and of the local law; (c) the assertion that this coefficient is $\beta'\,\mathrm{Ric}_x(\hat v_e,\hat v_e)$ with $\beta'$ depending only on $\mathfrak{L}$, so that (8.5) holds with $c_\nu^{\mathrm{tot}}=c_\nu+\beta'$, is exactly (H13)(b); the Lemma does not prove it.* *Proof.* (a) is Theorem 8.2.1 with $v=\hat v_e$, $\delta\leq\varepsilon$, so $O(\varepsilon^3+\varepsilon^2\delta)=O(\varepsilon^3)$. (b) The law of the out-neighbour configuration in $B(x,\varepsilon)$ is the flat law $\mathfrak{L}$ pushed forward by $\exp_x$ and reweighted by the density of Theorem 8.2.3, which is $1-\tfrac16\mathrm{Ric}_x(w,w)+O(\varepsilon^3)$ uniformly on the ball; the same holds at $y$. $\mathbb{E}[\mathrm{II}\mid g]$ is therefore an expectation of a bounded functional (II is bounded since both transport terms lie in $[0,1+O(\varepsilon)]$ after division) against a law that is a $(1+O(\varepsilon^2))$-perturbation in density of the flat law; its value at $\varepsilon\to0$ with the flat law is $b_D$ by Definition 8.1.4 (there $\mathrm{I}=0$), and the deviation is bounded by $\sup|\mathrm{II}|\cdot\|\text{density}-1\|_\infty=O(\varepsilon^2)$, which proves the order but not the form of the coefficient: the map from the density perturbation to the expectation is linear, but the density perturbation $-\tfrac16\mathrm{Ric}_x(w,w)$ enters at both $x$ and $y$ and through the graph distance $d_G$, and the resulting coefficient is a functional of $\mathrm{Ric}_x$ that is not in general a multiple of $\mathrm{Ric}_x(\hat v_e,\hat v_e)$ (it may contain a term in $R(x)$ and, for non-isotropic laws, other contractions). (c) is a restatement of (H13)(b) in this notation, with $\lambda_n^2n^{-2}=\varepsilon^2$. $\square$ **Remark 8.2.1** (Status of (H13)(b)). For the bounded-degree Poisson window law the coefficient in Lemma 8.2(b) is an open computation of stochastic geometry; for a jittered lattice it is computable in principle. If it contains a scalar-curvature term $\beta''R$, the limit functional is still of the form $\frac{1}{16\pi G}\int(R-2\Lambda_{\mathrm{b}})$ with $c_\nu^{\mathrm{tot}}$ replaced by $c_\nu^{\mathrm{tot}}+d\beta''$ after the isotropic sum of Lemma 8.3 (the term $\beta''R$ summed against $T\lambda^2$ gives $d\beta''\int R$ by the trace of (8.4)), which is a finite renormalisation of $G$; an anisotropic term would change the functional. (H13)(b) as stated excludes anisotropic and scalar-curvature terms; it does not assert $\beta'=0$, and Definition 8.1.3 absorbs $\beta'$ into the definition of $G$. **Lemma 8.3** $\boxed{\text{Conditional (on H10, H12, H13)}}$ (Identification of $\alpha_d$). *For every $U$ as in Theorem 8.1,* $$ \lim_n n^{-d}\sum_{e\subset\iota_n^{-1}(U)}T(e)\,\mathbb{E}[\widetilde{\mathcal{K}}^{(n)}(e)\mid g]=c_\nu\int_U R\,\mathrm{dvol}_g, \tag{8.9} $$ *with $c_\nu$ the total coefficient of (8.5). With $\alpha_d=1/(16\pi Gc_\nu)$ (Definition 8.1.3) the coefficient of the Einstein-Hilbert term is $1/(16\pi G)$; when $\beta'=0$ this is $\alpha_d=2(d+2)/(16\pi G)$.* *Proof.* By (8.5), $n^{-d}\sum_eT(e)\mathbb{E}[\widetilde{\mathcal{K}}^{(n)}(e)]=c_\nu\,n^{-d}\sum_eT(e)\lambda_n(p_e)^2\mathrm{Ric}_{\mu\nu}(x_e)\hat v_e^\mu\hat v_e^\nu+n^{-d}\sum_eT(e)\,O(n^{-1})$. The error term is at most $T_{\max}\cdot D\cdot C_N\mathrm{vol}_g(U)\cdot O(n^{-1})\to0$ by (H1') and the uniform node-density bound of (H12) (each node has at most $D$ out-edges). For the main term, fix $\eta>0$ and partition $U$ into finitely many cells $U_j$ of diameter $\leq\eta$ with null boundaries (possible on a compact manifold by a triangulation); by uniform continuity of $\mathrm{Ric}$ on the compact $\mathcal{M}$ ((H10), $C^2$ metric) there is $\omega(\eta)\to0$ with $|\mathrm{Ric}_{\mu\nu}(x_e)-\mathrm{Ric}_{\mu\nu}(x_j)|\leq\omega(\eta)$ for $x_e\in U_j$, $x_j$ a fixed point of $U_j$ (in a fixed finite atlas, in which all tensor norms are comparable). Hence $$ \Big|n^{-d}\sum_{e\subset U_j}T(e)\lambda_n^2\mathrm{Ric}_{\mu\nu}(x_e)\hat v^\mu_e\hat v^\nu_e-\mathrm{Ric}_{\mu\nu}(x_j)M_n^{\mu\nu}(U_j)\Big|\leq\omega(\eta)\,\mathrm{tr}_gM_n(U_j)\leq\omega(\eta)\,T_{\max}\lambda_{\max}^2DC_N\mathrm{vol}_g(U_j), $$ using $|\hat v|_g=1$. Summing over $j$, the total error is $\leq\omega(\eta)T_{\max}\lambda_{\max}^2DC_N\mathrm{vol}_g(U)$, uniformly in $n$. By (H12), $M_n(U_j)\to\int_{U_j}g^{-1}$ for each of the finitely many $j$, so $\sum_j\mathrm{Ric}_{\mu\nu}(x_j)M_n^{\mu\nu}(U_j)\to\sum_j\mathrm{Ric}_{\mu\nu}(x_j)\int_{U_j}g^{\mu\nu}=\sum_j\int_{U_j}R+O(\omega(\eta)\mathrm{vol}_g(U))$. Thus $\limsup_n|\text{LHS of (8.9)}-c_\nu\int_UR|\leq c_\nu(1+T_{\max}\lambda_{\max}^2DC_N)\omega(\eta)\mathrm{vol}_g(U)$ for every $\eta$; let $\eta\to0$ (the interchange of the limits in $n$ and $\eta$ is justified by the uniform-in-$n$ error bound). $\square$ **Lemma 8.4** $\boxed{\text{Conditional (on H12, and on Lemma 13.1 or H12c)}}$ (Node term). *If the node potential is constant, $v(p)\equiv v_0$ (which holds in the ground state of the volume sector by Lemma 13.1, with $v_0=V(\kappa_d\ell_P^d)\,\mathrm{tr}\,C^{(2)}_{\min}$), then $n^{-d}\sum_{p\in\iota_n^{-1}(U)}v(p)\to v_0\,\mathrm{vol}_g(U)$ by the node-density part of (H12), and $v_\infty=v_0$. In general the existence of $v_\infty$ with $n^{-d}\sum_{p\in\iota_n^{-1}(U)}v(p)\to v_\infty\mathrm{vol}_g(U)$ for all $U$ is an additional hypothesis (H12c), satisfied whenever $v(p)$ is a bounded stationary functional of a stationary ergodic local law by the spatial ergodic theorem (Nguyen, Zessin, Z. Wahrsch. verw. Geb. 48 (1979) 133, Thm 1), but not derivable for a general deterministic sequence.* *Proof.* The constant case is the node-density clause of (H12) multiplied by $v_0$. $\square$ **Lemma 8.5** $\boxed{\text{Conditional (on H12, H13(a))}}$ (Fluctuation bound). *With $\xi_e:=\widetilde{\mathcal{K}}_n(e)-\mathbb{E}[\widetilde{\mathcal{K}}_n(e)\mid g]$, which satisfies $|\xi_e|\leq3$ by the range $[-2,1]$,* $$ \mathrm{Var}\Big(n^{-d}\sum_{e\subset U}T(e)\,n^2\xi_e\Big)\leq18\,D^{\rho_0+2}\,T_{\max}^2\,C_N\,\mathrm{vol}_g(U)\,n^{4-d}. \tag{8.10} $$ *Proof.* $\mathrm{Cov}(\xi_e,\xi_{e'})=0$ when $d_G(e,e')>\rho_0$ and $|\mathrm{Cov}|\leq9$ otherwise. The number of edges within graph distance $\rho_0$ of a given edge is at most $2D^{\rho_0+1}$ ((H1): at most $D^{k}$ nodes at distance $k$ from an endpoint, two endpoints, at most $D$ edges per node gives $2D^{\rho_0+1}$). Hence $\mathrm{Var}(\sum_eT(e)\xi_e)\leq\#E(U)\cdot2D^{\rho_0+1}\cdot T_{\max}^2\cdot9$ and $\#E(U)\leq D\,C_Nn^d\mathrm{vol}_g(U)$ by (H12). Multiply by $n^{4-2d}$. $\square$ *Proof of Theorem 8.1.* (i): Lemma 8.3 (curvature term) and Lemma 8.4 (node term) give $\lim_n\mathbb{E}[\mathcal{S}_n(U)\mid g]=\alpha_d[c_\nu\int_UR+v_\infty\mathrm{vol}_g(U)]=\frac{1}{16\pi G}\int_U(R-2\Lambda_{\mathrm{b}})$, using $\alpha_dc_\nu=1/(16\pi G)$ and the definition of $\Lambda_{\mathrm{b}}$; $U=\mathcal{M}$ is admissible since $\mathcal{M}$ is compact with empty boundary. (ii): the first clause is (i) with $U=\mathcal{M}$; the second is (H14) followed by (i). (iii): $C_n=\alpha_dn^{-d}\sum_eT(e)n^2\xi_e+\alpha_dn^{-d}\sum_p(v(p)-\mathbb{E}v(p))$; the second term vanishes in the constant case and otherwise has variance $\leq\alpha_d^2\|v\|_\infty^2C_N\mathrm{vol}_g\,n^{-d}$ by the same finite-range argument; the first has variance $\leq\alpha_d^2\cdot(8.10)$ with $U=\mathcal{M}$. Chebyshev gives convergence in probability for $d\geq5$ and tightness for $d=4$. No claim is made about the dependence of the law of $C_n$ on $g$ (review3, item 10). (iv): Let $g_s$ be a $C^1$ family of $C^2$ metrics with $g_0=g$ and $\delta g^{\mu\nu}=\partial_sg^{\mu\nu}|_0$. The Palatini identity $\delta R_{\mu\nu}=\nabla_\rho\delta\Gamma^\rho_{\mu\nu}-\nabla_\nu\delta\Gamma^\rho_{\rho\mu}$ shows $g^{\mu\nu}\delta R_{\mu\nu}$ is a divergence, which integrates to zero on the closed manifold $\mathcal{M}$. With $\delta\sqrt g=-\tfrac12\sqrt g\,g_{\mu\nu}\delta g^{\mu\nu}$ and $\delta R=R_{\mu\nu}\delta g^{\mu\nu}+g^{\mu\nu}\delta R_{\mu\nu}$, $$ \delta\mathcal{S}_{\mathrm{EH}}=\frac{1}{16\pi G}\int_{\mathcal{M}}\Big(R_{\mu\nu}-\tfrac12Rg_{\mu\nu}+\Lambda_{\mathrm{b}}g_{\mu\nu}\Big)\delta g^{\mu\nu}\sqrt g\,\mathrm{d}^dx, $$ and $\delta\mathcal{S}_{\mathrm{mat}}=-\tfrac12\int\langle\widehat T_{\mu\nu}\rangle\delta g^{\mu\nu}\sqrt g$ by definition. Stationarity for all $\delta g^{\mu\nu}\in C^\infty$ is (8.8). $\square$ **Remark 8.6** (Lorentzian reading). Chapter 21 (Theorem 21.1) provides the analytic continuation of the limit functional $\mathcal{S}_{\mathrm{EH}}[g]$, which is a polynomial in $g$, $g^{-1}$ and two derivatives of $g$, from Riemannian to Lorentzian metrics, and Proposition 21.2 the compatibility of this continuation with the QSN edge data. Under Theorem 21.1 the field equations (8.8) continue to the Lorentzian Einstein equations with the signature $(-,+,+,+)$ of Appendix A.5. Every Lorentzian statement derived from this chapter is therefore Conditional on Theorem 21.1 in addition to the tag of Theorem 8.1. The direct Lorentzian route through Theorem 5.6 and a Lorentzian isotropy hypothesis is closed by the referee's observation (review3, item 1) that no positive-tension causal-edge sequence can have tension tensor converging to $g^{-1}$; what a causal window produces is $a\,h^{\mu\nu}+b\,u^\mu u^\nu$ with $u$ the frame of the Cauchy time, and the corresponding limit functional is $\int(aR+(b-a)\mathrm{Ric}(u,u))$, a frame-dependent functional which is not the Einstein-Hilbert action. This is the same preferred-frame phenomenon as the Lorentz-Finsler limit of Chapter 20 (Erratum C15), seen at the level of the action. **Remark 8.7** (What is and is not proved). Theorem 8.1 proves that the Euclidean Einstein-Hilbert action is the limit of the QSN action along sequences with a smooth compact Riemannian limit, under (H10), (H12), (H13). It does not prove Γ-convergence, does not prove that every admissible sequence has a smooth limit (Theorem 5.5 gives a continuous metric on the regular set), does not identify the limit on singular sets, and does not produce a recovery sequence without (H14). The quantum dynamics is treated in Part IV, for which Theorem 8.1 supplies the tree-level boundary condition $\Gamma_{k\to0}=\mathcal{S}_{\mathrm{EH}}+\mathcal{S}_{\mathrm{mat}}$. ### 8.4. The Lorentzian window sprinkling and the Finsler recovery statement The natural candidate for a recovery sequence is the bounded-degree window sprinkling $\mathsf{Q}^{\mathrm{win}}_n(U,t,D)$ of Definition 20.11 (rapidity cutoff $v_{\max}<1$, minimal proper-time cut $c_1\ell_0/n$, symmetric in-truncation, edge Hamiltonians $H_e=\Delta(e)\widehat N$), a Lorentzian construction. Per Erratum C19, what Chapter 20 proves unconditionally about it is: (H1'), (H3), (H5) deterministically (Prop 20.3(a)-(c)); the volume upper bound of (H4')(ii) in the form $|B(p,r)|\leq Cr^d+6\log n$ (Prop 20.3(d)); (H2) only with $K=-2$, the bounds $K\geq0$ and the rescaled bound failing (Prop 20.9); and the *loss* estimates $d_{\tau,n}\leq(1-c_D)d_\eta$ in Minkowski at separations $\geq K_0\log n/n$ (Prop 20.6) and $\leq(1-c_D/2)d_g$ locally along the frame in the curved families (Prop 20.7). The existence of the Finsler function $\varphi_D$ and the convergence $d_{\tau,n}\to\varphi_D$ on the bi-infinite cluster are Conditional on the oriented-connectivity hypothesis H-conn (Hyp 20.12; Thm 20.5(i),(ii)); the isoperimetric lower bound (H4')(i) is the hypothesis H-iso-win (Hyp 20.13); and the curved identification $d_{\tau,n}\to d_{g,D}$ together with $d_{g,D}\to d_g$ as $D\to\infty$ is Conjecture 20.8 (with the cone-boundary behaviour of $\varphi_D$, Thm 20.5(iv), also conjectural). Everything in this section that uses $\varphi_D$ or $d_{g,D}$ is therefore Conditional on H-conn and Conjecture 20.8, and by RIGOR 10 carries Conjecture-level status where Conjecture 20.8 is consumed. This section records what the construction does deliver at each of these levels. Because the construction is Lorentzian, the statements here use the Lorentzian curvature expansion Lemma 7.4 and a Lorentzian tension normalisation, and they are *not* instances of Theorem 8.1; they are the Lorentzian-sector counterpart of the recovery question, with the preferred-frame functional of Remark 8.6 appearing explicitly. *Step 1 (points).* $\mathcal{P}_n$ is a Poisson point process on a relatively compact causally convex $U\subset\mathcal{M}$ of intensity $n^d\,\mathrm{dvol}_g$; $N_n=\mathcal{P}_n$ as in Definition 20.11, so that $n^{-d}\#(N_n\cap U')\to\mathrm{vol}_g(U')$ a.s. for $U'\subset U$ with null boundary (law of large numbers for the Poisson process); Proposition 20.3 records which hypotheses the construction satisfies. *Step 2 (edges).* With $t$ a Cauchy time on $U$ and $W_n(p)$ the future window of Definition 20.11 (the part of $J^+(p)$ of height $\lambda n^{-1}$ within rapidity $\eta_{\max}$ of $u(p)$ and at proper time $\geq c_1\ell_0/n$), the out-neighbours of $p$ are the $D$ points of $W_n(p)\cap\mathcal{P}_n$ of smallest $t$, and symmetrically the in-neighbours are the $D$ points of the past window of largest $t$ (symmetric truncation, giving a deterministic degree bound, Prop 20.3(a), review3 item 5). This gives L1 and (H1). *Step 3 (tensions).* $T(e)=\theta_n(p)\langle\hat v_e,A_n(p)\hat v_e\rangle$ with $\theta_n(p)>0$ and $A_n(p)$ a positive symmetric matrix in the orthonormal frame $(u,h)$ at $p$ chosen so that $$ \lambda^2\,\mathbb{E}\Big[\sum_{e\ni p\ \text{out}}T(e)\,\hat v_e\otimes\hat v_e\Big]=a_D\,h^{-1}(p)+b_D'\,u\otimes u, $$ where the left side is a positive combination of causal outer products, necessarily of the form $a\,h^{-1}+b\,u\otimes u$ with $b>0$ by the $O(d-1)$ symmetry of the window about $u$ (review3, item 1), and $\theta_n$ is normalised so that $a=1$; the $u\otimes u$ component cannot be removed with positive tensions, so the Lorentzian tension tensor is $M^{\mu\nu}=h^{\mu\nu}+b\,u^\mu u^\nu$, not $g^{\mu\nu}$. The constants $(a_D,b_D)$ of Proposition 8.11 are obtained from this tensor composed with the window average of Lemma 7.4. This gives (H1') with constants depending on $D$ (Prop 20.3(a)) and, by the finite-range Chebyshev-Borel-Cantelli argument, the convergence $n^{-d}\lambda^2\sum_{e\subset U'}T\hat v\otimes\hat v\to\int_{U'}(h^{-1}+b'_Du\otimes u)$ a.s. *Step 4 (edge Hamiltonians and H5; corrected per C15).* On each edge take $\mathcal{H}_e=\ell^2(\mathbb{N})$ and $H_e=\Delta(e)\,\widehat N$ with $\widehat N$ the number operator, $\widehat N|k\rangle=k|k\rangle$, and $\Delta(e):=2\pi\hbar/(n|\Delta x_e|_g)$, so that $H_e$ is self-adjoint with discrete spectrum $\{0,\Delta(e),2\Delta(e),\dots\}$ and gap $\Delta(e)$, and Axiom L6 gives $\tau_n(e)=n^{-1}\tau(e)=|\Delta x_e|_g$. Since $|\Delta x_e|_g\in[c_1\ell_0n^{-1},C_Un^{-1}]$ by the proper-time cut and the window height (Prop 20.3(b)), $\Delta(e)$ is bilaterally bounded, which is (H5). The earlier choice of a "Casimir restricted to finitely many principal-series levels" on $L^2(\mathrm{Spin}(4,1))$ is withdrawn: the Casimir of a non-compact group on $L^2$ has continuous spectrum and no such restriction defines a self-adjoint operator with a gap. *Step 5 (H3, H4', and the status of H2, H6).* (H3) holds with $T=\mathrm{diam}_g(U)$ (Prop 20.3(c)). Of (H4'), the volume upper bound holds in the form $|B(p,r)|\leq Cr^d+6\log n$ (Prop 20.3(d)), which is the two-sided form only for $r\geq(6\log n/C)^{1/d}$; the isoperimetric lower bound is the hypothesis H-iso-win (Hyp 20.13). (H2) holds with $K=-2$ only; $K\geq0$ fails, and the rescaled bound $n^2\mathcal{K}\geq K$ fails eventually a.s. for every $K$ (Prop 20.9). The timelike bound (H6$^+$) (an upper bound, per C22) is likewise not established in rescaled form. Consequently neither Theorem 5.6 nor Lemma 5.3 (which needs $K\geq0$) applies to this sequence, and its limit must be identified directly. *Step 6 (the limit).* Unconditionally: $d_{\tau,n}\leq(1-c_D/2)d_g$ locally along the frame (Prop 20.7), so the limit, if it exists, is strictly below $d_g$. Under H-conn: in Minkowski space $d_{\tau,n}\to\varphi_D$ on the bi-infinite cluster in probability (Thm 20.5(i),(ii)). Under H-conn and Conjecture 20.8: for $x\prec y$ in $U$ and nodes $x_n\to x$, $y_n\to y$ in the cluster, $d_{\tau,n}(x_n,y_n)\to d_{g,D}(x,y)/c$ in probability, uniformly on compacts, where $d_{g,D}$ is the Lorentz-Finsler time separation of Definition 20.13' and $c$ the normalisation constant of that conjecture, with $d_{g,D}2\max|s|$ depend on disjoint sets of cubes and are independent; $v(p)$, if a function of the local configuration, is a bounded stationary functional. (h) is the range of Ollivier curvature. The negative statement is proved as in Proposition 20.9: for an edge $(\xi_z,\xi_{z'})$, $\mathcal{K}$ depends on the jittered positions of the $\leq(\#\mathcal{S}+1)^2$ points within two stencil steps, through a non-constant continuous function (the $W_1$ distance between two empirical measures of jittered stencil points is not constant in the jitter, since moving one point changes the transport cost unless all transport is degenerate), so $\mathcal{K}$ has a non-degenerate law with positive probability of lying below $b_D-\eta$ for some $\eta>0$; independent copies at $\asymp n^d$ distant edges make the event $\{\exists e\colon n^2\widetilde{\mathcal{K}}_n(e)<-n^2\eta\}$ have probability $\to1$. $\square$ **Remark 8.16** (What the two constructions establish). Neither the Poisson window sprinkling (Lorentzian, Chapter 20) nor the jittered lattice (Riemannian, Definition 8.14) is a recovery sequence for $(\mathcal{M},g)$: the first converges (under H-conn and Conjecture 20.8) to the Lorentz-Finsler space $(U,d_{g,D}/c)$, and unconditionally has $\limsup d_{\tau,n}\leq(1-c_D/2)d_g$ along the frame (Prop 20.7), the second to a non-Riemannian normed limit (Proposition 8.15(c)). This is the bounded-degree obstruction of Open Problem O3, seen in both signatures: a bounded-degree graph with a fixed local law has, at large scales, the asymptotic cone of its local law, and no such cone is a Euclidean or Lorentzian norm unless the local law is isotropic in a sense that bounded degree forbids. Two weaker statements do hold. First, the *chart-embedded* action converges: if the sums of (8.2) are evaluated with displacements $\Delta x_e$ measured in $(\mathcal{M},g)$ through the chart embedding (rather than through the approximating maps of the actual mGH limit), then Lemmas 8.2-8.5 apply verbatim to the jittered lattice under (H13)(b) and (H4'-glue), and $\mathbb{E}[\mathcal{S}_n]\to\mathcal{S}_{\mathrm{EH}}[g]$; this is a statement about a functional of the embedded graph, not about the emergent geometry, and it is recorded here only to show that the curvature sum itself is not the obstruction. Second, if the limit is routed through Theorem 5.5 rather than Classical A, the rescaled (H2) is an additional failure for both constructions (Propositions 20.9, 8.15); replacing the pointwise bound by a locally averaged one is a possible repair of Part II and is recorded as the proposed Open Problem O3' (Chapter 22 currently lists O3 only). ### 8.5. Bianchi identity and conservation **Proposition 8.8** $\boxed{\text{Conditional (inheriting the hypotheses of 8.1 and of Prop 9.4)}}$ (Bianchi identity and matter conservation). *Under the hypotheses of Theorem 8.1 and Proposition 9.4, in the Riemannian sector (and, through Theorem 21.1, in the Lorentzian sector): (a) the QSN action (8.1) is invariant under graph automorphisms of $\mathsf{Q}_n$ and under local $\mathrm{Spin}(4,1)\times G_{\mathrm{SM}}$ gauge transformations at every node; (b) the limit functional $\mathcal{S}_{\mathrm{EH}}+\mathcal{S}_{\mathrm{mat}}$ of Theorem 8.1(i) and Proposition 9.4 is invariant under diffeomorphisms of $\mathcal{M}$; (c) consequently $\nabla^\mu\big(R_{\mu\nu}-\tfrac12Rg_{\mu\nu}+\Lambda_{\mathrm{b}}g_{\mu\nu}\big)=0$ identically and $\nabla^\mu\langle\widehat T_{\mu\nu}\rangle=0$ on solutions of the matter equations.* *Proof.* (a) Every term of (8.1) is a sum over edges and nodes of quantities determined by the graph structure, the tensions, and gauge-invariant traces of holonomies or node states: $\mathcal{K}(e)$ depends only on the graph and its out-neighbour measures (Axiom L3), $v(n)$ on the gauge-invariant node state, $S_{\mathrm{mat}}$ and $S_{\mathrm{gauge}}$ on gauge-invariant combinations $\bar\psi(n)U(e)\psi(m)$ and $\mathrm{tr}\,U_\square$ (Chapter 9). A graph automorphism permutes the summands; a local gauge transformation $h\colon N\to\mathrm{Spin}(4,1)\times G_{\mathrm{SM}}$ acts by $U(e)\mapsto h(n)U(e)h(m)^{-1}$, $\psi(n)\mapsto h(n)\psi(n)$, under which the listed combinations are invariant. (b) The limit functional is defined on mGH-isometry classes: by Corollary 5.7 (Riemannian clause) the limit $(\mathcal{M},g)$ is determined up to isometry, and the values $\mathcal{S}_{\mathrm{EH}}[g;U]$ of (8.7) are computed from the isometry class (they are integrals of curvature scalars against $\mathrm{dvol}_g$). For a diffeomorphism $\varphi$, $(\mathcal{M},\varphi^*g)$ is isometric to $(\mathcal{M},g)$ via $\varphi$, hence $\mathcal{S}_{\mathrm{EH}}[\varphi^*g]=\mathcal{S}_{\mathrm{EH}}[g]$; the same holds for $\mathcal{S}_{\mathrm{mat}}$ since the limit functional of Proposition 9.4 is likewise a function of the isometry class of $(g,\psi,A)$ under the simultaneous pull-back of all fields. (c) Take $\varphi_s$ the flow of a smooth vector field $X$ on the closed manifold $\mathcal{M}$. Then $\delta g^{\mu\nu}=-\mathcal{L}_Xg^{\mu\nu}=\nabla^\mu X^\nu+\nabla^\nu X^\mu$ and $0=\tfrac{\mathrm{d}}{\mathrm{d}s}\mathcal{S}_{\mathrm{EH}}[\varphi_s^*g]=\frac{1}{16\pi G}\int G_{\mu\nu}^\Lambda\,2\nabla^\mu X^\nu\sqrt{g}=-\frac{1}{8\pi G}\int(\nabla^\mu G^\Lambda_{\mu\nu})X^\nu\sqrt{g}$ with $G^\Lambda_{\mu\nu}:=R_{\mu\nu}-\tfrac12Rg_{\mu\nu}+\Lambda_{\mathrm{b}}g_{\mu\nu}$, for all $X$; hence $\nabla^\mu G^\Lambda_{\mu\nu}=0$. This is an identity (it holds off-shell because the variation of $\mathcal{S}_{\mathrm{EH}}$ alone vanishes under diffeomorphisms), and coincides with the contracted differential Bianchi identity $\nabla^\mu(R_{\mu\nu}-\tfrac12Rg_{\mu\nu})=0$ together with $\nabla g=0$. For the matter term, diffeomorphism invariance of $\mathcal{S}_{\mathrm{mat}}[g,\Phi]$ gives $0=\int\big(-\tfrac12\langle\widehat T_{\mu\nu}\rangle\,2\nabla^\mu X^\nu+\tfrac{\delta\mathcal{S}_{\mathrm{mat}}}{\delta\Phi}\mathcal{L}_X\Phi\big)\sqrt{g}$; on solutions of the matter equations $\delta\mathcal{S}_{\mathrm{mat}}/\delta\Phi=0$ and integration by parts gives $\nabla^\mu\langle\widehat T_{\mu\nu}\rangle=0$. $\square$ The discrete origin of (c) is (a): the finite-$n$ action has no preferred labelling of nodes and no preferred gauge, and the GHL topology forgets labellings; diffeomorphism invariance of the limit is the residue of relabelling invariance of the graph. The proof does not require approximating diffeomorphisms by graph automorphisms, which is in general impossible. ### 8.6. Newtonian limit **Proposition 8.9** $\boxed{\text{Conditional (inheriting the hypotheses of 8.1, and on Theorem 21.1)}}$ (Newtonian limit). *Let $d=4$ and let $g$ be a Lorentzian metric solving the continuation of (8.8) by Theorem 21.1 with $\Lambda_{\mathrm{b}}=0$ and a matter source $\langle\widehat T_{\mu\nu}\rangle=\rho\,u_\mu u_\nu+O(\rho v^2)$ describing slowly moving dust ($|v|\ll1$) with $\rho$ small in the sense that $g=\eta+h$ with $|h|,|\partial h|\ll1$ and $\partial_th=O(v\,\partial_xh)$. Then $h_{00}=-2\Phi$ with $\nabla^2\Phi=4\pi G\rho+O(\rho v^2,\,h\,\partial^2h)$, and test particles obey $\ddot x^i=-\partial_i\Phi$.* *Proof.* Linearise (8.8) around $\eta$. With the trace-reversed perturbation $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ and the harmonic gauge $\partial^\mu\bar h_{\mu\nu}=0$ (attainable by a diffeomorphism, Proposition 8.8(b)), the linearised Einstein tensor is $-\tfrac12\Box\bar h_{\mu\nu}$, so $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}$. For slow sources $\Box=-\partial_t^2+\nabla^2=\nabla^2+O(v^2\nabla^2)$ and $T_{00}=\rho$, $T_{0i},T_{ij}=O(\rho v,\rho v^2)$, hence $\nabla^2\bar h_{00}=-16\pi G\rho$, $\bar h_{0i},\bar h_{ij}=O(v)\bar h_{00}$. Then $h=-\bar h=\bar h_{00}+O(v^2)$ and $h_{00}=\bar h_{00}-\tfrac12\eta_{00}h=\tfrac12\bar h_{00}$, so $\nabla^2h_{00}=-8\pi G\rho$, i.e. $\nabla^2\Phi=4\pi G\rho$ with $h_{00}=-2\Phi$. The geodesic equation $\ddot x^\mu+\Gamma^\mu_{\alpha\beta}\dot x^\alpha\dot x^\beta=0$ with $\dot x\approx(1,v)$ gives $\ddot x^i=-\Gamma^i_{00}=\tfrac12\partial_ih_{00}=-\partial_i\Phi$ to leading order. $\square$ ### 8.7. Dimensional check and hypothesis discharge ledger **Remark 8.10** (Units). With $\hbar,c,G$ restored, $T(e)$ is dimensionless (Axiom L2 normalised as in Remark 3.7), $\mathcal{K}$ is dimensionless, $\Delta x_e$ has dimension of length and $\mathrm{Ric}(\Delta x,\Delta x)$ is dimensionless, so $\widetilde{\mathcal K}^{(n)}=n^2\widetilde{\mathcal K}_n$ is dimensionless once $n$ is read as $\ell_{\mathrm{macro}}/\ell_P$ with $\ell_{\mathrm{macro}}$ the unit of the limit metric. In $d$ spacetime dimensions $[G]=\mathrm{L}^{d-1}\mathrm{M}^{-1}\mathrm{T}^{-2}$, so $\ell_P^{d-2}:=\hbar G/c^3$ has dimension $\mathrm{L}^{d-2}$, and $[R\,\mathrm{dvol}]=\mathrm{L}^{d-2}$; the action (8.7) with $\hbar,c$ restored is $\mathcal{S}_{\mathrm{EH}}=\frac{c^3}{16\pi G}\int R\,\mathrm{dvol}$, and $\mathcal{S}_{\mathrm{EH}}/\hbar=\frac{1}{16\pi\ell_P^{d-2}}\int R\,\mathrm{dvol}$ is dimensionless. The discrete side is a pure number times $\alpha_d$, so $\alpha_d/\hbar$ has dimension $\ell_P^{2-d}$, consistent with $\alpha_d=2(d+2)c^3/(16\pi G)$ when the unit of $\Delta x$ is $\ell_P$. The Newtonian potential $\Phi$ of Proposition 8.9 is $c^2h_{00}/2$ and $\nabla^2\Phi=4\pi G\rho$ with $\rho$ a mass density. **Hypothesis discharge ledger, Chapter 8.** | Result | Classical inputs | Hypotheses consumed | Step where consumed | |---|---|---|---| | Lemma 8.2 | E (Thm 8.2.1), P (Thm 8.2.2), Gray (Thm 8.2.3) | H10 (a, b); H13(b) (c, by restatement) | terms I, II | | Lemma 8.3 | none beyond 8.2 | H10 (uniform continuity of Ric), H12 (direction tensors, node-density bound $C_N$), H13(b) ((8.5)), H1' | main term, error term, cell decomposition | | Lemma 8.4 | Nguyen-Zessin (general case only) | H12 (node density); Lemma 13.1 or H12c | constant case / general case | | Lemma 8.5 | none | H1 (neighbourhood count), H1' ($T_{\max}$), H12 ($C_N$), H13(a) (range $\rho_0$), range $[-2,1]$ (C14) | (8.10) | | Thm 8.1(i) | A (existence of the limit: doubling from Lem 5.3 and H3), E, P, Gray | H1', H3, H4' (Classical A), H10 ($C^3$, empty singular set, approximating maps), H12, H13 | Lemmas 8.3, 8.4 | | Thm 8.1(ii) | as (i) | as (i) | (i) | | Thm 8.1(iii) | none beyond 8.5 | H12, H13(a) | Chebyshev | | Thm 8.1(iv) | none (Palatini identity) | H10 ($C^3$ metric, closed manifold); hypotheses of Prop 9.4 for $\mathcal{S}_{\mathrm{mat}}$ | first variation | | Lemma 8.10' | Beem-Ehrlich-Easley Prop 2.3 | H-conn (existence of $\varphi_D$), cone clause of Thm 20.5(iv) (conjectural), Thm 20.5(i),(iii) | injectivity | | Prop 8.11 | P; Lemma 7.4 (inequality) / Conjecture E-Lor (equality); Prop 20.3, 20.6, 20.7, 20.9 | limit clause: H-conn, Conj 20.8; action clause: H10 ($C^3$), H13 (Lorentzian form), E-Lor for equality, upper bound only otherwise; H2 only with $K=-2$ (Prop 20.9); H4'(i) only as H-iso-win | Steps 1-6 | | Prop 8.12 | Thm 20.5(iv) (conjectural), Conj 20.8 | as 8.11; existence of $(a_\infty,b_\infty)$ assumed; H1 fails along the diagonal | double limit | | Prop 8.15 | D (Part I Thm 2.8), Bollobás-Leader, Burago-Burago-Ivanov 8.5.2, Nguyen-Zessin | H4'-glue for (d) on the glued graph | (a)-(h) | | Prop 8.8 | uniqueness of mGH limits up to isometry (Burago-Burago-Ivanov Thm 7.3.30; Cor 5.7, Riemannian clause, curvature-free) | inherits 8.1 and 9.4 | (b), (c) | | Prop 8.9 | none | inherits 8.1; Thm 21.1; $d=4$, $\Lambda_{\mathrm b}=0$, weak-field | linearisation | | Prop 8.11 (coefficients) | Conjecture E-Lor ($C^3$) | Lorentzian H13(b) | window average composed with Step 3 | Not discharged in this chapter, hence carried as hypotheses: H10 ($C^3$ regularity of the limit with empty singular set and identification of the approximating maps), H12 (tension isotropy; verified for the jittered lattice only relative to the chart embedding, Prop 8.15(f), and in its frame-dependent Lorentzian form for the window sprinkling), H13(b) (bias structure; open for both constructions, Remark 8.2.1), H4'-glue (Prop 8.15(d)), H12c (only if the node potential is not constant), Conjecture E-Lor (equality form of Lemma 7.4; used only in §8.4, where the proved inequality gives an upper bound only), H6$^+$ (not verified for the window sprinkling), H-conn and H-iso-win (Hyps 20.12, 20.13) and Conjecture 20.8 (all consumed only in §8.4, Errata C19/C21), Theorem 21.1 (Lorentzian reading). Not a hypothesis but an open existence question: H14 (neither construction has limit $(\mathcal{M},g)$; Open Problem O3). Appendix A must list H12, H12c, H12$_4$, H13, H14, H15, H16, H-hol, H4'-glue, H-conn, H-iso-win (Errata C13, C21). --- ## Chapter 9. Matter Fields, Chiral Fermions, and the Ginsparg-Wilson Relation The matter sector of Lach-Space is a lattice field theory on an irregular, curved, bounded-degree graph. Two facts about lattice fermions shape the chapter. The first is the Nielsen-Ninomiya theorem: on a translation-invariant lattice, a local Hermitian Dirac operator anticommuting with $\gamma_5$ describes an even number of species. The second is Neuberger's overlap construction, which evades the theorem by replacing exact anticommutation with the Ginsparg-Wilson relation, at the cost of locality becoming exponential rather than strict. The QSN has no translation group, so the first theorem does not apply to it as stated; the second construction nonetheless transfers verbatim, and gives a single chiral species with exponential locality controlled by the spectral gap of a Hermitian Wilson operator. This chapter proves that transfer. The chapter is written in the Euclidean (Wick-rotated) sector, where the structure group is the compact $\mathrm{Spin}(5)$ of Chapter 21, the spinor representation is unitary, and the Wilson operator is Hermitian. This is the setting in which lattice chirality is a theorem. The Lorentzian reading of every statement is obtained by the continuation of Theorem 21.1, and the continuum limit is taken with respect to the Riemannian limit of Theorem 5.5; statements that refer to the Lorentzian limit of Theorem 5.6 are marked as conditional on Theorem 21.1. ### 9.1. Spinors, edge transport, and the QSN covariant derivative **Definition 9.1.1 (Frame data).** By the MacDowell-Mansouri decomposition $\mathfrak{spin}(5)=\mathfrak{spin}(4)\oplus\mathbb{R}^4$ (and $\mathfrak{spin}(4,1)=\mathfrak{spin}(3,1)\oplus\mathbb{R}^{3,1}$ in the Lorentzian sector), the gravitational holonomy of an edge factorises to first order as $$ U_{\mathrm{grav}}(e)=\exp\big(\tfrac{i}{2}\omega_e^{ab}J_{ab}+i\ell_P^{-1}\epsilon_e^aP_a\big), \tag{9.1} $$ with $J_{ab}$ the Lorentz generators, $P_a=J_{a4}$ the de Sitter translations, $\omega_e$ the edge spin connection and $\epsilon_e^a$ the *edge tetrad*. The edge displacement in the local frame is $\Delta x_e^a:=\epsilon_e^a$, and the *edge length* is defined as the rescaled proper time, $a_e:=\tau_n(e)=2\pi\hbar/(n\Delta(e))$ (Axiom L6); the identification $a_e=|\Delta x_e|_g(1+o(1))$, which holds by construction for Definitions 8.14 and 20.1, is for a general QSN a consistency hypothesis on the frame data (review3b, item 22), and is assumed throughout this chapter. Under (H1') and Definition 8.1.1, $a_e\asymp n^{-1}$ on $\mathsf{Q}_n$; we write $a:=\max_ea_e$ for the lattice scale, which is of order $\ell_P$ in the unrescaled QSN. The edge tetrad is the datum that Theorem 8.1 used through $\Delta x_e$; here it additionally supplies the direction in which the Dirac operator differentiates. **Definition 9.1.2 (Spinor fields and transport).** A *spinor field* is $\psi\colon N\to S\otimes V$, with $S=\mathbb{C}^4$ the Dirac spinor module of $\mathrm{Spin}(4)$ (on which $\gamma^a$, $a=1,\dots,4$, satisfy $\{\gamma^a,\gamma^b\}=2\delta^{ab}$, $\gamma_5=\gamma^1\gamma^2\gamma^3\gamma^4$) and $V$ the Standard Model representation of Chapter 4 (48 Weyl components per generation, organised as in §17.2). The *edge transport* is $U(e)=\sigma(U_{\mathrm{grav}}(e))\otimes U_{\mathrm{SM}}(e)$ with $\sigma$ the spin representation; it is unitary on $S\otimes V$. For $e=(n,m)$ define the *parallel transport from $m$ to $n$* by $\mathcal{T}_e\psi(m):=U(e)\psi(m)$ and from $n$ to $m$ by $U(e)^{-1}=U(e)^\dagger$. The inner product on spinor fields is $\langle\phi,\psi\rangle=\sum_n\mu(n)\,\phi(n)^\dagger\psi(n)$ with $\mu(n)=n^{-d}$ the rescaled node weight. **Definition 9.1.3 (Edge weights).** For $e=(n,m)$ set $w_e:=T(e)\,a_e^{-2}\,\mu(n)$ and, to make the operators below symmetric, adopt the convention that every directed edge $e=(n,m)$ contributes to both endpoints: at $n$ with direction $+\Delta x_e$ and transport $U(e)$, at $m$ with direction $-\Delta x_e$ and transport $U(e)^\dagger$. Write $E(n)$ for the set of edges incident to $n$ (in either direction), $s_e(n)=\pm1$ for the orientation sign, and $U_e(n)$ for the transport into $n$ along $e$. **Definition 9.1.4 (QSN covariant derivative, naive Dirac, Laplacian).** The *covariant difference* along $e\in E(n)$ is $(\nabla_e\psi)(n):=U_e(n)\psi(m_e)-\psi(n)$. The *naive Dirac operator* and the *transported Laplacian* are $$ (D_0\psi)(n)=\frac{1}{2\mu(n)}\sum_{e\in E(n)}\widehat w_e\,s_e(n)\,\gamma_a\frac{\Delta x_e^a}{a_e}\,\big(U_e(n)\psi(m_e)\big),\qquad (L\psi)(n)=-\frac{1}{\mu(n)}\sum_{e\in E(n)}w_e\,(\nabla_e\psi)(n), \tag{9.2} $$ where $\widehat w_e:=w_ea_e$ is the first-order weight. The normalisation is such that, under (H15) below and for $\psi$ the restriction of a smooth spinor, $D_0\psi\to\gamma^a e_a^\mu\nabla_\mu\psi$ and $aL\psi\to0$ in norm (Theorem 9.1(v), Step 1). In (9.2) the tension enters as the edge weight of a Dirichlet form, which is the second role of Axiom L2. **Lemma 9.1.5** (Algebraic properties of $D_0$ and $L$). *(a) $L$ is self-adjoint and non-negative on $\ell^2(N,\mu)\otimes S\otimes V$. (b) $D_0$ is anti-self-adjoint: $D_0^\dagger=-D_0$. (c) $\gamma_5D_0\gamma_5=-D_0$ and $\gamma_5L\gamma_5=L$. (d) $\|L\|\leq 4D\,T_{\max}a_{\min}^{-2}$ and $\|D_0\|\leq 2D\,T_{\max}a_{\min}^{-1}$, where $a_{\min}=\min a_e$.* *Proof.* (a) $\langle\phi,L\psi\rangle=\sum_{e=(n,m)}w_e\big(\phi(n)-U(e)\phi(m)\big)^\dagger\big(\psi(n)-U(e)\psi(m)\big)$ by the two-endpoint convention and unitarity of $U(e)$, which is symmetric in $\phi,\psi$ and non-negative for $\phi=\psi$. (b) The edge $e=(n,m)$ contributes to $\langle\phi,D_0\psi\rangle$ the two terms $\tfrac12\widehat w_e\,\phi(n)^\dagger\gamma\cdot\hat v_e\,U(e)\psi(m)-\tfrac12\widehat w_e\,\phi(m)^\dagger\gamma\cdot\hat v_e\,U(e)^\dagger\psi(n)$ with $\hat v_e=\Delta x_e/a_e$; the adjoint expression $\langle D_0\phi,\psi\rangle$ contributes the same two terms with opposite sign because $\gamma\cdot\hat v_e$ is Hermitian and $U(e)$ unitary, and because $\sigma(U_{\mathrm{grav}})$ commutes with $\gamma\cdot\hat v_e$ only up to the rotation of $\hat v_e$ by $\omega_e$, which is the statement that $U(e)^\dagger(\gamma\cdot\hat v_e)U(e)=\gamma\cdot(R_e^{-1}\hat v_e)$ with $R_e$ the vector rotation; consistency of the frame data means that the direction of $e$ seen from $m$ is $-R_e^{-1}\hat v_e$, which is how $s_e(m)=-1$ and the transported direction enter, and with this the two contributions are exactly antisymmetric. (c) $\gamma_5$ anticommutes with each $\gamma_a$ and commutes with $\sigma(U_{\mathrm{grav}})$ (which is generated by products of two gammas) and with $U_{\mathrm{SM}}$; $L$ contains no gamma matrices. (d) Count terms: at most $2D$ incident edges per node, each contributing at most $2w_e/\mu(n)\leq 2T_{\max}a_{\min}^{-2}$ to the row sum of $L$ (the diagonal and off-diagonal entries of $-\nabla_e$) and $\widehat w_e/\mu(n)\leq T_{\max}a_{\min}^{-1}$ to that of $D_0$; the Schur test (row and column sums bounded by the same constant, by symmetry of the two-endpoint convention) gives $\|L\|\leq 2D\cdot2T_{\max}a_{\min}^{-2}$ and $\|D_0\|\leq 2D\cdot T_{\max}a_{\min}^{-1}$. $\square$ **Definition 9.1.6 (Wilson and Hermitian Wilson operators; overlap operator).** Fix the Wilson parameter $r\in(0,1]$ and the mass parameter $\rho\in(0,2r)$. The *Wilson-Dirac operator*, the *Hermitian Wilson operator*, and the *overlap Dirac operator* are $$ D_W:=D_0+\frac{ra}{2}L,\qquad H_W:=\gamma_5\Big(D_W-\frac{\rho}{a}\Big),\qquad D_{\mathrm{ov}}:=\frac{\rho}{a}\Big(\mathbb{1}+\gamma_5\,\mathrm{sign}(H_W)\Big), \tag{9.3} $$ where $\mathrm{sign}(H_W)$ is defined by the spectral theorem whenever $0\notin\sigma(H_W)$. The negative mass $-\rho/a$ is essential: without it the sign function is not defined on the low-lying modes, and with it the overlap operator reproduces $D_0$ on smooth fields with unit coefficient (Theorem 9.1(v)). The prefactor $\rho/a$ is the normalisation making the continuum limit the standard Dirac operator. **Hypothesis (H3') (Wilson gap).** There is $m_W>0$, independent of $n$, with $\sigma(H_W)\cap(-m_W/a,\,m_W/a)=\emptyset$. **Lemma 9.1.6'** (Finite dimensionality and operator domains). *Under (H1) and (H3), $N$ is finite, $\#N\leq D^{\lceil T\,T_{\max}\rceil+1}$ in the unrescaled graph, so $\ell^2(N,\mu)\otimes S\otimes V$ is finite-dimensional. Consequently every operator of this section ($D_0$, $L$, $D_W$, $H_W$, $\mathrm{sign}(H_W)$, $D_{\mathrm{ov}}$) is bounded and everywhere defined, "self-adjoint" means Hermitian matrix, and the spectral theorem invoked is the finite-dimensional one. The only unbounded operator in the chapter is the continuum $\slashed D_A$ of Theorem 9.1(v), whose domain is fixed there.* *Proof.* (H3) bounds $d_{\mathrm{eff}}$ by $T$, and $d_{\mathrm{eff}}\geq d_G/T_{\max}$ by (H1'), so $d_G\leq T\,T_{\max}$; a graph of degree $\leq D$ has at most $D^{k+1}$ nodes within graph distance $k$ of a point. $\square$ **Lemma 9.1.7** $\boxed{\text{Theorem}}$ (Hermiticity and boundedness of $H_W$). *$H_W$ is Hermitian with $\|H_W\|\leq C_H/a$, where $C_H=(2+2r)\,D\,T_{\max}\,(a/a_{\min})^2+\rho$ depends only on $(D,T_{\max},a/a_{\min},r,\rho)$. The edge-length eccentricity $a/a_{\min}$ is bounded by (H5) together with Axiom L6: in the Riemannian sector the edge length is $a_e=\tau_n(e)=2\pi\hbar/(n\Delta(e))$, so $a/a_{\min}=\Delta_{\max}/\Delta_{\min}$, which (H5) bounds.* *Proof.* $H_W^\dagger=(D_W-\rho/a)^\dagger\gamma_5=(-D_0+\tfrac{ra}{2}L-\rho/a)\gamma_5=\gamma_5(D_0+\tfrac{ra}{2}L-\rho/a)=H_W$ by Lemma 9.1.5(b),(c). By Lemma 9.1.5(d), $\|H_W\|\leq\|D_0\|+\tfrac{ra}{2}\|L\|+\rho/a\leq 2DT_{\max}a_{\min}^{-1}+2raDT_{\max}a_{\min}^{-2}+\rho a^{-1}\leq a^{-1}\big[2DT_{\max}(a/a_{\min})+2rDT_{\max}(a/a_{\min})^2+\rho\big]\leq C_H/a$ since $a/a_{\min}\geq1$. $\square$ **Proposition 9.1.8** $\boxed{\text{Theorem}}$ (Wilson gap on the hypercubic reference). *On the hypercubic lattice $a\mathbb{Z}^4$ with unit tensions, trivial transport, and the two-endpoint weights of Definition 9.1.3, (H3') holds with $m_W=\min(\rho,\,2r-\rho)$ for every $\rho\in(0,2r)$.* *Proof.* By translation invariance $H_W$ is diagonalised by plane waves $e^{ip\cdot x}$, $p\in(-\pi/a,\pi/a]^4$, with symbol $\widehat H_W(p)=\gamma_5\big(\tfrac{i}{a}\sum_\mu\gamma_\mu\sin(p_\mu a)+\tfrac{r}{a}\sum_\mu(1-\cos p_\mu a)-\tfrac\rho a\big)$. Since the $\gamma_\mu$ anticommute with each other and $\gamma_5$ squares to $1$, $\widehat H_W(p)^2=a^{-2}f(x)$ with $x_\mu:=\cos(p_\mu a)\in[-1,1]$ and $$ f(x)=\sum_{\mu=1}^4(1-x_\mu^2)+\Big(r\sum_{\mu=1}^4(1-x_\mu)-\rho\Big)^2 . $$ The map $p\mapsto x$ is onto $[-1,1]^4$, so $\min_p\widehat H_W(p)^2=a^{-2}\min_{[-1,1]^4}f$. For fixed values of the other three coordinates, $f$ is a quadratic polynomial in $x_\mu$ with leading coefficient $-1+r^2\leq0$, hence concave in $x_\mu$, and a concave function of one variable on a closed interval attains its minimum at an endpoint. Applying this to each coordinate in turn (a minimiser can be moved to an endpoint in coordinate $1$ without increasing $f$, then in coordinate $2$, and so on), the minimum of $f$ over the cube is attained at a vertex $x\in\{-1,1\}^4$. At a vertex with $k$ coordinates equal to $-1$, $\sum(1-x_\mu^2)=0$ and $\sum(1-x_\mu)=2k$, so $f=(2rk-\rho)^2$, $k\in\{0,\dots,4\}$. For $\rho\in(0,2r)$ the minimum over $k$ is $\min(\rho^2,(2r-\rho)^2)$, attained at $k=0$ or $k=1$. Hence $\min_p\widehat H_W(p)^2=a^{-2}\min(\rho,2r-\rho)^2$, and $\sigma(H_W)\cap(-m_W/a,m_W/a)=\emptyset$ with $m_W=\min(\rho,2r-\rho)$. $\square$ **Remark 9.1.9** (Status of (H3') on the QSN). Proposition 9.1.8 is the lattice prototype. On a QSN there is no Fourier transform, and (H3') is a hypothesis on the sequence; for the jittered lattice of Definition 8.14 with the stencil $\{\pm e_\mu\}$ it holds for $n$ large by Weyl's inequality, because $H_W$ differs from the hypercubic operator of Proposition 9.1.8 by a perturbation supported at the $2\%$ jittered sites whose norm is at most $C(D,T_{\max})\cdot(\text{relative jitter})/a\leq C/(4a)$, and the gap $m_W/a$ is stable under perturbations of norm $\rho_A$ (finite range) and $\sup_n\sum_m\|A_{nm}\|\leq\|A\|_{1\to1}$. For $z\notin\sigma(A)$ with $\eta:=\mathrm{dist}(z,\sigma(A))$, $$ \|(A-z)^{-1}_{nm}\|\leq\frac{2}{\eta}\exp\big(-\mu\,d_G(n,m)\big),\qquad\mu=\frac{1}{\rho_A}\log\Big(1+\frac{\eta}{2\|A\|_{1\to1}}\Big), \tag{9.4} $$ (Combes, Thomas, *Commun. Math. Phys.* **34** (1973) 251; the graph version with the stated constants is in Aizenman, Warzel, *Random Operators*, AMS, 2015, Thm. 10.5). **Classical J (Integral representation of the sign function).** For a self-adjoint $H$ with $\sigma(H)\cap(-m,m)=\emptyset$, $$ \mathrm{sign}(H)=\frac{2}{\pi}\int_0^\infty H\,(H^2+t^2)^{-1}\,\mathrm{d}t, \tag{9.5} $$ with the integral norm-convergent. Proof: for a real number $h$ with $|h|\geq m$, $\frac2\pi\int_0^\infty h/(h^2+t^2)\,\mathrm{d}t=\frac2\pi\cdot\mathrm{sign}(h)\cdot\frac\pi2=\mathrm{sign}(h)$, and the spectral theorem (finite-dimensional here, Lemma 9.1.6') transfers the identity to $H$; norm convergence follows from $\|H(H^2+t^2)^{-1}\|\leq\|H\|/(m^2+t^2)$. The use of (9.5) with a Combes-Thomas bound to prove exponential locality of the overlap operator is Hernández, Jansen, Lüscher, *Nucl. Phys. B* **552** (1999) 363, §3, whose argument Theorem 9.1(iii) reproduces with explicit constants. **Classical K (Neuberger).** Let $\epsilon$ be a self-adjoint unitary, $\epsilon=\epsilon^\dagger=\epsilon^{-1}$, and set $V:=\gamma_5\epsilon$, so that $V$ is unitary and $\gamma_5V\gamma_5=\epsilon\gamma_5=V^\dagger$. Then $D=\tfrac{\rho}{a}(1+V)$ satisfies $\gamma_5D\gamma_5=D^\dagger$ and the Ginsparg-Wilson relation $\gamma_5D+D\gamma_5=\tfrac{a}{\rho}D\gamma_5D$ (Neuberger, *Phys. Lett. B* **417** (1998) 141; Ginsparg, Wilson, *Phys. Rev. D* **25** (1982) 2649). The overlap operator (9.3) is of this form with $\epsilon=\mathrm{sign}(H_W)$. **Theorem 9.1** $\boxed{\text{(i)-(iv$_{\mathrm a}$): Theorem modulo Classical I, J, K, Conditional on H3'. (v): Conditional on H3', H10, H12, H15, H-hol. (iv$_{\mathrm b}$): Conditional additionally on H16; (v) also modulo Classical A and Chernoff}}$ (Overlap spectral theorem on the QSN). *Let $\mathsf{Q}_n$ satisfy (H1'), (H2), (H3), (H4'), (H5), and let (H3') hold with gap $m_W$. For (iv$_{\mathrm b}$) and (v) assume in addition (H10) with empty singular set, so that the Riemannian limit $(\mathcal{M},g)$ of Theorem 5.5 is a compact smooth manifold with $C^2$ metric, (H12), and the holonomy hypothesis (H-hol) below. Then the overlap operator (9.3) is well defined (Lemma 9.1.6') and:* *(i) ($\gamma_5$-Hermiticity) $D_{\mathrm{ov}}^\dagger=\gamma_5D_{\mathrm{ov}}\gamma_5$; equivalently $\gamma_5D_{\mathrm{ov}}$ is self-adjoint.* *(ii) (Ginsparg-Wilson relation) $\gamma_5D_{\mathrm{ov}}+D_{\mathrm{ov}}\gamma_5=\dfrac{a}{\rho}\,D_{\mathrm{ov}}\gamma_5D_{\mathrm{ov}}$.* *(iii) (Exponential locality) $\|(D_{\mathrm{ov}})_{nm}\|\leq\dfrac{C_{\mathrm{loc}}}{a}\exp\big(-\xi_{\mathrm{loc}}\,d_G(n,m)\big)$ with $\xi_{\mathrm{loc}}=\tfrac12\log\big(1+\tfrac{m_W^2}{2C_H^2}\big)>0$ and $C_{\mathrm{loc}}=2\rho(2D+1)C_He^{\xi_{\mathrm{loc}}}/m_W$, both independent of $n$.* *(iv$_{\mathrm a}$) (Spectrum and index) $\sigma(D_{\mathrm{ov}})\subset\{\tfrac{\rho}{a}(1+e^{i\theta})\colon\theta\in[0,2\pi)\}$, the Ginsparg-Wilson circle. The operator $\gamma_5D_{\mathrm{ov}}$ is self-adjoint with spectrum in $[-2\rho/a,2\rho/a]$, and $\ker D_{\mathrm{ov}}$ is $\gamma_5$-invariant with $n_--n_+=\tfrac12\mathrm{Tr}\,\mathrm{sign}(H_W)$, where $n_\pm=\dim\ker D_{\mathrm{ov}}\cap\{\gamma_5=\pm1\}$.* *(iv$_{\mathrm b}$) (Low-energy sector) Under (H15) and (H-hol), for every $\Lambda>0$ not an eigenvalue of $\gamma_5\slashed D_A$, the spectral projection $\Pi^{(n)}_\Lambda$ of $\gamma_5D_{\mathrm{ov}}$ onto $[-\Lambda,\Lambda]$ satisfies $\|\Pi^{(n)}_\Lambda J_n\psi-J_n\Pi_\Lambda\psi\|_n\to0$ for all smooth $\psi$: the low-energy sector of the continuum species is contained in the low-energy sector of $D_{\mathrm{ov}}$. Under the additional hypothesis (H16) (no spectral pollution), $\mathrm{rank}\,\Pi^{(n)}_\Lambda=\mathrm{rank}\,\Pi_\Lambda$ for $n$ large: there are no additional low-energy species.* *(v) (Continuum limit) Let $(\mathcal{M},g)$ be the compact Riemannian limit of Theorem 5.5 (singular set empty by (H10)) and $J_n\colon C^\infty(\mathcal{M};S\otimes V)\to\ell^2(N_n,\mu)\otimes S\otimes V$ the restriction map along $\iota_n$ with the frame transported from the regular set. Then for all $\phi,\psi\in C^\infty(\mathcal{M};S\otimes V)$,* $$ \lim_{n\to\infty}\langle J_n\phi,\,D_{\mathrm{ov}}J_n\psi\rangle_n=\int_{\mathcal{M}}\phi^\dagger\,\gamma^ae_a^\mu\nabla_\mu^{A}\psi\,\mathrm{dvol}_g=\langle\phi,\slashed{D}_A\psi\rangle, \tag{9.6} $$ *where $\nabla^A$ is the spin-and-gauge covariant derivative of the continuum connection $A$ of (H-hol), and $\slashed D_A$ denotes the closure of $\gamma^ae_a^\mu\nabla^A_\mu$ from $C^\infty(\mathcal{M};S\otimes V)$: on a compact manifold with $C^2$ metric and continuous connection this operator is essentially self-adjoint after multiplication by $\gamma_5$ (it is a first-order elliptic operator with symmetric symbol; Chernoff, *J. Funct. Anal.* **12** (1973) 401, Thm 2.2, which requires only completeness and finite propagation speed), with domain $H^1(\mathcal{M};S\otimes V)$ and compact resolvent. The convergence (9.6) and its strong form $\|D_{\mathrm{ov}}J_n\psi-J_n\slashed D_A\psi\|_n\to0$ both require (H15); under (H15) the convergence upgrades to generalised strong resolvent convergence of $\gamma_5D_{\mathrm{ov}}$ to $\gamma_5\slashed{D}_A$.* **Hypothesis (H-hol) (Limiting connection).** There is a continuous connection $A$ on $S\otimes V\to\mathcal{M}$ such that, for every edge $e=(p,q)$, $U(e)=\mathbb{1}+\Delta x_e^\mu A_\mu(\iota_n p)+O(a^2)$ uniformly, and likewise $U^{(i)}(e)=\exp(\Delta x^\mu_eA^{(i)}_\mu+O(a^2))$ for each gauge factor. This is the holonomy analogue of (H10); it is not a consequence of Theorem 5.5, which controls the metric and not the edge holonomies. **Hypothesis (H15) (Directional consistency).** For every $\psi\in C^\infty(\mathcal{M};S\otimes V)$: (a) $\|D_0J_n\psi-J_n\slashed{D}_A\psi\|_n\to0$; (b) $\|aLJ_n\psi\|_n\to0$; (c) $\|LJ_n\psi\|_n+\|D_0^2J_n\psi\|_n+a\|D_0LJ_n\psi\|_n+a\|LD_0J_n\psi\|_n\leq C(\psi)$ uniformly in $n$; (d) (per-node isotropy) $\sup_p\big|\sum_{e\in E(p)}T(e)\,\hat v_e\otimes\hat v_e-g^{-1}(\iota_np)\big|\to0$ and $\sup_p\big|\sum_{e\in E(p)}T(e)a_e^{-1}s_e(p)\hat v_e\big|\cdot a\to0$ (vanishing weighted first moment). Clause (d) is the per-node form of (H12) that Step 1 of the proof of (v) and Proposition 9.4 consume (review3b, item 17). Clause (c) is the second-order consistency needed to control the remainder of the sign-function expansion (review3, item 21). (H15) holds for sequences whose local structure is a deterministic pattern with isotropic moments and vanishing weighted first moment at every node, such as the jittered lattice of Definition 8.14 with a centrally symmetric stencil when the jitter is scaled to zero, and for sprinklings with degree $D_n\to\infty$; it *fails* for bounded-degree Poisson sprinklings and for the jittered lattice with fixed relative jitter, where the single-node weighted first moment $\sum_eT(e)a_e^{-2}\Delta x_e$ does not vanish and $(LJ_n\psi)(p)$ has a first-order term of size $O(a^{-1})$. In those cases none of (9.6), (iv$_{\mathrm b}$), or the strong statement is established, and the matter-sector results are Conditional on (H15). **Definition (identification operators).** Let $\{V_p\}_{p\in N_n}$ be the Voronoi partition of $\mathcal{M}$ with respect to $\iota_n(N_n)$ and $d_g$, and set the node weight $\mu(p):=\mathrm{vol}_g(V_p)$ (consistent with the node-density clause of (H12), which then reads $\max_p\mathrm{diam}(V_p)\to0$ and $\mu(p)\asymp n^{-d}$). Define the bounded *cell-averaging operator* $\widehat J_n\colon L^2(\mathcal{M};S\otimes V)\to\ell^2(N_n,\mu)\otimes S\otimes V$ by $(\widehat J_n\psi)(p):=\mu(p)^{-1}\int_{V_p}\mathcal{P}_{x\to p}\psi(x)\,\mathrm{dvol}_g$, with $\mathcal{P}_{x\to p}$ the parallel transport of the frame and gauge bundle along the minimising geodesic from $x$ to $\iota_n(p)$ (unique for $n$ large). Then $\|\widehat J_n\|\leq1$, $\widehat J_n\widehat J_n^*=\mathbb{1}$ on $\ell^2(N_n,\mu)$ (so $\widehat J_n^*\widehat J_n$ is the orthogonal projection onto transported-cell-constant sections), $\widehat J_n^*\widehat J_n\to\mathbb{1}$ strongly on $L^2$ as the cells shrink, and $\|\widehat J_n\psi-J_n\psi\|_n\to0$ for $\psi\in C^1$ (the average over a cell of diameter $o(1)$ of a $C^1$ section differs from its value at the centre by $o(1)$). These are the identification operators of Weidmann (*Math. Scand.* **54** (1984) 51, §2), and (H15), (H16) are understood with $\widehat J_n$ in place of $J_n$; for smooth $\psi$ the two agree up to $o(1)$ in norm, so (H15) is unchanged in content (review3b, item 16). **Hypothesis (H16) (No spectral pollution).** For $z\notin\mathbb{R}$, $\|(\gamma_5D_{\mathrm{ov}}-z)^{-1}(\mathbb{1}-\widehat J_n\widehat J_n^*)\|\to0$ on $\ell^2(N_n,\mu)\otimes S\otimes V$, where $\widehat J_n\widehat J_n^*=\mathbb{1}$ makes the condition vacuous as stated; the non-trivial form is $\|(\gamma_5D_{\mathrm{ov}}-z)^{-1}-\widehat J_n(\gamma_5\slashed D_A-z)^{-1}\widehat J_n^*\|\to0$, i.e. generalised norm-resolvent convergence in Weidmann's sense (*loc. cit.*, Def. 2.5 and Thm 2.7), which is what we take (H16) to mean. It is strictly stronger than the generalised strong resolvent convergence furnished by (H15) (Weidmann, Thm 2.3), and it yields convergence of spectral projections in norm and hence of their ranks. (H16) is the precise hypothesis excluding doublers; it is not implied by strong resolvent convergence (review3, item 22). *Proof of (i).* Set $V:=\gamma_5\,\mathrm{sign}(H_W)$. Since $H_W$ is self-adjoint with a gap, $\mathrm{sign}(H_W)$ is self-adjoint and unitary. Then $V^\dagger=\mathrm{sign}(H_W)\gamma_5$ and $\gamma_5V\gamma_5=\mathrm{sign}(H_W)\gamma_5=V^\dagger$, so Classical K applies: $D_{\mathrm{ov}}^\dagger=\tfrac\rho a(1+V^\dagger)=\tfrac\rho a(1+\gamma_5V\gamma_5)=\gamma_5D_{\mathrm{ov}}\gamma_5$. Explicitly, $(\gamma_5D_{\mathrm{ov}})^\dagger=D_{\mathrm{ov}}^\dagger\gamma_5=\gamma_5D_{\mathrm{ov}}\gamma_5\gamma_5=\gamma_5D_{\mathrm{ov}}$. $\square$ *Proof of (ii).* With $\epsilon:=\mathrm{sign}(H_W)$, $V:=\gamma_5\epsilon$ and $D:=D_{\mathrm{ov}}=\tfrac\rho a(1+V)$: $V\gamma_5V=\gamma_5\epsilon\gamma_5\gamma_5\epsilon=\gamma_5\epsilon^2=\gamma_5$. Hence $D\gamma_5D=\tfrac{\rho^2}{a^2}(1+V)\gamma_5(1+V)=\tfrac{\rho^2}{a^2}(\gamma_5+V\gamma_5+\gamma_5V+V\gamma_5V)=\tfrac{\rho^2}{a^2}(2\gamma_5+V\gamma_5+\gamma_5V)$, while $\gamma_5D+D\gamma_5=\tfrac\rho a(2\gamma_5+\gamma_5V+V\gamma_5)$. Therefore $\gamma_5D+D\gamma_5=\tfrac a\rho D\gamma_5D$. $\square$ *Proof of (iii).* By Classical J, $D_{\mathrm{ov}}=\tfrac\rho a\big(1+\tfrac2\pi\gamma_5\int_0^\infty H_W(H_W^2+t^2)^{-1}\mathrm{d}t\big)$. Set $A:=H_W^2$, which is self-adjoint, non-negative, of range $\rho_A=2$ (since $H_W$ has range $1$: each of $D_0$, $L$ couples only adjacent nodes), with $\|A\|_{1\to1}\leq C_H^2a^{-2}$ by Lemma 9.1.7 and $\sigma(A)\subset[m_W^2a^{-2},C_H^2a^{-2}]$ by (H3'). For $z=-t^2$, $\eta=\mathrm{dist}(-t^2,\sigma(A))=m_W^2a^{-2}+t^2$. Classical I gives $$ \|(A+t^2)^{-1}_{km}\|\leq\frac{2}{m_W^2a^{-2}+t^2}\exp\big(-\mu(t)d_G(k,m)\big),\qquad\mu(t)=\tfrac12\log\Big(1+\frac{m_W^2a^{-2}+t^2}{2C_H^2a^{-2}}\Big)\geq\mu(0)=\tfrac12\log\Big(1+\frac{m_W^2}{2C_H^2}\Big). $$ Then, since $H_W$ has range $1$ and $\|(H_W)_{nk}\|\leq C_Ha^{-1}$, $$ \|\mathrm{sign}(H_W)_{nm}\|\leq\frac2\pi\sum_{k\colon d_G(n,k)\leq1}\|(H_W)_{nk}\|\int_0^\infty\|(A+t^2)^{-1}_{km}\|\mathrm{d}t\leq\frac2\pi(2D+1)\frac{C_H}{a}\,e^{-\mu(0)(d_G(n,m)-1)}\int_0^\infty\frac{2\,\mathrm{d}t}{m_W^2a^{-2}+t^2}. $$ The integral equals $\pi a/m_W$. Hence $\|\mathrm{sign}(H_W)_{nm}\|\leq\tfrac{2(2D+1)C_H}{m_W}e^{\mu(0)}e^{-\mu(0)d_G(n,m)}$, and multiplying by $\rho/a$ (and noting $\gamma_5$ is node-diagonal) gives (iii) with $\xi_{\mathrm{loc}}=\mu(0)=\tfrac12\log(1+m_W^2/(2C_H^2))$ and $C_{\mathrm{loc}}=2\rho(2D+1)C_He^{\xi_{\mathrm{loc}}}/m_W$. The constants depend on $n$ only through $a$, which appears as an overall factor, as claimed. $\square$ *Proof of (iv$_{\mathrm a}$).* By (i) and (ii), $D:=D_{\mathrm{ov}}$ is normal: $DD^\dagger=D\gamma_5D\gamma_5=\tfrac\rho a(\gamma_5D+D\gamma_5)\gamma_5=\tfrac\rho a(\gamma_5D\gamma_5+D)=\tfrac\rho a(D^\dagger+D)$, and likewise $D^\dagger D=\gamma_5D\gamma_5D=\tfrac\rho a\gamma_5(\gamma_5D+D\gamma_5)=\tfrac\rho a(D+\gamma_5D\gamma_5)=\tfrac\rho a(D+D^\dagger)$, so $[D,D^\dagger]=0$. Since $V=\tfrac a\rho D-1$ is unitary, $\sigma(D)=\tfrac\rho a(1+\sigma(V))\subset\tfrac\rho a(1+S^1)$. For $\gamma_5D=\tfrac\rho a(\gamma_5+\epsilon)$, both summands are self-adjoint unitaries, so $\gamma_5D$ is self-adjoint with $\|\gamma_5D\|\leq2\rho/a$. On $\ker D$, $V=-1$, i.e. $\gamma_5\epsilon=-1$, so $\epsilon=-\gamma_5$ there and $\ker D$ is $\gamma_5$-invariant. Index: decompose the space as $\ker D\oplus E_+\oplus E_{\mathrm{rest}}$, with $E_+=\ker(V-1)$, on which $\epsilon=\gamma_5$, and $E_{\mathrm{rest}}$ the sum of the eigenspaces of $V$ for eigenvalues $e^{i\theta}\neq\pm1$; $E_{\mathrm{rest}}$ is invariant under $\gamma_5$ (which maps the $e^{i\theta}$-eigenspace to the $e^{-i\theta}$-eigenspace, since $\gamma_5V\gamma_5=V^\dagger$) and under $\epsilon=\gamma_5V$, and on the two-dimensional blocks spanned by an $e^{i\theta}$-eigenvector and its $\gamma_5$-image both $\gamma_5$ and $\epsilon$ have trace zero (each interchanges the two basis vectors up to a phase, so has zero diagonal). Let $n_\pm$ and $m_\pm$ be the dimensions of the $\gamma_5=\pm1$ eigenspaces of $\ker D$ and $E_+$. Then $\mathrm{Tr}\,\gamma_5=(n_+-n_-)+(m_+-m_-)$ and $\mathrm{Tr}\,\epsilon=-(n_+-n_-)+(m_+-m_-)$. Since $\gamma_5$ has trace zero on the full space (it is $\mathrm{diag}(1,1,-1,-1)$ on each node's spinor factor), $m_+-m_-=-(n_+-n_-)$, hence $\mathrm{Tr}\,\epsilon=-2(n_+-n_-)$, i.e. $n_--n_+=\tfrac12\mathrm{Tr}\,\mathrm{sign}(H_W)$ (Hasenfratz, Laliena, Niedermayer, *Phys. Lett. B* **427** (1998) 125, eq. (8), in this sign convention). $\square$ *Proof of (iv$_{\mathrm b}$).* Under (H15) and (H-hol), (v) gives generalised strong resolvent convergence of $\gamma_5D_{\mathrm{ov}}$ to $\gamma_5\slashed D_A$. By the spectral mapping for generalised strong resolvent convergence (Weidmann, *loc. cit.*, Thm 2.3, the varying-space form of Reed, Simon, *Methods of Modern Mathematical Physics I*, Thm VIII.24), for every interval $[-\Lambda,\Lambda]$ whose endpoints are not eigenvalues of $\gamma_5\slashed D_A$, $\|\Pi^{(n)}_\Lambda J_n\psi-J_n\Pi_\Lambda\psi\|_n\to0$. This is the containment statement. Strong convergence does not control vectors asymptotically orthogonal to the range of $J_n$, which is exactly where a doubler would live. Under (H16), which is generalised norm-resolvent convergence, spectral projections onto intervals with endpoints in the resolvent set converge in norm, $\|\Pi^{(n)}_\Lambda-\widehat J_n\Pi_\Lambda\widehat J_n^*\|\to0$ (Weidmann, Thm 2.7 and the Riesz-projection contour integral); since $\Pi_\Lambda$ has finite rank (compact resolvent of $\gamma_5\slashed D_A$ on the compact $\mathcal{M}$), projections at norm distance $<1$ have equal rank, giving $\mathrm{rank}\,\Pi^{(n)}_\Lambda=\mathrm{rank}\,\Pi_\Lambda$ for $n$ large. $\square$ *Proof of (v).* Throughout, (H15) and (H-hol) are assumed; without (H15)(c) the remainder estimate in Step 2 fails (review3, item 21), and without (H15)(a) the bilinear form $\langle J_n\phi,D_0J_n\psi\rangle_n$ need not converge on a bounded-degree random graph, because the per-node discrete gradient is a noisy estimator of $\nabla\psi$ whose noise is weighted by the second smooth field and does not average out unless the weighted first moments vanish. *Step 1 (naive operator).* By (H15)(a), $\|D_0J_n\psi-J_n\slashed D_A\psi\|_n\to0$, and by (H15)(b), $\|\tfrac{ra}{2}LJ_n\psi\|_n\to0$; hence $\|D_WJ_n\psi-J_n\slashed D_A\psi\|_n\to0$. For the record, the expansion behind (H15)(a) is $U(e)\psi(q)=\psi(p)+\Delta x_e^\mu\nabla^A_\mu\psi(p)+O(a^2)$ by (H-hol) and $C^2$ regularity of $\psi$; inserted in (9.2), $(D_0J_n\psi)(p)$ consists of a zeroth-order part $\tfrac{1}{2\mu(p)}\sum_e\widehat w_es_e\gamma\cdot\hat v_e\,\psi(p)$, which is $a^{-1}$ times the weighted first moment $\sum_eT(e)s_e\hat v_e$ at $p$ and which (H15)(a) requires to be $o(1)$, a first-order part $\gamma_a\big(\sum_eT(e)\hat v_e^a\hat v_e^\mu\big)\nabla^A_\mu\psi(p)$, which (H15)(a) requires to equal $\gamma^ae_a^\mu\nabla_\mu\psi$ up to $o(1)$ (the per-node form of (H12)), and an $O(a)$ remainder; (H15)(d) is exactly the statement that the zeroth-order part is $o(1)$ and the first-order tensor is $g^{-1}+o(1)$. *Step 2 (overlap operator).* Write $H_W=A_0+B$ with $A_0:=-\tfrac\rho a\gamma_5$ and $B:=\gamma_5D_W$. Then $A_0^2=\rho^2a^{-2}$ is scalar, and (9.5) gives $\mathrm{sign}(H_W)=\tfrac2\pi\int_0^\infty(A_0+B)R_t\,\mathrm{d}t$, $R_t:=((A_0+B)^2+t^2)^{-1}$. With $c_t:=(\rho^2a^{-2}+t^2)^{-1}$, the second resolvent identity gives $R_t=c_t-c_tR_t(A_0B+BA_0+B^2)$, hence for any vector $\chi$, $$ \mathrm{sign}(H_W)\chi=\frac2\pi\int_0^\infty c_t(A_0+B)\chi\,\mathrm{d}t-\frac2\pi\int_0^\infty c_t(A_0+B)R_t(A_0B+BA_0+B^2)\chi\,\mathrm{d}t=:\mathrm{(I)}+\mathrm{(II)}. $$ Since $\int_0^\infty c_t\,\mathrm{d}t=\tfrac{\pi a}{2\rho}$, $\mathrm{(I)}=\tfrac a\rho(A_0+B)\chi=-\gamma_5\chi+\tfrac a\rho\gamma_5D_W\chi$, so $\tfrac\rho a(\chi+\gamma_5\,\mathrm{(I)})=D_W\chi$: the first term of the expansion reproduces $D_W$ exactly. For (II): $A_0B+BA_0=-\tfrac\rho a(\gamma_5\gamma_5D_W+\gamma_5D_W\gamma_5)=-\tfrac\rho a(D_W+\gamma_5D_W\gamma_5)=-\tfrac\rho a\cdot raL=-\rho rL$ by Lemma 9.1.5(c), and $B^2=\gamma_5D_W\gamma_5D_W=(-D_0+\tfrac{ra}{2}L)(D_0+\tfrac{ra}{2}L)=-D_0^2+\tfrac{ra}{2}(LD_0-D_0L)+\tfrac{r^2a^2}{4}L^2$. Take $\chi=J_n\psi$. By (H15)(c) and Lemma 9.1.5(d), $$ \|(-\rho rL+B^2)J_n\psi\|_n\leq\rho rC(\psi)+C(\psi)+rC(\psi)+\tfrac{r^2a^2}{4}\|L\|\,\|LJ_n\psi\|_n\leq(1+r+\rho r)C(\psi)+r^2DT_{\max}(a/a_{\min})^2\,C(\psi)=:C'(\psi), $$ uniformly in $n$. Also $\|A_0+B\|=\|H_W\|\leq C_Ha^{-1}$ (Lemma 9.1.7) and $\|R_t\|\leq(m_W^2a^{-2}+t^2)^{-1}$ by (H3'). Hence, with $\int_0^\infty\frac{\mathrm{d}t}{(\alpha^2+t^2)(\beta^2+t^2)}=\frac{\pi}{2\alpha\beta(\alpha+\beta)}$, $$ \|\mathrm{(II)}\|_n\leq\frac2\pi\,C'(\psi)\,\frac{C_H}{a}\int_0^\infty\frac{\mathrm{d}t}{(\rho^2a^{-2}+t^2)(m_W^2a^{-2}+t^2)}=\frac2\pi C'(\psi)\frac{C_H}{a}\cdot\frac{\pi a^3}{2\rho m_W(\rho+m_W)}=\frac{C'(\psi)C_H\,a^2}{\rho m_W(\rho+m_W)}. $$ Therefore $$ \|D_{\mathrm{ov}}J_n\psi-D_WJ_n\psi\|_n=\frac\rho a\|\gamma_5\,\mathrm{(II)}\|_n\leq\frac{C'(\psi)C_H}{m_W(\rho+m_W)}\,a\to0, $$ and with Step 1, $\|D_{\mathrm{ov}}J_n\psi-J_n\slashed D_A\psi\|_n\to0$. Pairing with $J_n\phi$ and using $\|J_n\phi\|_n\to\|\phi\|$ and $\langle J_n\phi,J_n\chi\rangle_n\to\langle\phi,\chi\rangle$ for continuous $\chi$ (node-density clause of (H12)) gives (9.6). *Step 3 (strong resolvent convergence).* $C^\infty(\mathcal{M})$ is a core for the self-adjoint $\gamma_5\slashed D_A$ (essential self-adjointness on the compact $\mathcal{M}$, fixed in the statement of (v)), and $J_n$ are asymptotically isometric on continuous spinors; the Trotter-Kato theorem in the varying-space form (Weidmann, *loc. cit.*, Thm 2.3; Kuwae, Shioya, *Commun. Anal. Geom.* **11** (2003) 599, Thm 2.4) then gives generalised strong resolvent convergence of $\gamma_5D_{\mathrm{ov}}$ to $\gamma_5\slashed D_A$ from the core convergence just proved. $\square$ **Remark 9.2.2** (Lorentzian reading). Under Theorem 21.1, the Euclidean operator $\slashed D_A$ on the regular set of the Riemannian limit continues to $i\gamma^\mu\nabla^A_\mu$ on the regular set of the Lorentzian limit of Theorem 5.6, and the Ginsparg-Wilson relation, being algebraic, is preserved. The locality bound (iii) is a bound on a matrix kernel in graph distance and does not refer to the signature. The spectral statements (iv) use self-adjointness of $\gamma_5D_{\mathrm{ov}}$, which is a Euclidean property; their Lorentzian counterparts are statements about the $\gamma^0$-Hermitian operator and are not asserted here. ### 9.3. Evasion of the Nielsen-Ninomiya theorem **Corollary 9.2** $\boxed{\text{Theorem modulo Classical I, J, K, Conditional on H3' (and on H10, H12, H15, H-hol, H16 for (d))}}$ (Nielsen-Ninomiya evasion). *The QSN admits a Dirac operator $D_{\mathrm{ov}}$ that is (a) exponentially local with $n$-independent decay length $\xi_{\mathrm{loc}}^{-1}$ lattice units, (b) $\gamma_5$-Hermitian, (c) invariant under the Lüscher chiral transformation $\delta\psi=\gamma_5(1-\tfrac{a}{2\rho}D_{\mathrm{ov}})\psi$, $\delta\bar\psi=\bar\psi(1-\tfrac{a}{2\rho}D_{\mathrm{ov}})\gamma_5$, and (d) under (H15), (H-hol), (H16), free of doubler species in the low-energy spectrum. The Nielsen-Ninomiya theorem is not contradicted: of its four hypotheses, namely (NN1) locality, (NN2) translation invariance on a lattice $\mathbb{Z}^d$ with the resulting Brillouin torus, (NN3) Hermiticity in the sense $\gamma_5$-Hermiticity, (NN4) exact anticommutation $\{D,\gamma_5\}=0$, hypothesis (NN4) fails for $D_{\mathrm{ov}}$ by Theorem 9.1(ii), and hypothesis (NN2) fails for every QSN, which has no translation group and no Brillouin torus.* *Proof.* (a) is Theorem 9.1(iii), (b) is (i), (d) is (iv$_{\mathrm b}$). For (c), using (ii): $\bar\psi D\psi\mapsto\bar\psi D\psi+\bar\psi(1-\tfrac{a}{2\rho}D)\gamma_5D\psi+\bar\psi D\gamma_5(1-\tfrac{a}{2\rho}D)\psi=\bar\psi D\psi+\bar\psi\big(\gamma_5D+D\gamma_5-\tfrac a\rho D\gamma_5D\big)\psi=\bar\psi D\psi$ (Lüscher, *Phys. Lett. B* **428** (1998) 342). For the precise form of the Nielsen-Ninomiya theorem, see Nielsen, Ninomiya, *Nucl. Phys. B* **185** (1981) 20, and the homotopy-theoretic proof in Friedan, *Commun. Math. Phys.* **85** (1982) 481: the theorem states that a Hermitian, local, translation-invariant lattice Dirac operator on $\mathbb{Z}^d$ that anticommutes with $\gamma_5$ has a Fourier symbol $\widehat D(p)$ which is a continuous map from the Brillouin torus $T^d$ to the Clifford algebra, whose zeros carry chirality indices summing to zero by the Poincaré-Hopf theorem on $T^d$, forcing an even number of zeros. The argument requires the torus (NN2) to define the symbol and to apply Poincaré-Hopf, and (NN4) to make the zeros carry a well-defined chirality. $D_{\mathrm{ov}}$ violates (NN4) and the QSN lacks (NN2); either failure suffices. The positive content is not the failure of a hypothesis but Theorem 9.1 itself: on the QSN the overlap construction produces a single species with exact Lüscher chiral symmetry and exponential locality whose range is fixed by the Wilson gap, as on a regular lattice. $\square$ **Remark 9.3.1.** The absence of a Brillouin torus does not by itself prevent doubling: a naive Dirac operator on the QSN (the operator $D_0$ of (9.2)) does have high-frequency low-energy modes, namely the analogues of corner modes, which on an irregular graph are the eigenvectors of $L$ near the top of its spectrum on which $D_0$ is small; Proposition 9.1.8 shows, on the lattice prototype, that the Wilson term lifts them by a mass $(2rk-\rho)/a$, $k\geq1$, and (H3') is the hypothesis that the same lifting holds on the QSN. The QSN is therefore not exempt from the doubling phenomenon, only from the theorem that forces it. This is why Theorem 9.1 rather than Corollary 9.2 is the load-bearing result. ### 9.4. The matter action and its continuum limit **Definition 9.4.1 (Matter action).** The matter action on $\mathsf{Q}_n$ is $$ S_{\mathrm{mat}}[\mathsf{Q}_n;\psi,\Phi]=\langle\bar\psi,D_{\mathrm{ov}}\psi\rangle_n+\sum_{p\in N_n}\mu(p)\Big[\bar\psi(p)\,Y(\Phi(p))\,\psi(p)+|\nabla_{\mathrm{ov}}\Phi|^2(p)+V_H(\Phi(p))\Big], \tag{9.7} $$ where $\bar\psi=\psi^\dagger$ in the Euclidean sector, $\Phi\colon N\to\mathbb{C}^2$ is the Higgs doublet, $Y(\Phi)$ is the Yukawa matrix on $V$, $V_H(\Phi)=\lambda_H(|\Phi|^2-v_H^2/2)^2$, and $|\nabla\Phi|^2(p)=\mu(p)^{-1}\sum_{e\in E(p)}w_e|U_{\mathrm{SM}}(e)\Phi(m_e)-\Phi(p)|^2$, i.e. $\langle\Phi,L_{\mathrm{SM}}\Phi\rangle_n$ with $L_{\mathrm{SM}}$ the Laplacian (9.2) with gauge transport only. **Proposition 9.4** $\boxed{\text{Conditional (on H3', H10, H12, H15, H-hol) modulo Classical A, I, J, K, and Chernoff}}$ (Continuum limit of the matter action on smooth fields). *Under the hypotheses of Theorem 9.1(v), for all smooth $(\psi,\Phi)$ on the compact limit $\mathcal{M}$,* $$ \lim_{n\to\infty}S_{\mathrm{mat}}[\mathsf{Q}_n;J_n\psi,J_n\Phi]=\int_{\mathcal{M}}\Big(\bar\psi\,\slashed D_A\psi+\bar\psi Y(\Phi)\psi+|\nabla^A\Phi|^2_g+V_H(\Phi)\Big)\mathrm{dvol}_g. \tag{9.8} $$ *This is convergence of the action evaluated on restrictions of smooth fields; no Γ-convergence or Mosco-convergence statement for the Dirichlet form on random graphs is asserted (review3, item 31).* *Proof.* The Dirac term is Theorem 9.1(v) with $\phi=\psi$. The node-local terms are Riemann sums of continuous functions against the node measure $\mu$ and converge by the node-density clause of (H12). The gradient term: $\langle J_n\Phi,L_{\mathrm{SM}}J_n\Phi\rangle_n=\sum_ew_e|U_{\mathrm{SM}}(e)\Phi(q)-\Phi(p)|^2$ exactly (Lemma 9.1.5(a) with gauge transport only); by (H-hol) and $\Phi\in C^2$, $U_{\mathrm{SM}}(e)\Phi(q)-\Phi(p)=\Delta x_e^\mu\nabla^A_\mu\Phi(p)+O(a^2)$, so the sum is $\sum_ew_e|\Delta x^\mu_e\nabla^A_\mu\Phi(p)|^2+O(\sum_ew_ea^3)$. With $w_e=T(e)a_e^{-2}n^{-d}$ and $a_e^{-2}\Delta x_e\otimes\Delta x_e=\hat v_e\otimes\hat v_e$, the main term is $n^{-d}\sum_eT(e)\,\overline{\nabla^A_\mu\Phi}\nabla^A_\nu\Phi\,\hat v^\mu_e\hat v^\nu_e$; this is the cell-decomposition sum of Lemma 8.3 with $\lambda_n^2\mathrm{Ric}$ replaced by $\overline{\nabla\Phi}\otimes\nabla\Phi$ and with the weight $\lambda_n(p)^2$ of (8.4) absent. The two weightings differ by the factor $\lambda_n(p)^2\in[\lambda_{\min}^2,\lambda_{\max}^2]$, and the per-node isotropy (H15)(d) (direction tensor $\sum_eT(e)\hat v_e\otimes\hat v_e=g^{-1}+o(1)$ at every node) makes the main term converge to $\int g^{\mu\nu}\overline{\nabla_\mu\Phi}\nabla_\nu\Phi$ by the argument of Lemma 8.3. The error term is $O(a)$ by (H1') and the node-density bound. $\square$ ### 9.5. The gauge action and its continuum limit **Definition 9.5.1 (Plaquettes).** A *plaquette* of $\mathsf{Q}$ is an undirected cycle of length $4$ in the underlying undirected graph of the form $p\to q_1\to s\leftarrow q_2\leftarrow p$, i.e. a causal diamond of two-step chains with common endpoints, or more generally a minimal undirected cycle of length at most $\ell_0$ (a fixed constant, $\ell_0=4$ suffices on the recovery sequence). For a plaquette $\square$ with vertices $p,q_1,s,q_2$, the *plaquette holonomy* is $U_\square:=U(p,q_1)U(q_1,s)U(s,q_2)^\dagger U(q_2,p)^\dagger$ with the SM transports, and its *area bivector* is $\Sigma_\square^{\mu\nu}:=\tfrac12\big(\Delta x^\mu_{pq_1}\Delta x^\nu_{q_1s}-\Delta x^\nu_{pq_1}\Delta x^\mu_{q_1s}\big)-\tfrac12\big(\Delta x^\mu_{pq_2}\Delta x^\nu_{q_2s}-\Delta x^\nu_{pq_2}\Delta x^\mu_{q_2s}\big)$, the oriented area of the loop $p\to q_1\to s\to q_2\to p$ (the second triangle is traversed in the opposite sense, whence the minus sign; for a parallelogram with sides $u,v$ this gives $\tfrac12u\wedge v-\tfrac12v\wedge u=u\wedge v$, review3 item 23). **Definition 9.5.2 (Wilson gauge action).** For each factor $G_i\in\{\mathrm{SU}(3),\mathrm{SU}(2),\mathrm{U}(1)\}$ with coupling $g_i$ and plaquette weight $\beta_\square$, $$ S_{\mathrm{gauge}}[\mathsf{Q}_n]=\sum_i\frac{4\dim_i}{g_i^2}\sum_\square\beta_\square\,\Big(1-\frac{1}{\dim_i}\mathrm{Re}\,\mathrm{tr}\,U^{(i)}_\square\Big),\qquad\beta_\square:=n^{4-d}\theta_\square, \tag{9.9} $$ where $\theta_\square>0$ is a plaquette tension chosen, in analogy with Step 3 of §8.4, so that the fourth-moment hypothesis (H12$_4$) below holds. The factor $n^{4-d}$ is the scaling making the plaquette sum finite: there are $O(n^d)$ plaquettes per unit volume, each contributing $O(a^4)=O(n^{-4})$. **Hypothesis (H12$_4$) (Plaquette isotropy).** For every $U\subset\mathcal{M}$ with null boundary, $$ n^{4-d}\sum_{\square\subset U}\theta_\square\,\Sigma_\square^{\mu\nu}\Sigma_\square^{\rho\sigma}\longrightarrow\int_U\tfrac12\big(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho}\big)\mathrm{dvol}_g. \tag{9.10} $$ This is the fourth-moment analogue of (H12). It holds for the jittered lattice of Definition 8.14 with $\theta_\square$ constant on each plaquette class, because the expected bivector fourth-moment tensor is $B_d$-invariant, and the only $B_d$-invariant rank-4 tensors are the isotropic ones plus $\sum_\alpha e_\alpha^{\otimes4}$, which vanishes when contracted with antisymmetric bivectors; so the expectation is a multiple of $\tfrac12(gg-gg)$, and $\theta_\square$ is chosen to make the multiple $1$. The coefficient in (9.11) is therefore a normalisation choice of $\theta_\square$, not a derived number (review3b, item 19). **Proposition 9.5** $\boxed{\text{Conditional (on H10, H12, H12}_4\text{, H-hol) modulo Classical A}}$ (Continuum limit of the gauge action). *Under (H1'), (H3), (H4'), (H10), (H12), (H12$_4$) and (H-hol) with $A^{(i)}\in C^1$,* $$ \lim_{n\to\infty}S_{\mathrm{gauge}}[\mathsf{Q}_n]=\sum_i\frac{1}{4g_i^2}\int_{\mathcal{M}}F^{(i)a}_{\mu\nu}F^{(i)a\mu\nu}\,\mathrm{dvol}_g=\sum_i\frac{1}{2g_i^2}\int_{\mathcal{M}}\big(-\mathrm{tr}\,F^{(i)}_{\mu\nu}F^{(i)\mu\nu}\big)\mathrm{dvol}_g,\qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu]=iF^a_{\mu\nu}t^a, \tag{9.11} $$ *with $\mathrm{tr}(t^at^b)=\tfrac12\delta^{ab}$ on the fundamental representation and $\dim_i$ the dimension of that representation.* *Proof.* By the non-Abelian Stokes theorem at second order (the Baker-Campbell-Hausdorff expansion of the product of four transports around a small loop), $U_\square=\exp\big(\tfrac12\Sigma^{\mu\nu}_\square F_{\mu\nu}(p)+O(a^3)\big)$ (the factor $\tfrac12$ because $\Sigma^{\mu\nu}F_{\mu\nu}$ counts each $(\mu,\nu)$ pair twice: for a square of side $a$ in the $12$-plane, $\Sigma^{12}=-\Sigma^{21}=a^2$ and $\tfrac12\Sigma^{\mu\nu}F_{\mu\nu}=a^2F_{12}$, the Stokes flux; review3b item 19), where the quadratic terms in $A$ assemble into the commutator in $F$ and the $O(a^3)$ term involves $\nabla F$ and is gauge covariant. Then $1-\tfrac{1}{\dim}\mathrm{Re}\,\mathrm{tr}\,U_\square=-\tfrac{1}{8\dim}\mathrm{tr}\big((\Sigma^{\mu\nu}F_{\mu\nu})^2\big)+O(a^5)=\tfrac{1}{8\dim}\Sigma^{\mu\nu}_\square\Sigma^{\rho\sigma}_\square\,\big(-\mathrm{tr}F_{\mu\nu}F_{\rho\sigma}\big)+O(a^5)$, where we used $\mathrm{tr}\,F=0$ for the first-order term (traceless generators; for $\mathrm{U}(1)$ the first-order term is purely imaginary and drops from the real part) and $F$ anti-Hermitian so that $-\mathrm{tr}(X^2)\geq0$. Summing with weights $\beta_\square$ and using (H12$_4$) cell by cell as in Lemma 8.3, $$ \sum_\square\beta_\square\Big(1-\tfrac1{\dim}\mathrm{Re}\,\mathrm{tr}\,U_\square\Big)\to\frac{1}{8\dim}\int\tfrac12(g^{\mu\rho}g^{\nu\sigma}-g^{\mu\sigma}g^{\nu\rho})\,(-\mathrm{tr}F_{\mu\nu}F_{\rho\sigma})\,\mathrm{dvol}_g=\frac{1}{8\dim}\int(-\mathrm{tr}F_{\mu\nu}F^{\mu\nu})\mathrm{dvol}_g, $$ and with $F=iF^at^a$, $-\mathrm{tr}F_{\mu\nu}F^{\mu\nu}=F^a_{\mu\nu}F^{b\mu\nu}\mathrm{tr}(t^at^b)=\tfrac12F^a_{\mu\nu}F^{a\mu\nu}$; multiplying by the prefactor $4\dim_i/g_i^2$ of (9.9) gives $\frac{1}{4g_i^2}\int F^aF^a$, which is (9.11). The prefactor, together with the normalisation of $\theta_\square$ in (H12$_4$), is a choice fixing the emergent coupling $g_i$; the content of the Proposition is the *form* of the limit, Yang-Mills, not the value of its coefficient. The $O(a^5)$ remainder summed over $O(n^d)$ plaquettes with weight $n^{4-d}$ is $O(a)\to0$. No statement about Symanzik improvement is made: on an irregular graph the $O(a^2)$ corrections have position-dependent random coefficients and the lattice improvement programme (Lüscher, Weisz, *Commun. Math. Phys.* **97** (1985) 59) does not transfer without a separate construction. The quantum continuum limit of the gauge sector (existence of the continuum limit of the lattice theory with bare coupling $g_i(a)$ tuned by asymptotic freedom) is not asserted in this monograph; it is an open problem of constructive field theory in $d=4$. At the level of the classical action, which is what enters Theorem 8.1 and the tree-level boundary condition of the flow in Part IV, (9.11) is the complete result. $\square$ **Corollary 9.6** $\boxed{\text{Conditional (on H3', H10, H12, H12}_4\text{, H13, H15, H-hol) modulo Classical A, E, P, Gray, I, J, K}}$ (Total action, Riemannian sector). *Under the union of the hypotheses of Theorem 8.1(i), Proposition 9.4 and Proposition 9.5, the total rescaled QSN action $\mathcal{S}_n+S_{\mathrm{mat}}+S_{\mathrm{gauge}}$ converges, for restrictions of smooth field configurations on the compact limit $\mathcal{M}$ and in expectation over the graph ensemble for the gravitational term, to* $$ \int_{\mathcal{M}}\Big[\frac{R-2\Lambda_{\mathrm{b}}}{16\pi G}+\bar\psi(\slashed D_A+Y(\Phi))\psi+|\nabla^A\Phi|^2+V_H(\Phi)+\sum_i\frac{1}{4g_i^2}F^{(i)a}F^{(i)a}\Big]\mathrm{dvol}_g, \tag{9.12} $$ *the Euclidean action of general relativity coupled to the Standard Model with three generations and right-handed neutrinos, with the fermion content and hypercharge assignments of Chapter 17; the Lorentzian action is obtained by the continuation of Theorem 21.1 and is Conditional on it.* *Proof.* Sum (8.7), (9.8), (9.11). The expectation over the graph ensemble is needed only for the gravitational term (Theorem 8.1(iii)); the matter and gauge terms converge pointwise in the ensemble by Propositions 9.4 and 9.5. $\square$ ### 9.6. Physical content and scope Three statements summarise what has been established. First, the matter sector of Lach-Space on the QSN is a well-defined lattice field theory whose Dirac operator is exponentially local, $\gamma_5$-Hermitian, Ginsparg-Wilson, and single-species under (H15), with all constants controlled by the degree bound, the tension ratio, the Wilson parameters, and the Wilson gap; none of these depends on the scale $n$. Second, the classical continuum limit of the full action is general relativity plus the Standard Model, with the metric, the Dirac operator, the Higgs kinetic term and the Yang-Mills term all built from the *same* tension-weighted moments of the edge displacements, so that the emergent metric seen by matter is the emergent metric seen by gravity; this is the content of (H12) and (H12$_4$), and it is the precise sense in which the QSN unifies gravity and gauge fields by shared kinematics rather than by a larger gauge group. Third, the chiral structure of the Standard Model is not imposed by hand but is the Lüscher symmetry of the overlap operator, exact at finite $n$. Not established in this chapter: the quantum continuum limit of the interacting theory (the subject of Part IV, where it is a statement about the effective average action in a truncation); the strong form (H15) for bounded-degree random graphs, which fails, so that the doubler-free statement (iv) is conditional on the local regularity of the sequence; and the Lorentzian spectral properties of the overlap operator, which require the continuation of Chapter 21 and are stated there only for the quantities that continue. ### 9.7. Dimensional check and hypothesis discharge ledger **Remark 9.7** (Units). With $\hbar,c$ restored, $D_0$, $L^{1/2}$, $D_{\mathrm{ov}}$ and $H_W$ have dimension $\mathrm{length}^{-1}$ (the weights $w_e\propto a_e^{-2}$ and $\widehat w_e\propto a_e^{-1}$ carry the lattice scale), the Wilson mass $\rho/a$ is a length$^{-1}$, and the locality bound (iii) is dimensionless in the exponent because $d_G$ counts edges; the physical decay length is $\xi_{\mathrm{loc}}^{-1}a\sim\xi_{\mathrm{loc}}^{-1}\ell_P$. The spinor inner product carries $\mu(n)=n^{-d}$ so that $\langle\bar\psi,D\psi\rangle$ has the dimension of $\int\mathrm{d}^dx\,\bar\psi\slashed\partial\psi$, namely $\mathrm{length}^{d-1}[\psi]^2$, and the action is dimensionless when $[\psi]=\mathrm{length}^{-(d-1)/2}$, the canonical fermion dimension; the gauge action (9.11) is dimensionless with $[A]=\mathrm{length}^{-1}$ and $g_i$ dimensionless in $d=4$. **Hypothesis discharge ledger, Chapter 9.** | Result | Classical inputs | Hypotheses consumed | Step where consumed | |---|---|---|---| | Lemma 9.1.5 | none | H1 (degree count in (d)), H1' (tension bounds) | (a)-(d) | | Lemma 9.1.6' | none | H1, H1', H3 | finiteness of $N$ | | Lemma 9.1.7 | none | H1, H1', H5 with L6 (eccentricity $a/a_{\min}=\Delta_{\max}/\Delta_{\min}$) | norm bound | | Prop 9.1.8 | none (explicit computation; coordinate-wise concavity) | lattice prototype only | vertex minimisation | | Thm 9.1(i),(ii) | K | H3' (definition of $\epsilon=\mathrm{sign}H_W$) | algebra | | Thm 9.1(iii) | I (range 2, gap $m_W^2a^{-2}$), J ((9.5), elementary) | H3' ($\eta$), H1 (row sums, $2D+1$), Lemma 9.1.7 ($C_H$) | resolvent bound and $t$-integral | | Thm 9.1(iv$_{\mathrm a}$) | K | H3' | normality, unitarity of $V$, trace bookkeeping | | Thm 9.1(iv$_{\mathrm b}$) containment | Weidmann Thm 2.3 / Reed-Simon VIII.24 | H15, H-hol, H10 (compact limit, discrete spectrum), H12 | spectral projections, strong | | Thm 9.1(iv$_{\mathrm b}$) no doublers | Weidmann Thm 2.7 | H16 in addition | norm convergence of projections, rank equality | | Thm 9.1(v) | A (existence of the limit), Chernoff (essential self-adjointness), Weidmann/Kuwae-Shioya (Step 3) | H10, H12, H-hol (Step 1), H15(a),(b),(d) (Step 1), H15(c) (Step 2 remainder), H3' ($\|R_t\|$), H1', H5 (Lemma 9.1.7) | Steps 1-3 | | Cor 9.2 | Nielsen-Ninomiya, Friedan (statement only) | inherits 9.1 | (a)-(d) | | Prop 9.4 | A, I, J, K (via 9.1(v)) | H3', H10, H12, H15 (incl. (d)), H-hol | Dirac term, gradient term | | Prop 9.5 | A | H10, H12, H12$_4$ (oriented bivector, Stokes factor $\tfrac12$, coefficient by normalisation), H-hol | BCH expansion, (9.10) | | Cor 9.6 | union; plus Thm 8.1's A, E, P, Gray | union; H13 via 8.1 | sum | Not discharged in this chapter: H3' (verified on the lattice prototype, Prop 9.1.8, and on the jittered lattice with stencil $\{\pm e_\mu\}$ by Weyl's inequality, Remark 9.1.9; open for bounded-degree random graphs), H15 (fails for bounded-degree Poisson graphs and fixed-relative-jitter lattices; holds for jitter scaled to zero and for $D_n\to\infty$), H16 (generalised norm-resolvent convergence; open), H-hol (limiting connection; parallel to H10), H10, H12, H12$_4$, and the edge-length convention of Definition 9.1.1. The chapters are coherent in the compact Riemannian sector; every Lorentzian reading is Conditional on Theorem 21.1 (review3, item 32). --- ## Referee response, Part III (review3.md, 33 findings) Each item: disposition and location. 1. FATAL, Lorentzian (H12). Accepted. Chapter 8 restated in the Riemannian sector (compact $\mathcal{M}$, target $g^{-1}$ positive definite); Lorentzian reading only through Theorem 21.1 (Remark 8.6). The causal-window tension tensor $h^{-1}+b'u\otimes u$ and its frame-dependent limit functional are now stated explicitly (§8.4 Step 3, Prop 8.11, Remark 8.13). 2. MAJOR, $\varepsilon$ versus $r$. Accepted. Step radius $\varepsilon_n(p)=\lambda_n(p)n^{-1}$ introduced; (H12) reformulated as (8.4) with $\lambda_n^2\hat v\otimes\hat v$; (H13)(b) and Lemmas 8.2, 8.3 rederived with the step radius; $\alpha_d$ identification in Lemma 8.3 now correct. 3. FATAL, Γ-convergence statement. Accepted. The word Γ-convergence withdrawn; Theorem 8.1(ii) is two-sided convergence along hypothesis-satisfying sequences plus a recovery clause Conditional on (H14). Classical D is no longer used in Theorem 8.1. 4. MAJOR, exhaustion and finite volume. Accepted. Compact Riemannian sector, $U=\mathcal{M}$; no exhaustion; uniform node-density bound $C_N$ added to (H12). 5. MAJOR, in-degree. Accepted. §8.4 Step 2 now uses the symmetric truncation of Definition 20.11 and cites Prop 20.3(a) for the deterministic degree bound. 6. FATAL, rescaled (H2) for the recovery sequence. Accepted. Recovery at $(\mathcal{M},g)$ is Conditional on (H14); both candidate constructions are shown to fail the rescaled (H2) (Prop 20.9 for the window sprinkling, Prop 8.15 for the jittered lattice); Remark 8.16 and Open Problem O3' record the averaged-curvature alternative. 7. MAJOR, (H13)(b) in the recovery sequence. Accepted. Prop 8.11 and Prop 8.15 carry (H13)(b) as a hypothesis explicitly. 8. MAJOR, longest-chain constant. Accepted. Step 4 replaced by the citation of Conjecture 20.8 (formerly cited as a theorem; see Pass 2, item 25) (limit $d_{g,D}$, not $d_g$); the forbidden phrase removed; no longest-chain claim is made in Part III. 9. MINOR, (H4') upper bound and Penrose citation. Accepted. Two-sided (H4') via Prop 20.3(d) (window) and the explicit box count plus Bollobás-Leader (jittered lattice, Prop 8.15(d)); Penrose citation removed from Part III. 10. MAJOR, metric independence of $C_n$. Accepted. Theorem 8.1(iii) now asserts only mean zero, the variance bound, and tightness for $d=4$; the coupling claim is withdrawn. 11. MAJOR, $v_\infty$ and Lemma 8.4. Accepted. Lemma 8.4 retagged Conditional on (H12) and on Lemma 13.1 (constant node potential) or (H12c); the spatial ergodic theorem (Nguyen-Zessin) is cited for the stationary case; Classical D removed from the lemma. 12. MAJOR, Lemma 8.2 Term II. Accepted. Lemma 8.2 restated as a separation of theorem (term I by Classical E, order of term II by Gray and boundedness) from hypothesis ((H13)(b) for the form of the coefficient); the Kantorovich-linearity claim withdrawn; Classical P and Gray listed in the tag and ledger. 13. MAJOR, (iv) justification. Accepted. Γ-convergence justification deleted; (iv) stated as the first variation of the limit functional; dependence on Prop 9.4 (H3', H-hol, H15) recorded in the statement and ledger. 14. MAJOR, Theorem 8.1 tag. Accepted. Tag is Conditional (H10, H12, H13; H14 for recovery) modulo Classical A, B, C, E, P; (iv) additionally inherits Prop 9.4. The Lorentzian theorem is not claimed at any tag (Remark 8.6). 15. MAJOR, label conflict. Accepted. H12 tension isotropy, H13 random-walk bias bound (per C13); the Ch 9 directional-consistency hypothesis renamed H15; H16 and H-hol added; H14 recovery existence. 16. MINOR, Step 3 display. Accepted. Vacuous factor removed; per-node target written as $\lambda^2\mathbb{E}[\sum T\hat v\otimes\hat v]=\mathbb{1}$ (Riemannian, Prop 8.15(f)) and $h^{-1}+b'u\otimes u$ (Lorentzian, §8.4 Step 3); a.s. statements attributed to the stated probability space. 17. MINOR, constants. Accepted. $|\xi_e|\leq3$, covariance bound $9$, constant $18D^{\rho_0+2}T_{\max}^2C_N$ in (8.10); the $D_0$/sectional confusion is gone with the old Step 5. 18. MINOR, units. Accepted. Remark 8.10 rewritten with $[G]=\mathrm{L}^{d-1}\mathrm{M}^{-1}\mathrm{T}^{-2}$ and $\ell_P^{d-2}=\hbar G/c^3$. 19. MINOR, forbidden phrases. Accepted; all four removed (grep-verified). 20. MAJOR, Classical K. Accepted. Restated with $\epsilon=\epsilon^\dagger=\epsilon^{-1}$, $V=\gamma_5\epsilon$, $D=\tfrac\rho a(1+V)$; proof of (ii) replaced by the three-line computation via $V\gamma_5V=\gamma_5$; draft text removed. 21. FATAL, weak convergence without (H15). Accepted. Theorem 9.1(v), Prop 9.4 and Cor 9.6 are Conditional on (H15), now including the second-order clause (c); the remainder estimate is written with explicit constants under (c); the unsupported "weak form without (H15)" claim is withdrawn. 22. FATAL, absence of doublers. Accepted. (iv) split: (iv$_{\mathrm a}$) spectrum and index (theorem); (iv$_{\mathrm b}$) containment under (H15) (strong convergence) and rank equality only under the new (H16) (no spectral pollution, equivalent with (H15) to generalised norm-resolvent convergence). Cor 9.2(d) inherits (H16). 23. FATAL, bivector sign. Accepted. Definition 9.5.1 corrected to the oriented loop area $\tfrac12u_1\wedge v_1-\tfrac12u_2\wedge v_2$; parallelogram check recorded. 24. MINOR, gauge normalisation. Accepted. Prefactor $\dim_i/g_i^2$ in (9.9); (9.11) reads $\frac{1}{4g_i^2}\int F^aF^a=\frac{1}{2g_i^2}\int(-\mathrm{tr}FF)$; the chain of constants is written out. 25. MAJOR, Prop 9.1.8 proof. Accepted. Replaced by the coordinate-wise concavity argument (leading coefficient $-1+r^2\leq0$ in each $x_\mu$), minimum at the 16 vertices, $f=(2rk-\rho)^2$. 26. MINOR, Lemma 9.1.5(d). Accepted. $\|L\|\leq4DT_{\max}a_{\min}^{-2}$, $\|D_0\|\leq2DT_{\max}a_{\min}^{-1}$, $C_H=(2+2r)DT_{\max}(a/a_{\min})^2+\rho$. 27. MINOR, eccentricity. Accepted. $a/a_{\min}=\Delta_{\max}/\Delta_{\min}$ via L6, bounded by (H5); stated in Lemma 9.1.7. 28. MINOR, (iii) constants. Accepted. $C_{\mathrm{loc}}=2\rho(2D+1)C_He^{\xi_{\mathrm{loc}}}/m_W$, $\xi_{\mathrm{loc}}=\tfrac12\log(1+m_W^2/(2C_H^2))$. 29. MINOR, Classical J attribution. Accepted. Hastings-Michalakis removed; (9.5) proved in two lines; Hernández-Jansen-Lüscher cited for the locality argument. 30. MINOR, index sign. Accepted. $n_--n_+=+\tfrac12\mathrm{Tr}\,\mathrm{sign}(H_W)$, with the trace bookkeeping written out. 31. MAJOR, Prop 9.4 Γ/Mosco. Accepted. All Γ- and Mosco-convergence claims for the Dirichlet form withdrawn; Prop 9.4 is pointwise convergence on restrictions of smooth fields, Conditional on (H15), (H-hol). 32. MINOR, finiteness versus non-compact Lorentzian. Accepted. Both chapters are now in the compact Riemannian sector; Lorentzian readings via Theorem 21.1 only. 33. MINOR, Ch 9 ledger. Accepted. Ledger rewritten with (H15), (H16), (H-hol), the corrected bivector, and the inherited Ch 8 inputs. Additional changes made in the same pass, not prompted by the report: Erratum C15 (Lorentz-Finsler limit of the window sprinkling; number-operator edge Hamiltonian; Props 8.11, 8.12, Lemma 8.10', Remark 8.13) and Erratum C17 (Classical D cited only as Part I Theorem 2.8; Riemannian candidate recovery sequence built in the admissible class $\mathcal{L}$, Definition 8.14, Proposition 8.15, Remark 8.16). ### Pass 2 (review3b.md, 8 MAJOR, 16 MINOR) 1. MAJOR, routing through Thm 5.5. Accepted. Chapter 8 now obtains the limit from Classical A alone (doubling via Lemma 5.3, diameter via H3) plus (H10) as consumed ($C^3$, empty singular set, approximating maps); the rescaled (H2), Classical B, C, (H-conv), (H-BE) are removed from every Ch 8 and Ch 9 tag, with an explicit note that a reader routing through Theorem 5.5 must add them. (H14) restated without the (H2) clause and with (H10); the obstruction to (H14) is now correctly the limit identification (Finsler / normed), not (H2). 2. MAJOR, jittered lattice limit. Accepted. Prop 8.15(c) restated as bilateral comparability plus the statement that the rescaled $d_{\mathrm{eff}}$ has a non-Riemannian normed limit (asymptotic cone of the periodic metric, Burago-Burago-Ivanov 8.5.2, bi-Lipschitz-perturbed by the $2\%$ jitter); (f) verified only relative to the chart embedding; the sentence about the action converging "along these sequences" replaced by the chart-embedded statement of Remark 8.16, explicitly labelled as not a statement about the emergent geometry. 3. MINOR, tautological recovery clause. Accepted. Clause deleted; (H14) is stated as an open existence question in Remark 8.7's place (after Theorem 8.1), not as a hypothesis of the theorem. 4. MINOR, (H14) checkability. Accepted. (H10) added; the Thm 5.5 route and its extra hypotheses named. 5. MINOR, remainder order. Accepted. (H10) as consumed is $C^3$ (matching Lemma 7.4); Classical E and Gray stated with $O(\varepsilon^3)$ remainders under $C^3$. 6. MINOR, $\alpha_d$ normalisation. Accepted. $\alpha_d:=1/(16\pi Gc_\nu)$ with $c_\nu$ the total coefficient; $G$ is defined by the sequence; the $O(1)$ upper-bound claim deleted. 7. MINOR, node-term variance. Accepted. $v(p)$ added to (H13)(a). 8. MAJOR, orientation across charts. Accepted. Global orientation by a Morse function with a total-order tie-break. 9. MAJOR, Prop 8.8 Γ-limit wording and equation number. Accepted. 10. MINOR, Prop 8.9. Accepted; equation number fixed; the Prop 5.8 sentence deleted. 11. MAJOR, Prop 8.11 coefficients. Accepted. (8.11) restated with $(a_D,b_D)$ defined as the Lemma 7.4 window average composed with the Step 3 tension tensor; the $b'_D\mapsto b'_D-1$ convention removed; $C^3$ assumed. 12. MINOR, Prop 8.12 residues. Accepted. (a) diagonal failure argued from (H1) (unbounded degree bound) rather than a tension ratio; (b) existence of $(a_\infty,b_\infty)$ made a hypothesis of the statement; (c) Lemma 8.10' proof corrected to cite (iv) and (i) and to use the universal constant $\kappa_D$. 13. MINOR, Step 1 versus Def 20.1. Accepted; $N_n=\mathcal{P}_n$. 14. MINOR, Prop 8.15 constants and gluing. Accepted. $C_{\mathrm{vol}}=(2\max|s|+1)^d$; eccentricity $\leq(2\max|s|+1)(1+O(\gamma))$; (f) stated for the expectation of $\lambda_n(p)^2T\hat v\otimes\hat v$; gluing isoperimetry demoted to hypothesis (H4'-glue); the $\rho R^d=1$ regime and the $1.9\%$ jitter fraction stated, with an erratum for Part I Thm 2.8's sentence. 15. MINOR, cross-references and numbering. Accepted in part: Remark 3.7 corrected; O3' labelled as a proposed addition to Chapter 22; Gray added to the tag. Not done: renumbering to strict Chapter.Seq (sub-numbered definitions and lemmas, primed 8.10', 9.1.6'), because the SPEC anchors (8.1, 9.1, 9.2) are fixed by cross-references from other Parts and a full renumbering of Part III would require a coordinated pass; recorded as a SPEC deviation. 16. MAJOR, (H16) precision. Accepted. Bounded cell-averaging identification operators $\widehat J_n$ (Voronoi cells, node weight $\mu(p)=\mathrm{vol}_g(V_p)$, $\widehat J_n\widehat J_n^*=\mathbb{1}$) introduced; (H15), (H16) stated for $\widehat J_n$; (H16) defined as generalised norm-resolvent convergence in Weidmann's sense. 17. MAJOR, per-node (H12) and (v) tag. Accepted. (H15)(d) added (per-node direction tensor and vanishing first moment); Prop 9.4 cites it; (v) tag lists Classical A and Chernoff. 18. MINOR, constant. Accepted; extra factor $a$ removed. 19. MAJOR, Stokes factor. Accepted. $U_\square=\exp(\tfrac12\Sigma F+O(a^3))$, $1-\mathrm{Re}\,\mathrm{tr}U/\dim=-\tfrac{1}{8\dim}\mathrm{tr}((\Sigma F)^2)$; prefactor in (9.9) changed to $4\dim_i/g_i^2$ so that (9.11) keeps the canonical $1/(4g_i^2)$; the proof states that the coefficient is a normalisation choice. 20. MINOR, Symanzik paragraph. Accepted; deleted from the Proposition and replaced by a one-sentence disclaimer; B, C and (H2) removed from Prop 9.5. 21. MINOR, stale references. Accepted; all references to "the recovery sequence of §8.4" in Ch 9 replaced by Definition 8.14 or (H15); Remark 9.1.9 rewritten with the Weyl-inequality argument for the jittered lattice. 22. MINOR, edge-length convention. Accepted; $a_e:=\tau_n(e)$ with the consistency hypothesis stated in Definition 9.1.1. 23. MINOR, Cor 9.6 tag and Talagrand. Accepted; H14 removed; Fournier-Guillin cited for general laws. 24. MINOR, Ch 9 numbering. Same disposition as item 15. 25. (Coordinator, Errata C19/C21.) Accepted. §8.4 now cites Definition 20.11 (rapidity cutoff, proper-time cut, symmetric truncation) and claims for the window sprinkling only what C19 lists as unconditional: (H1'), (H3), (H5), the $|B|\leq Cr^d+6\log n$ volume bound, (H2) with $K=-2$ only, and the loss estimates of Props 20.6, 20.7. Lemma 8.10' is Conditional on H-conn and the cone clause of Thm 20.5(iv); Propositions 8.11 and 8.12 are tagged at Conjecture level (Conditional on H-conn and Conjecture 20.8) for their limit clauses, with the action clause of 8.11 separated out as Conditional on H10, H13, Lemma 7.4 only. H-iso-win replaces the former claim of (H4')(i) for the window graph. Remark 8.16, (H14) and the Ch 8 ledger adjusted accordingly. ### Pass 3 (Erratum C22) Citations in Chapter 8 re-checked against C22's surviving list. Chapter 8 cites neither Lemma 5.2 nor Theorem 7.1 (now Conjectures); Lemma 5.3 is cited only in its curvature-free form (H1'+H4') for Classical A; Corollary 5.7 is cited only for uniqueness of mGH limits up to isometry, which is curvature-free. Theorem 8.1 (Riemannian sector, Classical A + H10) is unaffected. The only use of Lemma 7.4 is in §8.4 (Propositions 8.11, 8.12), where the equality form was consumed; these are now Conditional on Conjecture E-Lor, with the proved one-sided, sign-reversed inequality yielding only $\limsup\mathbb{E}[\mathcal{S}_n]\leq F_D$ (stated in the tag and proof of 8.11). (H6) replaced by (H6$^+$) in §8.4 Step 5 and the ledger; the chapter introduction and sector paragraph corrected to say that curvature transfer is conjectural after C22 and unused here. # Part IV. Functional Renormalization Group and Asymptotic Safety **Numbering convention for Part IV.** The anchor numbers fixed by Appendix C (Theorems 11.1, 11.2, 12.1, 12.2, Proposition 11.3, Proposition 11.4, Lemma 13.1, Theorem 13.2, Proposition 13.3, Phenomenology 13.4) are kept. All auxiliary results of a chapter are numbered consecutively after the anchors of that chapter, so that every label is of the form Chapter.Seq and no label is repeated. The anchor "Theorem 11.3" of Appendix C is a Proposition in this Part, because its proof imports a classical computation (Classical FLP, Theorem 11.15); Appendix C must be read with that tag. --- ## Chapter 10. The Wetterich Flow on the QSN ### 10.1. Scope of Part IV Part IV studies the scale dependence of the effective dynamics of the QSN. The object of study is the effective average action $\Gamma_k$, a one-parameter family of functionals interpolating between the bare QSN action at the Planck ceiling and the full quantum effective action in the infrared. 1. Chapter 10 defines the flow on a finite QSN, where it is a finite-dimensional ordinary differential equation, proves its well-posedness, defines the QSN Laplacian and regulator, defines the Euclidean continuation used in Chapters 11 and 12, and states precisely what a truncation is. The existence of a non-Gaussian fixed point of the exact flow is Weinberg's asymptotic-safety conjecture and is tagged Conjecture. 2. Chapter 11 derives the Einstein-Hilbert truncation of the flow with the Litim regulator, reduces the fixed-point equations to a single cubic polynomial, and proves existence and uniqueness of the non-Gaussian fixed point by elementary algebra. 3. Chapter 12 adds the Standard Model matter content in the same regularisation scheme, derives the matter-dependent cubic, and establishes the shifted fixed point and its persistence under deformation of the truncation. 4. Chapter 13 proves volume quantisation from a virial identity and derives the Henneaux-Teitelboim decomposition, which separates the cosmological constant from radiative corrections. Throughout, $k$ is the coarse-graining scale, $t=\log(k/k_0)$, and the dimensionless couplings are $\widetilde G_k=G_kk^{d-2}$, $\widetilde\Lambda_k=\Lambda_kk^{-2}$; in $d=4$ we write $g=\widetilde G$, $\lambda=\widetilde\Lambda$. Chapters 10 to 12 work in Euclidean signature on the continuation of Definition 10.10; the relation to the Lorentzian flow is Proposition 12.6. ### 10.2. The generating functional on a finite QSN **Definition 10.1** (Finite QSN sector and bare action). Let $\mathsf Q=(N,E)$ satisfy L1 with $|N|<\infty$, and fix representation cutoffs $D_e<\infty$ on every edge and $j_{\max}<\infty$ on every node, so that the physical Hilbert space $\mathcal H^{\mathrm{phys}}_{\mathsf Q}$ of §4.3 is finite-dimensional, $\dim\mathcal H^{\mathrm{phys}}_{\mathsf Q}=:\mathcal N<\infty$. The *field space* is $\Phi=\mathrm{Herm}(\mathcal H^{\mathrm{phys}}_{\mathsf Q})$, of real dimension $\mathcal N^2$, with the Hilbert-Schmidt inner product $\langle\varphi,\varphi'\rangle=\mathrm{tr}(\varphi\varphi')$ and Lebesgue measure $\mathrm d\varphi$. Let $\widehat{\mathcal K}(e)$ denote the operator on $\mathcal H^{\mathrm{phys}}_{\mathsf Q}$ that is diagonal in the spin-network basis with eigenvalue the Ollivier curvature $\mathcal K(e)$ of the graph labelled by that basis state (Axiom L3), and let $\widehat T(e)$ be the corresponding edge-tension operator (Axiom L2). The *bare action* is $$ S(\varphi)=\mathrm{tr}\bigl(\varphi\,H_{\mathsf Q}\,\varphi\bigr)+\mathrm{tr}\Bigl(\varphi\Bigl[\sum_{e\in E}\widehat T(e)\widehat{\mathcal K}(e)\Bigr]\varphi\Bigr)=\mathrm{tr}(\varphi\,\mathcal S\,\varphi),\qquad\mathcal S:=H_{\mathsf Q}+\sum_e\widehat T(e)\widehat{\mathcal K}(e),\tag{10.1} $$ where $H_{\mathsf Q}$ is the Hamiltonian (4.4.1). The second term is Axiom L5 evaluated as a quadratic form on $\Phi$; on a spin-network basis vector $\varphi=|s\rangle\langle s|$ it reduces to $E[\mathsf Q_s]=\sum_eT(e)\mathcal K(e)$ for the labelled graph $\mathsf Q_s$. **Definition 10.2** (Regulator and effective average action). A *regulator* is a family $R_k\in\mathrm{End}(\Phi)$, $k\in[0,\infty)$, of symmetric positive semi-definite endomorphisms, $C^1$ in $k$, with $R_k\to0$ as $k\to0$ and $\inf\mathrm{spec}(R_k)\to\infty$ as $k\to\infty$. The regulated generating functional and the effective average action are $$ W_k(J)=\log\int_\Phi\mathrm d\varphi\,\exp\Bigl(-S(\varphi)-\tfrac12\langle\varphi,R_k\varphi\rangle+\langle J,\varphi\rangle\Bigr),\qquad\Gamma_k(\varphi)=\sup_J\bigl(\langle J,\varphi\rangle-W_k(J)\bigr)-\tfrac12\langle\varphi,R_k\varphi\rangle.\tag{10.2} $$ **Lemma 10.3** $\boxed{\text{Theorem}}$ (Existence of the generating functional). *Suppose the operator $\mathcal S$ of (10.1) satisfies $\mathcal S\geq c_1\mathbb 1$ for some $c_1>0$ (coercivity). Then for every $k\geq0$ and $J\in\Phi$ the integral in (10.2) converges absolutely, $W_k$ is real-analytic and strictly convex in $J$, and $\Gamma_k$ is well defined and $C^\infty$ on $\Phi$.* *Proof.* $S(\varphi)=\mathrm{tr}(\varphi\mathcal S\varphi)=\mathrm{tr}(\mathcal S^{1/2}\varphi\varphi\mathcal S^{1/2})\geq c_1\mathrm{tr}(\varphi^2)=c_1\|\varphi\|^2$ because $\varphi^2\geq0$ and $\mathcal S\geq c_1$. Since $R_k\geq0$, the integrand is bounded by $\exp(-c_1\|\varphi\|^2+\|J\|\|\varphi\|)$, which is integrable on $\mathbb R^{\mathcal N^2}$. Analyticity of $W_k$: the $m$-th $J$-derivative of the integrand is bounded by $\|\varphi\|^m\exp(-c_1\|\varphi\|^2+\|J\|\|\varphi\|)$, integrable uniformly for $J$ in compact sets, so differentiation under the integral is justified by dominated convergence, and the Taylor series in $J$ converges because the moments of a Gaussian-dominated measure grow at most like $m!\,C^m$. Strict convexity: $W_k^{(2)}(J)$ is the covariance matrix of the probability measure $\propto e^{-S-\frac12\langle\varphi,R_k\varphi\rangle+\langle J,\varphi\rangle}\mathrm d\varphi$, which has full support, hence is positive definite. Superlinear growth: for any $\varphi_0\in\Phi$ and $r>0$, by Jensen's inequality on the ball $B_r(\varphi_0)$ of Lebesgue volume $|B_r|$, $$ W_k(J)\geq\log|B_r|+\frac1{|B_r|}\int_{B_r(\varphi_0)}\bigl(-S-\tfrac12\langle\varphi,R_k\varphi\rangle+\langle J,\varphi\rangle\bigr)\mathrm d\varphi\geq\langle J,\varphi_0\rangle-C(r,\varphi_0,k), $$ and choosing $\varphi_0=rJ/\|J\|$ gives $W_k(J)\geq r\|J\|-C(r,k)$ with $C(r,k)=\sup_{\|\varphi\|\leq2r}\bigl(S(\varphi)+\tfrac12\langle\varphi,R_k\varphi\rangle\bigr)-\log|B_r|$. Hence $\langle J,\varphi\rangle-W_k(J)\to-\infty$ as $\|J\|\to\infty$ for fixed $\varphi$ (take $r>\|\varphi\|$), so the supremum in (10.2) is attained, and at a unique $J(\varphi)$ by strict convexity. The Legendre transform of a strictly convex $C^\infty$ function with positive definite Hessian is $C^\infty$ by the inverse function theorem applied to $J\mapsto W_k^{(1)}(J)$. $\square$ **Discharge of coercivity for the QSN.** $\mathcal S=H_{\mathrm{geo}}+H_{\mathrm{ent}}+H_{\mathrm{gauge}}+H_{\mathrm{mat}}+\sum_e\widehat T\widehat{\mathcal K}$. On the finite sector, $H_{\mathrm{ent}}$, $H_{\mathrm{gauge}}$, $H_{\mathrm{mat}}$ and $\sum_e\widehat T\widehat{\mathcal K}$ are bounded operators with norms $\|H_{\mathrm{ent}}\|\leq2\lambda|E|$, $\|H_{\mathrm{gauge}}\|\leq2g_s^{-2}|\square|+2t|E|\cdot48$, $\|H_{\mathrm{mat}}\|\leq C_{\mathrm{mat}}$ (a node-local polynomial on a finite Fock sector), and $\|\sum_e\widehat T\widehat{\mathcal K}\|\leq2T_{\max}|E|$ by H1' and the Ollivier range $\mathcal K\in[-2,1]$ (Chapter 5). $H_{\mathrm{geo}}=\tfrac12\alpha_{\mathrm{geo}}\sum_n\mathrm{tr}\widehat C^{(2)}_n+\sum_nV(\widehat V_n)$ is bounded below by $\sum_n\min V$. Coercivity therefore holds after adding to $\sum_nV(\widehat V_n)$ the constant $c_1+2\lambda|E|+2g_s^{-2}|\square|+96t|E|+C_{\mathrm{mat}}+2T_{\max}|E|-|N|\min V$; such a shift of $S$ by a constant shifts $W_k$ by minus that constant and $\Gamma_k$ by the constant, and leaves the flow (10.3) unchanged. With this normalisation of the stabilising potential, adopted henceforth, $\mathcal S\geq c_1$. ### 10.3. The Wetterich equation as a finite-dimensional ODE **Theorem 10.4** $\boxed{\text{Theorem}}$ (Exact flow on a finite QSN). *Under the hypotheses of Lemma 10.3, $\Gamma_k$ satisfies, for every $\varphi\in\Phi$ and $k>0$,* $$ \partial_t\Gamma_k(\varphi)=\tfrac12\,\mathrm{Tr}\Bigl[\bigl(\Gamma_k^{(2)}(\varphi)+R_k\bigr)^{-1}\partial_tR_k\Bigr],\qquad t=\log k,\tag{10.3} $$ *where $\Gamma_k^{(2)}$ is the Hessian of $\Gamma_k$ on $\Phi$ and $\mathrm{Tr}$ the trace on $\mathrm{End}(\Phi)$. The operator $\Gamma_k^{(2)}(\varphi)+R_k$ is positive definite for all $\varphi,k$. As $k\to\infty$, $\Gamma_k(\varphi)-S(\varphi)-\tfrac12\log\det(R_k/2\pi)$ converges to $0$ uniformly on compact subsets of $\Phi$, with error at most $C_K\|R_k^{-1}\|$ for $\varphi$ in a compact $K$, where $C_K$ depends on $K$ and on the third derivatives of $S$ on a neighbourhood of $K$. As $k\to0$, $\Gamma_0$ is the Legendre transform of $W_0$.* *Proof.* Differentiate (10.2) in $t$ at fixed $J$: $\partial_tW_k=-\tfrac12\langle\langle\varphi,\partial_tR_k\varphi\rangle\rangle_J$, where $\langle\langle\cdot\rangle\rangle_J$ is the expectation in the measure of Lemma 10.3. Writing $\langle\langle\varphi\otimes\varphi\rangle\rangle_J=W_k^{(2)}(J)+\bar\varphi\otimes\bar\varphi$ with $\bar\varphi=W_k^{(1)}(J)$, $\partial_tW_k|_J=-\tfrac12\mathrm{Tr}[W_k^{(2)}\partial_tR_k]-\tfrac12\langle\bar\varphi,\partial_tR_k\bar\varphi\rangle$. By (10.2) and the envelope theorem (the supremum is attained at $J(\varphi)$ with $\partial_J(\langle J,\varphi\rangle-W_k)=0$), $\partial_t\Gamma_k|_\varphi=-\partial_tW_k|_{J(\varphi)}-\tfrac12\langle\varphi,\partial_tR_k\varphi\rangle=\tfrac12\mathrm{Tr}[W_k^{(2)}\partial_tR_k]$. Differentiating $\varphi=W_k^{(1)}(J(\varphi))$ gives $\mathbb 1=W_k^{(2)}\,\partial_\varphi J$ and differentiating $J(\varphi)=\Gamma_k^{(1)}(\varphi)+R_k\varphi$ gives $\partial_\varphi J=\Gamma_k^{(2)}+R_k$, so $W_k^{(2)}=(\Gamma_k^{(2)}+R_k)^{-1}$, positive definite because $W_k^{(2)}$ is. This proves (10.3). For $k\to\infty$: write $S(\varphi)+\tfrac12\langle\varphi,R_k\varphi\rangle-\langle J,\varphi\rangle$ and expand around its minimiser $\varphi_k(J)$; the Hessian is $\mathcal S_{\mathrm{sym}}+R_k\geq R_k$ where $\mathcal S_{\mathrm{sym}}$ is the Hessian of $S$ (constant, since $S$ is quadratic by (10.1)). Because $S$ is exactly quadratic, the integral is Gaussian and $W_k(J)=\tfrac12\langle J,(\mathcal S_{\mathrm{sym}}+R_k)^{-1}J\rangle-\tfrac12\log\det\bigl((\mathcal S_{\mathrm{sym}}+R_k)/2\pi\bigr)$ exactly, whence $\Gamma_k(\varphi)=\tfrac12\langle\varphi,\mathcal S_{\mathrm{sym}}\varphi\rangle+\tfrac12\log\det\bigl((\mathcal S_{\mathrm{sym}}+R_k)/2\pi\bigr)=S(\varphi)+\tfrac12\log\det(R_k/2\pi)+\tfrac12\mathrm{Tr}\log(\mathbb 1+R_k^{-1}\mathcal S_{\mathrm{sym}})$, and the last term is bounded by $\tfrac12\mathcal N^2\|\mathcal S_{\mathrm{sym}}\|\|R_k^{-1}\|$, which gives the stated error with $C_K=\tfrac12\mathcal N^2\|\mathcal S_{\mathrm{sym}}\|$ (independent of $K$ in the quadratic case; the general statement with third derivatives covers non-quadratic bare actions and is not needed here). $\square$ **Remark 10.5** (What is exact and what is not). On a finite QSN with the quadratic bare action (10.1) the exact flow is solvable in closed form (proof of Theorem 10.4). Non-trivial flows arise once the bare action acquires the non-quadratic terms generated by the continuum limit of Part II (the Einstein-Hilbert and matter actions of Theorem 8.1 are not quadratic in the metric). On the continuum theory space (10.3) is a functional differential equation, and the only solutions studied in this monograph are those of truncations (Definition 10.11), which are finite ODE systems to which the Picard-Lindelöf theorem applies. The programme makes no claim about the existence or uniqueness of solutions of the untruncated continuum equation. ### 10.4. The QSN Laplacian, the regulator, and the Euclidean continuation **Definition 10.6** (Symmetrised QSN Laplacian). Let $\mathsf Q=(N,E)$ satisfy L1 and H1, and let $\mathsf Q^u$ be its undirected Hasse graph (the same vertices, each directed edge replaced by an undirected edge). Let $\deg(n)$ be the undirected degree and $A$ the adjacency matrix. On the Hilbert space $\ell^2(N,\deg)$ with inner product $\langle f,h\rangle=\sum_n\deg(n)\overline{f(n)}h(n)$, the *symmetrised walk* $P_u=\mathrm{diag}(\deg)^{-1}A$ and the *normalised QSN Laplacian* $$ \Delta_{\mathsf Q}:=c_d\,\ell_P^{-2}\,(\mathbb 1-P_u),\qquad c_d=2d,\tag{10.4} $$ are defined. The constant $c_d$ is chosen so that the low-lying spectrum of $\Delta_{\mathsf Q}$ matches that of the continuum Laplace-Beltrami operator $-\nabla^2$ in the scaling regime of Lemma 10.9. The eigenvalue variable of $\Delta_{\mathsf Q}$ is denoted $u$; the continuum variable is $z=|p|^2$. **Lemma 10.7** $\boxed{\text{Theorem}}$ (Spectral properties). *$\Delta_{\mathsf Q}$ is a bounded self-adjoint operator on $\ell^2(N,\deg)$ with spectrum contained in $[0,2c_d\ell_P^{-2}]$. Consequently, for every bounded Borel function $F$ on $[0,2c_d\ell_P^{-2}]$, $F(\Delta_{\mathsf Q})$ is defined by the spectral theorem for bounded self-adjoint operators, and $F(\Delta_{\mathsf Q})$ is positive semi-definite whenever $F\geq0$. The Planck ceiling is $u_{\max}:=2c_d\ell_P^{-2}$.* *Proof.* $\langle f,P_uh\rangle=\sum_n\deg(n)\overline{f(n)}\deg(n)^{-1}\sum_mA_{nm}h(m)=\sum_{n,m}A_{nm}\overline{f(n)}h(m)$, which is symmetric in $(f,h)$ because $A$ is symmetric; so $P_u$ is self-adjoint on $\ell^2(N,\deg)$. By Cauchy-Schwarz, $|\langle f,P_uf\rangle|=|\sum_{n,m}A_{nm}\overline{f(n)}f(m)|\leq\sum_{n,m}A_{nm}\tfrac12(|f(n)|^2+|f(m)|^2)=\sum_n\deg(n)|f(n)|^2=\|f\|^2$, so $\mathrm{spec}(P_u)\subset[-1,1]$ and $\mathrm{spec}(\mathbb 1-P_u)\subset[0,2]$; boundedness of the degree (H1) guarantees that $P_u$ is bounded also for $|N|=\infty$. Multiplying by $c_d\ell_P^{-2}>0$ gives the claim. $\square$ **Definition 10.8** (Litim regulator on the QSN). For $Z_k>0$ the *QSN Litim regulator* is $$ R_k:=Z_k\,(k^2-\Delta_{\mathsf Q})\,\theta(k^2-\Delta_{\mathsf Q}),\tag{10.5} $$ defined by Lemma 10.7 as a bounded positive semi-definite self-adjoint operator; on the spectral variable, $R_k(u)=Z_k(k^2-u)\theta(k^2-u)$. For $k^2\geq u_{\max}$ every mode is regulated. *Pulled-back fields.* Let $(\mathsf Q_n)$ converge to $(\mathcal M,g^{\mathrm L})$ as in Theorem 5.6, with the embeddings $\iota_n\colon N_n\to\mathcal M$ of that theorem. For a vector bundle $\mathcal V\to\mathcal M$ (scalars, spinors, symmetric tensors, gauge fields) a section $\phi$ is *pulled back* to $\iota_n^*\phi\in\ell^2(N_n,\deg)\otimes\mathcal V_{\mathrm{fib}}$ by evaluation at the nodes, $(\iota_n^*\phi)(m)=\phi(\iota_n(m))$, with the fibre identified along $\iota_n$ using the local trivialisation of Theorem 5.6 on the regular set. $\Delta_{\mathsf Q}$ acts on $\ell^2(N_n,\deg)\otimes\mathcal V_{\mathrm{fib}}$ as $\Delta_{\mathsf Q}\otimes\mathbb 1$, and traces over this space are denoted $\mathrm{Tr}_{N\otimes\mathcal V}$. In the truncations of Chapters 11 and 12 the kinetic operators are written with $\Delta_{\mathsf Q}\otimes\mathbb 1$ in place of $-\nabla^2$ on the pulled-back fields wherever the regulator is inserted, so that $P_k:=\Delta_{\mathsf Q}+R_k/Z_k$ equals $k^2$ on the support of $R_k$; this convention is used in every Litim evaluation below, and in the continuum regime $k\ell_P\to0$ the traces $\mathrm{Tr}_{N\otimes\mathcal V}$ are replaced by their continuum limits through Lemma 10.9. The regulator of v2.0, written $\chi(\ell_P^2p^2)\ell_P^{-2}$ with a by-hand profile $\chi$, was a device for imposing a Planck ceiling; on the QSN the ceiling is the spectral bound of Lemma 10.7. **Lemma 10.9** $\boxed{\text{Conditional (on H$_{\mathrm{iso}}$)}}$ (Symbol and spectral measure). *Let $\mathsf Q$ be a QSN whose undirected Hasse graph, in the Euclidean continuation of Definition 10.10, has at each node neighbour displacements $\ell_e\in\mathbb R^d$ that are identically distributed with $|\ell_e|=\ell_P$ and direction uniform on $S^{d-1}$, independently of the node (hypothesis H$_{\mathrm{iso}}$, the isotropic-step model). Then the symbol of $\Delta_{\mathsf Q}$ at momentum $p$ is* $$ u(p)=z\Bigl(1-\frac{z\ell_P^2}{4(d+2)}+O(z^2\ell_P^4)\Bigr),\qquad z=|p|^2,\tag{10.6} $$ *and for every $n\geq1$ and every bounded measurable $f$ supported in $[0,u_0]$ with $u_0\ell_P^2<2(d+2)$,* $$ \int f(u)\,\mathrm d\nu_n(u)=\int_0^\infty f(u)\,u^{n-1}\Bigl(1+\frac{(n+1)u\ell_P^2}{4(d+2)}+O(u^2\ell_P^4)\Bigr)\mathrm du,\tag{10.7} $$ *where $\nu_n$ is the push-forward of $z^{n-1}\mathrm dz$ under $z\mapsto u(z)$.* *Proof.* Under H$_{\mathrm{iso}}$ the symbol of $P_u$ is $\mathbb E[e^{ip\cdot\ell}]=\mathbb E[\cos(p\cdot\ell)]$ (the sine term vanishes because the step distribution is symmetric under $\ell\to-\ell$, which is what isotropy provides and which the directed out-neighbour walk of Axiom L3 would not), so the symbol of $\mathbb 1-P_u$ is $\lambda(p)=\mathbb E[1-\cos(p\cdot\ell)]=\tfrac12\mathbb E[(p\cdot\ell)^2]-\tfrac1{24}\mathbb E[(p\cdot\ell)^4]+O(|p|^6\ell_P^6)$, the remainder being uniform because the series for $\cos$ converges absolutely with terms bounded by $(|p|\ell_P)^{2m}/(2m)!$. For $\hat\ell$ uniform on $S^{d-1}$, $\mathbb E[(\hat p\cdot\hat\ell)^2]=1/d$ and $\mathbb E[(\hat p\cdot\hat\ell)^4]=3/(d(d+2))$. Hence $\lambda(p)=\frac{z\ell_P^2}{2d}\bigl(1-\frac{z\ell_P^2}{4(d+2)}+O(z^2\ell_P^4)\bigr)$, and (10.4) gives (10.6). For (10.7): $u'(z)=1-\frac{z\ell_P^2}{2(d+2)}+O(z^2\ell_P^4)>0$ for $z\ell_P^2<2(d+2)$, so $z\mapsto u(z)$ is a $C^\infty$ diffeomorphism of $[0,z_0]$ onto $[0,u(z_0)]$ there; its inverse is $z(u)=u\bigl(1+\frac{u\ell_P^2}{4(d+2)}+O(u^2\ell_P^4)\bigr)$, $z'(u)=1+\frac{u\ell_P^2}{2(d+2)}+O(u^2\ell_P^4)$, and $$ z^{n-1}z'=u^{n-1}\Bigl(1+\frac{(n-1)u\ell_P^2}{4(d+2)}\Bigr)\Bigl(1+\frac{u\ell_P^2}{2(d+2)}\Bigr)+O(u^{n+1}\ell_P^4)=u^{n-1}\Bigl(1+\frac{(n+1)u\ell_P^2}{4(d+2)}\Bigr)+O(u^{n+1}\ell_P^4).\qquad\square $$ The inequality $1-\cos\theta\leq\theta^2/2$ shows $\lambda(p)\leq\tfrac12\mathbb E[(p\cdot\ell)^2]$ for *every* symmetric step distribution, isotropic or not: discreteness softens the dispersion. This sign is used in Proposition 11.4. H$_{\mathrm{iso}}$ is a model hypothesis: the displacement distribution of a sprinkled Hasse graph is concentrated near the light cone and is not isotropic; the lemma is used only to fix a definite value of the coefficient in Proposition 11.4(b). **Definition 10.10** (Euclidean continuation; hypothesis H$_{\mathrm W}$). Let $(\mathcal M,g^{\mathrm L})$ be the Lorentzian continuum limit of Theorem 5.6. Hypothesis H$_{\mathrm W}$: there is a global time function $\mathsf t$ on $\mathcal M$ such that, on the open set $U\subset\mathcal M$ on which the backgrounds of Chapters 11 and 12 are taken, $g^{\mathrm L}=-N^2\mathrm d\mathsf t^2+h_{ij}\mathrm dx^i\mathrm dx^j$ with $N$ and $h$ independent of $\mathsf t$. The *Euclidean continuation* is the Riemannian metric $g=N^2\mathrm d\mathsf t_E^2+h_{ij}\mathrm dx^i\mathrm dx^j$ on $U$ obtained by $\mathsf t=-i\mathsf t_E$. All heat-kernel computations of Chapters 11 and 12 are carried out on $(U,g)$; the relation between the Euclidean and Lorentzian flows is the content of Proposition 12.6 and is not used to define $g$. ### 10.5. Truncations and theory space **Definition 10.11** (Truncation). Let $\mathcal T$ be the space of background-gauge-invariant functionals of $(h,a,\psi;\bar g,\bar A)$ on $(U,g)$. The theory space on which the truncated flows of Chapters 11 and 12 are solved is this continuum space of functionals on $(U,g)$, with regulators inserted on the pulled-back fields (Definition 10.8); it is not the finite-dimensional space $\Phi$ of Definition 10.1, and the exact equation (10.3) on $\Phi$ is not the equation whose truncations are studied. The content of §10.2-10.3 used later is Lemma 10.7 (spectral calculus for the regulator) and the general form (10.3), which is the same identity on any field space where Lemma 10.3 holds. A *truncation* is a finite-dimensional affine subspace $\mathcal T_{\mathrm{tr}}\subset\mathcal T$ spanned by invariants $I_1,\dots,I_m$ with $k$-dependent coefficients $u_i(k)$, together with the projection $\Pi$ defined by expanding the right-hand side of (10.3) in the same invariants and discarding the remainder. The *truncated flow* is the ODE system $\partial_tu_i=\beta_i(u)$ obtained by applying $\Pi$. The Einstein-Hilbert truncation is $m=2$, $I_1=\int\sqrt g$, $I_2=\int\sqrt g\,R$. **Definition 10.12** (Fixed points, critical exponents). A *fixed point* is a zero $u^*$ of $\beta$ in the dimensionless couplings; its *stability matrix* is $M_{ij}=\partial\beta_i/\partial u_j|_{u^*}$; the *critical exponents* are $\theta_I=-\mathrm{eig}_I(M)$; a direction with $\mathrm{Re}\,\theta_I>0$ is UV-relevant. **Conjecture 10.13** $\boxed{\text{Conjecture}}$ (Asymptotic safety of the exact flow). *The exact flow (10.3) on the continuum theory space of the QSN admits a fixed point with finitely many UV-relevant directions.* This is Weinberg's conjecture (Weinberg, in *General Relativity: An Einstein Centenary Survey*, Cambridge, 1979). Nothing in this monograph proves it. ### 10.6. Heat-kernel evaluation of traces For a Laplace-type operator $\Delta_E=-\nabla^2+E$ on sections of a bundle of fibre dimension $\mathrm{tr}\,\mathbb 1$ over a closed $d$-manifold, and for $W$ with compact support in the spectral variable, $$ \mathrm{Tr}\,W(\Delta_E)=\frac1{(4\pi)^{d/2}}\sum_{m\geq0}Q_{d/2-m}[W]\int\mathrm d^dx\sqrt g\,\mathrm{tr}\,b_{2m}(x),\qquad Q_n[W]=\frac1{\Gamma(n)}\int_0^\infty\mathrm dz\,z^{n-1}W(z)\ (n>0),\quad Q_0[W]=W(0),\tag{10.8} $$ with $b_0=\mathbb 1$, $b_2=\tfrac R6\mathbb 1-E$ (Vassilevich, *Phys. Rep.* **388** (2003) 279, eqs. (4.26)-(4.27)). When the regulator is a function of $-\nabla^2$ alone and $E$ is kept in the propagator (the type Ia prescription used throughout Chapters 11 and 12), the expansion is applied to $W(-\nabla^2)$ and the $E$-dependence is obtained from $(P_k+E)^{-1}=P_k^{-1}-EP_k^{-2}+\ldots$, i.e. through $Q_n[W/(P_k+w)^2]$, not through $b_2$. **Lemma 10.14** $\boxed{\text{Theorem}}$ (Litim threshold functions). *For $R_k=Z_k(k^2-z)\theta(k^2-z)$, $P_k=z+R_k/Z_k$, $\eta=-\partial_t\log Z_k$, and $w>-k^2$, define (Reuter, *Phys. Rev. D* **57** (1998) 971, eq. (4.33))* $$ \Phi^p_n(\widetilde w):=\frac1{2k^{2(n-p+1)}}\,Q_n\Bigl[\frac{\partial_tR_k/Z_k+\eta R_k/Z_k}{(P_k+w)^p}\Bigr],\qquad\widetilde\Phi^p_n(\widetilde w):=\frac1{k^{2(n-p+1)}}\,Q_n\Bigl[\frac{R_k/Z_k}{(P_k+w)^p}\Bigr],\qquad\widetilde w=w/k^2.\tag{10.9a} $$ *Then* $$ \Phi^p_n(\widetilde w)=\frac1{\Gamma(n+1)(1+\widetilde w)^p},\qquad\widetilde\Phi^p_n(\widetilde w)=\frac1{\Gamma(n+2)(1+\widetilde w)^p},\tag{10.9b} $$ *and, for the $\eta$-independent part, $Q_n\bigl[2k^2\theta(k^2-z)/(P_k+w)^p\bigr]=2k^{2(n-p+1)}\Phi^p_n(\widetilde w)$. In particular $Q_2[\partial_tR_k/(Z_kP_k)]|_{\eta=0}=k^4$, $Q_1[\ldots]=2k^2$, and $Q_2[2k^2\theta/(P_k+w)^2]=k^2(1+\widetilde w)^{-2}$.* *Proof.* $\partial_tR_k=Z_k[2k^2\theta(k^2-z)]-\eta R_k$, so $\partial_tR_k/Z_k+\eta R_k/Z_k=2k^2\theta(k^2-z)$. On the support $z0,\ x>0,\ 1-gB_2(\lambda)>0\}$, $\partial_tg=0$ is equivalent to $\eta_N=-2$, which is equivalent to* $$ g=g_c(\lambda):=\frac{-2}{B_1(\lambda)-2B_2(\lambda)}=\frac{24\pi x^2}{28x^2-30x+48}.\tag{11.5} $$ *The denominator has discriminant $900-5376=-4476<0$, so it is positive on $\mathbb R$, $g_c>0$ for $x>0$, and $g_c$ is real-analytic there.* *Proof.* On $\mathcal D$, $(2+\eta_N)g=0$ forces $\eta_N=-2$; by (11.3) this is $g(B_1-2B_2)=-2$. From (11.4), $B_1-2B_2=\frac1{12\pi}\bigl[\frac{20}x-\frac{36}{x^2}-28\bigr]+\frac1{12\pi}\bigl[\frac{10}x-\frac{12}{x^2}\bigr]=\frac{30x-48-28x^2}{12\pi x^2}$, whence (11.5). $\square$ **Lemma 11.10** $\boxed{\text{Theorem}}$ (Reduced fixed-point function). *On $g=g_c(\lambda)$ the right-hand side of $\partial_t\lambda$ in (11.3) equals* $$ h(\lambda)=-4\lambda+\frac{16x(5-3x)}{28x^2-30x+48},\qquad x=1-2\lambda,\tag{11.6} $$ *and $h(\lambda)=0$ if and only if $p(x):=14x^3-41x^2+59x-24=0$.* *Proof.* At $\eta_N=-2$: $-(2-\eta_N)\lambda=-4\lambda$ and the bracket is $\frac5x-4+\frac5{3x}=\frac{20}{3x}-4$; multiplying by $\frac{g_c}{2\pi}=\frac{12x^2}{28x^2-30x+48}$ gives $\frac{80x-48x^2}{28x^2-30x+48}$. With $\lambda=\frac{1-x}2$, $h=0$ reads $-2(1-x)(28x^2-30x+48)+80x-48x^2=0$; expanding, $56x^3-116x^2+156x-96+80x-48x^2=56x^3-164x^2+236x-96=4p(x)$. $\square$ ### 11.4. Existence and uniqueness of the vacuum NGFP **Theorem 11.1** $\boxed{\text{Theorem modulo Classical (Reuter)}}$ (Non-Gaussian fixed point of the Einstein-Hilbert truncation). *The system (11.3)-(11.4) has exactly one fixed point in $\mathcal D$. It is $x^*=1-2\lambda^*$, the unique real root of $p(x)=14x^3-41x^2+59x-24$, and $g^*=g_c(\lambda^*)$:* $$ x^*=0.6135989857\ldots,\qquad\lambda^*=0.1932005072\ldots,\qquad g^*=0.7073208810\ldots,\tag{11.7} $$ *all displayed digits correct, and $1-g^*B_2(\lambda^*)=0.853890>0$.* *Proof.* *Root.* $p'(x)=42x^2-82x+59$ has discriminant $6724-9912=-3188<0$, so $p$ is strictly increasing on $\mathbb R$; $p(0)=-24$, $p(1)=8$; exactly one real root, in $(0,1)$ (intermediate value theorem, the one-dimensional case of Classical L). An axis-aligned rectangle in the $(g,\lambda)$ plane does not satisfy the Poincaré-Miranda sign conditions for (11.3), since the zero set of $\partial_tg$ is the curve (11.5) and not a vertical line; the v2.0 rectangle $[0.4,0.9]\times[0.05,0.4]$ moreover contains the singular locus $1-gB_2=0$ (at $g=0.9$ it is crossed at $\lambda=0.338$, the solution of $1-0.9B_2(\lambda)=0$), so the v2.0 appeal to Brouwer on it was invalid. The reduction of Lemmas 11.9 and 11.10 replaces it. *Membership and uniqueness in $\mathcal D$.* $x^*\in(0,1)$ gives $\lambda^*\in(0,\tfrac12)$ and $g^*>0$ by Lemma 11.9. Conversely a fixed point in $\mathcal D$ satisfies $g=g_c(\lambda)$ (Lemma 11.9) and $p(1-2\lambda)=0$ (Lemma 11.10), hence is this one. *Digits and error bounds.* Newton on $p$ from $x_0=0.6$: $p(0.6)=-0.336$, $p'(0.6)=24.92$, $x_1=0.61348$; converged value $x^*=0.613598985698\ldots$. Since $p'\geq p'(82/84)=18.976$ on $\mathbb R$, $|x-x^*|\leq|p(x)|/18.976$ for all $x$; at $x=0.6135989857$, $|p(x)|=3.85\cdot10^{-11}$, so $|x-x^*|<2.1\cdot10^{-12}$, certifying ten digits of $x^*$ and of $\lambda^*=(1-x^*)/2$. For $g^*$: $g_c'(x)=24\pi\frac{2x(28x^2-30x+48)-x^2(56x-30)}{(28x^2-30x+48)^2}$, with $g_c'(x^*)=2.229$ and $\max_{[0.6,0.62]}|g_c'|=2.243$, so $|g_c(x)-g^*|\leq2.25\,|x-x^*|<5\cdot10^{-12}$. Finally $B_2(\lambda^*)=-\frac1{24\pi}\bigl[\frac{10}{0.6135990}-\frac{12}{0.3765037}\bigr]=-\frac1{24\pi}[16.2973-31.8722]=0.2065686$ and $1-g^*B_2=1-0.1461103=0.853890$. $\square$ **Remark 11.11** (Newton residual in the original variables). At $(g,\lambda)=(0.7072,0.1932)$ the residuals of (11.3) are $(\partial_tg,\partial_t\lambda)=(2.884\cdot10^{-4},-1.156\cdot10^{-4})$, $\|M^{-1}\|_2=0.936$ for the Jacobian $M$ there, and the Newton step $-M^{-1}(\partial_tg,\partial_t\lambda)^{\mathsf T}=(+1.209\cdot10^{-4},+5.1\cdot10^{-7})$ moves the point to $(0.70732,0.19320)$, consistent with (11.7). ### 11.5. Critical exponents **Theorem 11.2** $\boxed{\text{Theorem modulo Classical (Reuter)}}$ (Stability matrix and critical exponents). *At the fixed point (11.7),* $$ M^*=\frac{\partial(\partial_tg,\partial_t\lambda)}{\partial(g,\lambda)}\Big|_*=\begin{pmatrix}-2.342223&-10.439826\\0.959082&-0.608382\end{pmatrix},\qquad\mathrm{tr}\,M^*=-2.950605,\quad\det M^*=11.437619,\tag{11.8} $$ *and the critical exponents are the complex pair $\theta_{1,2}=1.4753024\pm3.0432058\,i$, both with positive real part: two UV-relevant directions, spiral approach with period $2\pi/3.0432=2.0647$ in $t$.* *Proof.* With $\eta_N=gB_1/(1-gB_2)$: $\partial_g\eta_N=B_1/(1-gB_2)^2$, $\partial_\lambda\eta_N=[gB_1'(1-gB_2)+g^2B_1B_2']/(1-gB_2)^2$, $B_1'=\frac1{12\pi}\bigl[\frac{40}{x^2}-\frac{144}{x^3}\bigr]$, $B_2'=-\frac1{24\pi}\bigl[\frac{20}{x^2}-\frac{48}{x^3}\bigr]$ (using $\mathrm dx/\mathrm d\lambda=-2$). Then $M_{gg}=2+\eta_N+g\partial_g\eta_N$, $M_{g\lambda}=g\partial_\lambda\eta_N$, $M_{\lambda g}=\lambda\partial_g\eta_N+\frac1{2\pi}\bigl[\frac5x-4-\frac{5\eta_N}{6x}\bigr]-\frac{5g}{12\pi x}\partial_g\eta_N$, $M_{\lambda\lambda}=-(2-\eta_N)+\lambda\partial_\lambda\eta_N+\frac g{2\pi}\bigl[\frac{10}{x^2}-\frac{5\partial_\lambda\eta_N}{6x}-\frac{10\eta_N}{6x^2}\bigr]$. At the fixed point: $\eta_N=-2$, $x^*=0.6135990$, $B_1=-2.414434$, $B_2=0.2065686$, $1-gB_2=0.853890$, $B_1'=\frac1{12\pi}[106.24-623.28]=-13.7159$, $B_2'=-\frac1{24\pi}[53.12-207.76]=2.05113$; $\partial_g\eta_N=-2.414434/0.729128=-3.311401$; $\partial_\lambda\eta_N=[0.707321\cdot(-13.7159)\cdot0.853890+0.500303\cdot(-2.414434)\cdot2.05113]/0.729128=-14.759675$. Hence $M_{gg}=-0.707321\cdot3.311401=-2.342223$; $M_{g\lambda}=0.707321\cdot(-14.759675)=-10.439826$; $M_{\lambda g}=-0.639764+1.092576+0.506270=0.959082$; $M_{\lambda\lambda}=-4-2.851577+0.112574\cdot[26.5602+20.0452+8.8534]=-0.608382$. Characteristic polynomial $\mu^2+2.950605\mu+11.437619$, discriminant $8.70607-45.75048<0$, roots $\mu=-1.4753024\pm i\sqrt{11.437619-2.176517}=-1.4753024\pm3.0432058\,i$, $\theta=-\mu$. Stability of the displayed digits: the entries of $M$ along the curve (11.5) have $\|\mathrm dM/\mathrm dx\|_2=14.0$ at $x^*$ (central finite difference, $h=10^{-5}$), so an error $2\cdot10^{-12}$ in $x^*$ perturbs the entries by less than $3\cdot10^{-11}$. $\square$ **Remark 11.12** (Correction to v2.0). The v2.0 text quoted real exponents $\theta_G\approx2.3$, $\theta_\Lambda\approx-0.3$ alongside $\det J^*\approx11.26$; for real exponents the determinant would be negative. The real-exponent claim is withdrawn; the determinant is consistent with the complex pair, $|\theta|^2=11.44$. The fixed point and exponents coincide with Litim (*Phys. Rev. Lett.* **92** (2004) 201301). ### 11.6. Gravitational contribution to the gauge anomalous dimension Extend the truncation by $\frac{Z_{A,k}}4\int\sqrt g\,F^a_{\mu\nu}F^{a\,\mu\nu}$ and set $\eta_A=-\partial_t\log Z_{A,k}$; $\eta_A^{\mathrm{grav}}$ is its part of order $g$ at zeroth order in the gauge coupling. Two one-loop diagrams contribute: the graviton tadpole (the $O(F^2)$ part of the graviton Hessian inserted once in the graviton loop) and the mixed graviton-gauge loop (two $hAA$ vertices, one graviton and one gauge propagator). **Lemma 11.13** $\boxed{\text{Theorem}}$ (Structure; conformal-mode decoupling). *In $d=4$, $\alpha=1$, Litim regulator, at one graviton loop,* $$ \eta_A^{\mathrm{grav}}=-\frac g\pi\Bigl[\frac{c_1}{1-2\lambda}+\frac{c_2}{(1-2\lambda)^2}\Bigr]+O(g^2),\tag{11.9} $$ *with $c_1,c_2$ rational, independent of $\lambda$ and of the gauge group; the trace mode $\phi$ contributes to neither.* *Proof.* Both diagrams are traces of products of graviton propagators $(P_k-2\Lambda)^{-1}$ (Proposition 11.5 with $\bar R=0$ after projecting on $F^2$), gauge propagators $P_k^{-1}$, and $\partial_tR_k$ insertions. The tadpole contains the graviton propagator squared: $Q_2[\partial_tR/(P-2\Lambda)^2]\propto(1-2\lambda)^{-2}$ by Lemma 10.14. The mixed loop contains one gauge propagator and one graviton propagator, with the $\partial_tR_k$ insertion on either line: with the insertion on the gauge line it contains one graviton propagator, giving $(1-2\lambda)^{-1}$ times threshold functions at argument zero, and with the insertion on the graviton line it contains two graviton propagators, giving $(1-2\lambda)^{-2}$; so the mixed loop contributes to both $c_1$ and $c_2$. $\lambda$ enters only through the graviton mass term, so no other dependence arises. Group independence: the graviton couples to $F^aF^a$ through the stress tensor, diagonal in the adjoint index, whose trace reproduces the normalisation of the $F^2$ term. Decoupling of $\phi$: $\int\sqrt g\,g^{\mu\alpha}g^{\nu\beta}F_{\mu\nu}F_{\alpha\beta}$ is invariant under $g\to\Omega^2g$ in $d=4$, hence its expansion in $h$ at flat $\bar g$ has no $\phi$-dependence; explicitly, with $(F^2)_{\mu\alpha}=F_{\mu\nu}F_\alpha{}^\nu$ and $F^2=F_{\mu\nu}F^{\mu\nu}$, the second-order Lagrangian is $\mathcal L_2=\tfrac14(\tfrac18\phi^2-\tfrac14h_{ab}h^{ab})F^2-\tfrac14\phi h^{\mu\alpha}(F^2)_{\mu\alpha}+\tfrac12h^{\mu\rho}h_\rho{}^\alpha(F^2)_{\mu\alpha}+\tfrac14h^{\mu\alpha}h^{\nu\beta}F_{\mu\nu}F_{\alpha\beta}$, and at $h_{\mu\nu}=\tfrac14\delta_{\mu\nu}\phi$ the four terms are $F^2\phi^2(\tfrac1{16}-\tfrac14+\tfrac18+\tfrac1{16})=0$; the first-order Lagrangian $\tfrac18\phi F^2-\tfrac12h^{\mu\alpha}(F^2)_{\mu\alpha}$ likewise vanishes there. $\square$ **Lemma 11.14** $\boxed{\text{Theorem}}$ (Tadpole coefficient). *The tadpole contributes $-\frac{3g}{2\pi(1-2\lambda)^2}$ to $\partial_t(Z_A/4)$, i.e. $c_2^{\mathrm{tad}}=-6$ in (11.9) at $Z_A=1$.* *Proof.* Let $E=\partial^2\mathcal L_2/\partial h\partial h$ at flat background. Summing the quadratic form $\mathcal L_2$ over an orthonormal basis $\{e^{(i)}\}$ of symmetric tensors with the projector $\sum_ie^{(i)}_{\mu\nu}e^{(i)}_{\alpha\beta}=\tfrac12(\delta_{\mu\alpha}\delta_{\nu\beta}+\delta_{\mu\beta}\delta_{\nu\alpha})$ gives, term by term, $-\tfrac12F^2,-\tfrac14F^2,+\tfrac54F^2,-\tfrac18F^2$, so $\sum_i\mathcal L_2(e^{(i)})=\tfrac38F^2$; since $\mathcal L_2(h)=\tfrac12\langle h,Eh\rangle$, the diagonal of $E$ is $2\mathcal L_2$ and $\mathrm{tr}_{\mathrm{all}}E=2\cdot\tfrac38F^2=\tfrac34F^2$; the trace mode contributes $0$ (Lemma 11.13), so $\mathrm{tr}_{\hat h}E=\tfrac34F^2$. The $\hat h$ kinetic normalisation is $\frac1{32\pi G}\cdot\tfrac12$, so $\widetilde E=64\pi G\,E$. Expanding $(\Gamma^{(2)}+R)^{-1}=(P-2\Lambda)^{-1}-(P-2\Lambda)^{-1}\widetilde E(P-2\Lambda)^{-1}+\ldots$ in (10.3): $\partial_t\Gamma\supset-\tfrac12\mathrm{Tr}[(P-2\Lambda)^{-2}\partial_tR\,\widetilde E]=-\tfrac12\cdot\frac1{16\pi^2}\cdot k^2(1-2\lambda)^{-2}\cdot64\pi G\cdot\tfrac34\int F^2=-\frac{3Gk^2}{2\pi(1-2\lambda)^2}\int F^2$, using $Q_2[2k^2\theta/(P+w)^2]=k^2(1+\widetilde w)^{-2}$ (Lemma 10.14). $\square$ **Theorem 11.15** (Classical input FLP: Folkerts, Litim, Pawlowski, *Phys. Lett. B* **709** (2012) 234, eq. (17); Daum, Harst, Reuter, *Phys. Lett. B* **683** (2010) 196, eq. (4.5)). *In the truncation $\Gamma_k^{\mathrm{EH}}+\frac{Z_A}4\int\sqrt gF^2$ on a flat background, with background de Donder gauge $\alpha=1$ for the graviton, background Feynman gauge for the gauge field, Litim regulators on both the graviton and the gauge fluctuation at the same scale $k$, and at one loop in $g$ and zeroth order in the gauge coupling, the sum of the tadpole and mixed diagrams gives $\eta_A^{\mathrm{grav}}=-\frac{5g}{3\pi(1-2\lambda)^2}+O(g^2)$, i.e. $c_1=0$, $c_2=\tfrac53$ in (11.9).* **Proposition 11.3** $\boxed{\text{Theorem modulo Classical (FLP)}}$ (Gravitational anomalous dimension of gauge fields). *Under the hypotheses of Theorem 11.15,* $$ \eta_A^{\mathrm{grav}}=-\frac{5g}{3\pi(1-2\lambda)^2}+O(g^2),\qquad\eta_A^{\mathrm{grav}}\big|_*=-\frac{5\cdot0.7073209}{3\pi\cdot0.3765037}=-0.9967.\tag{11.10} $$ *Proof.* Lemma 11.13 gives the form (11.9) and Lemma 11.14 the tadpole value $c_2^{\mathrm{tad}}=-6$; both are proved here. Theorem 11.15 supplies the sum $c_1=0$, $c_2=\tfrac53$, so the mixed diagram contributes $c_2^{\mathrm{mix}}=\tfrac53-(-6)=\tfrac{23}3$ and $c_1^{\mathrm{mix}}=0$: the mixed diagram over-compensates the tadpole and fixes the overall sign. The hypotheses of Theorem 11.15 are those of the present truncation: flat background (Definition 10.10 on the Minkowski family of Chapter 20), gauges as stated, regulator of Definition 10.8 on both lines. The mixed-diagram algebra is not reproduced in this monograph; the sign of (11.10) at the fixed point therefore rests on Theorem 11.15 and is cited, not derived. The numerical value uses (11.7). $\square$ **Remark 11.16** (Withdrawn intermediate expression). The v2.0 Companion displayed a coefficient $\tfrac5{24}$ and asserted its equivalence to $\tfrac53$ "up to normalisation"; this is not an equivalence, and the expression is withdrawn. ### 11.7. The discreteness correction **Proposition 11.4** (Discreteness correction). *(a)* $\boxed{\text{Theorem}}$ *Let the truncation be written with $\Delta_{\mathsf Q}\otimes\mathbb 1$ in place of $-\nabla^2$ on pulled-back fields (Definition 10.8), let the background be flat, and assume (H$_{\mathrm{sym}}$) that $\Delta_{\mathsf Q}$ is unitarily equivalent, on the scaling window $u\leq u_0$, to a multiplication operator by a symbol $u(p)$ on $L^2(\mathbb R^4,\mathrm d^4p)$ with $u(z)=z(1+a\,z\ell_P^2+O(z^2\ell_P^4))$, $z=|p|^2$, and $u'(z)>0$ there (H$_{\mathrm{iso}}$ of Lemma 10.9 is a sufficient condition). Then every trace $\mathrm{Tr}_{N\otimes\mathcal V}[W(\Delta_{\mathsf Q}\otimes\mathbb 1)]$ on the flat background, evaluated through $Q_n$ with $n\geq1$ and a Litim regulator, equals its continuum value multiplied by* $$ 1-n\,a\,(k\ell_P)^2+O\bigl((k\ell_P)^4\bigr).\tag{11.11} $$ *For any symmetric random-walk Laplacian, $a\leq0$, hence the correction factor is $\geq1$.* *(b)* $\boxed{\text{Conditional (on H$_{\mathrm{iso}}$, and on Proposition 11.3 together with the single-$Q_2$ hypothesis stated in the proof)}}$ *In the isotropic-step model in $d=4$, $a=-\tfrac1{24}$. If the sum of diagrams in Theorem 11.15 is, on the QSN, a single $Q_2$-trace of the form covered by (a), then* $$ \eta_A^{\mathrm{grav,QSN}}=\eta_A^{\mathrm{grav}}\Bigl[1+\tfrac1{12}(k\ell_P)^2+O((k\ell_P)^4)\Bigr].\tag{11.12} $$ *Proof of (a).* Under H$_{\mathrm{sym}}$ the trace is $\mathrm{tr}\mathbb 1_{\mathcal V}\cdot\frac1{16\pi^2}\int\mathrm d^4p\,W(u(p))=\mathrm{tr}\mathbb 1_{\mathcal V}\cdot\frac1{16\pi^2}\cdot\pi^2\int z\,W(u(z))\mathrm dz$ for $n=2$, and similarly with $z^{n-1}$ for the $Q_n$ contributions of lower-dimensional or tensor-weighted traces; by the push-forward argument of Lemma 10.9 with general $a$, $z^{n-1}\mathrm dz=u^{n-1}(1-(n+1)a\,u\ell_P^2+O(u^2\ell_P^4))\mathrm du$. With the Litim regulator the integrand is supported on $u\leq k^2$ and equals $2k^2(k^2+w)^{-p}$ there (Definition 10.8), so $Q_n^{\mathsf Q}=\frac{2k^2}{\Gamma(n)(k^2+w)^p}\bigl[\frac{k^{2n}}n-a\ell_P^2k^{2n+2}\bigr]+O(k^{2n+6}\ell_P^4)=Q_n^{\mathrm{cont}}[1-na(k\ell_P)^2+O((k\ell_P)^4)]$. Sign: for a symmetric step law $\lambda(p)=\mathbb E[1-\cos(p\cdot\ell)]\leq\tfrac12\mathbb E[(p\cdot\ell)^2]$, so $u(z)\leq z$ and $a\leq0$. The remainder is uniform for $k^2\leq u_0$ with $u_0\ell_P^2<2(d+2)$ by Lemma 10.9. Traces multiplying curvature ($b_2$ and higher) are not covered: their $O((k\ell_P)^2)$ corrections involve the heat-kernel coefficients of the lattice Laplacian and are not computed here. *Proof of (b).* Lemma 10.9 gives $a=-\tfrac1{24}$. Under the single-$Q_2$ hypothesis, (a) with $n=2$ multiplies (11.10) by $1+\tfrac1{12}(k\ell_P)^2$. The tag is the minimum over the inputs (Rigor standard 10): H$_{\mathrm{iso}}$, Proposition 11.3 (modulo Classical FLP), and the single-$Q_2$ hypothesis, which is an assumption on the momentum structure of the mixed-diagram vertices and is not verified here. $\square$ **Remark 11.17** (Scheme dependence; correction to v2.0). The v2.0 value $\alpha=-\tfrac23$ cannot arise from a symmetric random-walk Laplacian with a Litim regulator, for which (a) proves a non-negative coefficient; it came from the by-hand profile $\chi$, for which the correction vanishes identically for $k\ell_P<1/\sqrt2$ and no $(k\ell_P)^2$ coefficient exists. It is withdrawn. The coefficient depends on the regulator profile and on the microscopic step law through $a$; the form (11.11) and the sign are scheme independent within symmetric random-walk Laplacians and monotone regulators. Phenomenological statement P1 of Chapter 18 must be read with $+\tfrac1{12}$ in the isotropic model and the sign as its robust content. **Remark 11.18** (Order of limits; dimensions). Proposition 11.4 holds at fixed $\ell_P$ as $k\ell_P\to0$; the continuum limit of Part II rescales $\ell_P\to\ell_P/n$ at fixed physical $k$ and removes the correction. The two limits commute on the coefficient because (10.6) is an expansion in the single variable $z\ell_P^2$. With units restored, $(k\ell_P)^2=\hbar Gk^2/c^3$ is dimensionless, as is $a\ell_P^2z$. ### 11.8. Hypothesis discharge ledger, Chapter 11 | Result | Classical inputs | Hypotheses used | Where consumed | |---|---|---|---| | Prop 11.5 | Thm 11.6(a) (Reuter) | Def 10.10 background; $\alpha=1$ | Hessian | | Thm 11.7 | Thm 11.6(b) (Reuter); Lem 10.14 | type Ia Litim (Def 10.8), $k\ell_P\to0$ | projection | | Lem 11.9, 11.10 | none | domain $\mathcal D$ | reduction | | Thm 11.1 | Classical L (one-dimensional) | via Thm 11.7 | existence, uniqueness, digits | | Thm 11.2 | none | via Thm 11.7 | exponents | | Lem 11.13, 11.14 | Weyl invariance of Maxwell in $d=4$ | one graviton loop, $\alpha=1$, Litim, flat background | structure, tadpole | | Prop 11.3 | Thm 11.15 (FLP) | hypotheses of Thm 11.15 | coefficient and sign | | Prop 11.4(a) | none | symmetric random-walk Laplacian, Litim, flat background, Def 10.8 | form and sign | | Prop 11.4(b) | Lem 10.9; Prop 11.3 | H$_{\mathrm{iso}}$; single-$Q_2$ hypothesis | value $\tfrac1{12}$ | --- ## Chapter 12. Standard-Model-Shifted Fixed Point and Truncation Refinements ### 12.1. Minimally coupled matter in the type Ia scheme Add to (11.1) $N_S$ real scalars, $N_D$ Dirac fermions and $N_V$ gauge vectors, minimally coupled, with kinetic operators $-\nabla^2$, $-\slashed D^2=-\nabla^2+\tfrac R4$ (Lichnerowicz), and $-\nabla^2\delta^\mu_\nu+R^\mu{}_\nu$ in Feynman gauge with two real scalar ghosts per vector; matter wave-function renormalisations are set to one. The regulator is of type Ia in every sector, as in Theorem 11.6: a function of $-\nabla^2$ only, with the endomorphisms kept in the propagator. **Lemma 12.3** $\boxed{\text{Theorem}}$ (Matter contributions, type Ia, Litim, $d=4$). *The matter fields add to the right-hand sides of (11.3) the terms* $$ \partial_t\lambda\supset\frac g{4\pi}V_{\mathrm{mat}},\qquad V_{\mathrm{mat}}=N_S-4N_D+2N_V,\tag{12.1} $$ $$ \eta_N=\frac{g\bigl[B_1(\lambda)+\frac{M_{\mathrm{mat}}}{6\pi}\bigr]}{1-gB_2(\lambda)},\qquad M_{\mathrm{mat}}=N_S-N_D-N_V.\tag{12.2} $$ *Proof.* Each field contributes $\pm\tfrac12\mathrm{Tr}[(P_k+E)^{-1}\partial_tR_k]$ with $+$ for bosons, $-$ for fermions and ghosts, and $P_k$ built from $-\nabla^2$. Expand $(P_k+E)^{-1}=P_k^{-1}-EP_k^{-2}+O(E^2)$ and apply (10.8) to the $-\nabla^2$ traces: $\mathrm{Tr}[\partial_tR/P]=\frac1{16\pi^2}\bigl[k^4\,\mathrm{tr}\mathbb 1\int\sqrt g+2k^2\,\mathrm{tr}\mathbb 1\int\sqrt g\tfrac R6\bigr]$ and $\mathrm{Tr}[E\,\partial_tR/P^2]=\frac1{16\pi^2}k^2\int\sqrt g\,\mathrm{tr}E$ at leading order in $R$ (Lemma 10.14: $Q_2[\partial_tR/P]=k^4$, $Q_1[\partial_tR/P]=2k^2$, $Q_2[\partial_tR/P^2]=k^2$). Per field the coefficients of $\frac1{32\pi^2}\int\sqrt g$ and $\frac1{32\pi^2}\int\sqrt gR$ are: scalar ($\mathrm{tr}\mathbb 1=1$, $E=0$): $+k^4$ and $+\tfrac{k^2}3$; Dirac ($\mathrm{tr}\mathbb 1=4$, $E=\tfrac R4$): $-4k^4$ and $-\bigl[4\cdot\tfrac{2k^2}6-k^2\cdot4\cdot\tfrac14\bigr]=-\bigl[\tfrac{4k^2}3-k^2\bigr]=-\tfrac{k^2}3$; vector ($\mathrm{tr}\mathbb 1=4$, $\mathrm{tr}E=\mathrm{tr}\,\mathrm{Ric}=R$) plus two ghosts ($\mathrm{tr}\mathbb 1=1$, $E=0$, sign $-$): $(4-2)k^4=2k^4$ and $\bigl[4\cdot\tfrac{2k^2}6-k^2\bigr]-2\cdot\tfrac{k^2}3=\tfrac{4k^2}3-k^2-\tfrac{2k^2}3=-\tfrac{k^2}3$. Matching to $\partial_t\bigl(\frac{\Lambda_k}{8\pi G_k}\bigr)=\frac{k^4}{8\pi g}[\partial_t\lambda+(2-\eta_N)\lambda]$ gives $\partial_t\lambda+(2-\eta_N)\lambda\supset\frac{8\pi g}{32\pi^2}(N_S-4N_D+2N_V)$, which is (12.1). Matching to $\partial_t\bigl(-\frac1{16\pi G_k}\bigr)=\frac{\eta_Nk^2}{16\pi g}$ gives $\eta_N\supset\frac{16\pi g}{32\pi^2}\cdot\tfrac13(N_S-N_D-N_V)=\frac g{6\pi}M_{\mathrm{mat}}$ as an additive term in the numerator before solving the linear relation for $\eta_N$, which yields (12.2). $\square$ **Remark 12.7** (Scheme and correction to v2.0). In the type II scheme (regulator a function of the full $\Delta_E$), the Dirac and vector $R$-coefficients change sign and magnitude and one obtains $N_S+2N_D-4N_V$; the sensitivity of the fermion sign to the scheme is discussed by Dona, Eichhorn, Percacci (*Phys. Rev. D* **89** (2014) 084035, §III.B). Since the gravitational sector of Theorem 11.7 is type Ia, Lemma 12.3 uses type Ia for consistency. The v2.0 "stability bound" $n_S+2n_F-4n_V=52<100$ (with $n_F$ counting Weyl fermions against a coefficient that multiplies Dirac fermions, and an underived threshold) is withdrawn. With $N_S=4$, $N_D=24$ (48 Weyl fermions, three right-handed neutrinos included), $N_V=12$: $M_{\mathrm{SM}}=4-24-12=-32$, $V_{\mathrm{SM}}=4-96+24=-68$; without right-handed neutrinos, $N_D=22.5$, $M=-30.5$, $V=-62$. ### 12.2. The matter-dependent cubic **Proposition 12.4** $\boxed{\text{Theorem}}$ (Matter-dependent fixed-point cubic). *With $x=1-2\lambda$, on $\mathcal D_M=\{g>0,\ x>0,\ D_M(x)>0,\ 1-gB_2>0\}$ the fixed points are in bijection with the positive real roots of* $$ p_M(x)=-2(1-x)D_M(x)+80x+(6V_{\mathrm{mat}}-48)x^2,\qquad D_M(x)=(28-2M_{\mathrm{mat}})x^2-30x+48,\tag{12.3} $$ *satisfying $D_M(x)>0$ and $1-g_c(x)B_2>0$, via $g=g_c(x)=24\pi x^2/D_M(x)$. If $M_{\mathrm{mat}}<\tfrac{4476}{384}=11.656$ then $\mathrm{disc}(D_M)=900-192(28-2M_{\mathrm{mat}})<0$ and $D_M>0$ on $\mathbb R$.* *Proof.* As in Lemma 11.9, $\eta_N=-2$ is $g\bigl[B_1+\frac M{6\pi}-2B_2\bigr]=-2$, i.e. $g=\frac{24\pi x^2}{28x^2-30x+48-2Mx^2}$. On this curve the $\lambda$-equation reads $-2(1-x)+\frac{12x^2}{D_M}\bigl[\frac{20}{3x}-4+\frac V2\bigr]=0$, i.e. (12.3). The discriminant statement is arithmetic. $\square$ **Proposition 12.5** $\boxed{\text{Theorem}}$ (Existence criterion). *For $M_{\mathrm{mat}}<11.656$, $p_M$ has a positive real root, and each positive root with $1-g_cB_2>0$ is a fixed point in $\mathcal D_M$. For $11.656\leq M_{\mathrm{mat}}<14$, $D_M$ has real roots and $g_c$ changes sign on $(0,\infty)$; for $M_{\mathrm{mat}}\geq14$ the leading coefficient $2(28-2M_{\mathrm{mat}})$ of $p_M$ is non-positive. In both latter cases existence of a positive root with $D_M>0$ and $1-g_cB_2>0$ must be checked case by case.* *Proof.* $p_M(x)=2(28-2M)x^3+[-2(28-2M)+6V-108]x^2+236x-96$; for $M<14$ the leading coefficient is positive and $p_M(0)=-96<0$, so the intermediate value theorem gives a positive root; for $M<11.656$, $D_M>0$ everywhere by Proposition 12.4. $\square$ ### 12.3. The Standard-Model-shifted fixed point **Theorem 12.1** $\boxed{\text{Theorem modulo Classical (Reuter)}}$ (SM-shifted NGFP, type Ia). *For $M_{\mathrm{SM}}=-32$, $V_{\mathrm{SM}}=-68$ the cubic (12.3) is $4\,q(x)$, $q(x)=46x^3-175x^2+59x-24$. It has exactly one positive real root,* $$ x^*_{\mathrm{SM}}=3.4787637\ldots,\qquad\lambda^*_{\mathrm{SM}}=-1.2393818\ldots,\qquad g^*_{\mathrm{SM}}=0.8632468\ldots,\tag{12.4} $$ *with $1-g^*B_2(\lambda^*)=1.0216>0$; it is the unique fixed point in $\mathcal D_M$. Its stability matrix and exponents are* $$ M^*_{\mathrm{SM}}=\begin{pmatrix}-1.957792&0.007688\\-2.857399&-3.859959\end{pmatrix},\qquad\theta_1=1.9694115,\quad\theta_2=3.8483397,\qquad\det M^*_{\mathrm{SM}}=7.5789647,\tag{12.5} $$ *two real relevant directions. For $N_D=22.5$ ($M=-30.5$, $V=-62$) the cubic is $2(89x^3-329x^2+118x-48)$ with unique positive root $x^*=3.3488060$, $(g^*,\lambda^*)=(0.8941728,-1.1744030)$, $\theta=(1.9624190,3.8306139)$, $1-g^*B_2=1.0227$.* *Proof.* With $M=-32$: $D_M=92x^2-30x+48$, $6V-48=-456$, $p_M=-2(1-x)(92x^2-30x+48)+80x-456x^2=184x^3-700x^2+236x-96=4q(x)$. $q'(x)=138x^2-350x+59$ has roots $x=\frac{350\pm\sqrt{122500-32568}}{276}$, i.e. $0.18157$ and $2.3547$; $q(0.18157)=-18.78<0$ at the local maximum, so $q<0$ on $(-\infty,2.3547]$ and is strictly increasing on $[2.3547,\infty)$; $q(3)=-180$, $q(4)=356$; exactly one real root, in $(3,4)$. Newton from $x_0=3.5$: $q(3.5)=11.0$, $q'(3.5)=524.5$, $x_1=3.47903$; converged value $3.478763657\ldots$; since $q'\geq q'(3)=251$ on $[3,\infty)$, $|x-x^*|\leq|q(x)|/251$, and at $x=3.4787637$, $|q(x)|=2.2\cdot10^{-5}$, so $|x-x^*|<9\cdot10^{-8}$, certifying seven digits. Then $\lambda^*=(1-x^*)/2$; $g^*=24\pi x^{*2}/(92x^{*2}-30x^*+48)=912.45/1057.0=0.86325$; $B_2(\lambda^*)=-\frac1{24\pi}\bigl[\frac{10}{3.47876}-\frac{12}{12.1018}\bigr]=-\frac1{24\pi}[2.87459-0.99159]=-0.024970$, $1-g^*B_2=1.02156$. The stability matrix uses the derivatives of Theorem 11.2 with $B_1\to B_1+\frac M{6\pi}$ and the extra constant $\frac V{4\pi}$ in $M_{\lambda g}$; eigenvalues of $-M^*$ from $\mu^2-5.8177513\mu+7.5789647$: $2.9088756\pm\sqrt{8.46156-7.57896}=2.9088756\pm0.9394641$. The $N_D=22.5$ case is identical with $M=-30.5$, $V=-62$: $p_M=178x^3-658x^2+236x-96$. $\square$ **Remark 12.8** (Correction to v2.0; reading of the result). The v2.0 values $(1.05,0.21)$ on a rectangle not containing them are withdrawn. The shifted fixed point (12.4) has negative $\lambda^*$ and $g^*$ close to the vacuum value; its exponents are real. Dona-Eichhorn-Percacci report, for the Standard Model in their type Ia scheme, a fixed point with negative $\lambda^*$ as well (*loc. cit.*, Table II); the precise coordinates differ because their gauge-field and ghost treatment is not identical to Lemma 12.3, and no agreement beyond the sign of $\lambda^*$ is claimed. The result establishes existence and uniqueness within the truncation and scheme, not a precise location. ### 12.4. Truncation persistence **Theorem 12.2** $\boxed{\text{Theorem modulo Classical (Reuter) and Classical M}}$ (Persistence under deformation). *Let $\beta_\epsilon=\beta_0+\epsilon\,\delta\beta$ with $\beta_0$ the system (11.3) (or its matter extension) and $\delta\beta$ any $C^1$ map on a neighbourhood of the fixed point. Classical M (implicit function theorem): if $F$ is $C^1$, $F(0,u_0)=0$, and $\partial_uF(0,u_0)$ is invertible, there is a unique $C^1$ branch $u(\epsilon)$ with $F(\epsilon,u(\epsilon))=0$, $u(0)=u_0$, $u'(0)=-(\partial_uF)^{-1}\partial_\epsilon F$. Since $\det M^*=11.437619\neq0$ (Theorem 11.2) and $\det M^*_{\mathrm{SM}}=7.5789647\neq0$ (Theorem 12.1), there are $\epsilon_0>0$ and $C^1$ branches of fixed points through (11.7) and (12.4), with* $$ \frac{\mathrm d}{\mathrm d\epsilon}\begin{pmatrix}g^*_\epsilon\\\lambda^*_\epsilon\end{pmatrix}\Big|_0=-(M^*)^{-1}\delta\beta(g^*,\lambda^*),\qquad(M^*)^{-1}=\begin{pmatrix}-0.053191&0.912762\\-0.083853&-0.204782\end{pmatrix}\ \text{(vacuum)}.\tag{12.6} $$ *The exponents depend continuously on $\epsilon$ and their real parts stay positive for $|\epsilon|<\epsilon_1\leq\epsilon_0$.* *Proof.* Apply Classical M with $F=\beta_\epsilon$, $u=(g,\lambda)$, $\partial_uF(0,u_0)=M^*$. The inverse is $\frac1{\det M^*}\begin{pmatrix}M_{\lambda\lambda}&-M_{g\lambda}\\-M_{\lambda g}&M_{gg}\end{pmatrix}$, evaluated from (11.8). Continuity of eigenvalues in the entries and openness of $\mathrm{Re}\,\theta>0$ give the last claim. $\square$ **Remark 12.9** (Quantitative persistence). $\epsilon_0$ is not computed here. Literature values in the same regulator class: Lauscher, Reuter (*Phys. Rev. D* **66** (2002) 025026) with $R^2$: $(g^*,\lambda^*)=(0.722,0.193)$; Benedetti, Machado, Saueressig (*Mod. Phys. Lett. A* **24** (2009) 2233) with $R^2+C^2$: product $g^*\lambda^*\in[0.14,0.19]$ across regulators (against $0.137$ for (11.7)), three relevant directions with $\mathrm{Re}\,\theta\in[1.4,2.7]$. These are cited as evidence that $\epsilon$ is small in the sense of Theorem 12.2; they carry no tag. ### 12.5. Lorentzian continuation **Proposition 12.6** $\boxed{\text{Conditional (on H$_{\mathrm W}$ and H$_{\mathrm{rot}}$)}}$ (Relation to the Lorentzian flow). *Let $(U,g)$ be the Euclidean continuation of Definition 10.10 (H$_{\mathrm W}$). Manrique, Rechenberger, Saueressig (*Phys. Rev. Lett.* **106** (2011) 251302) construct, on a foliated background with the regulator a function of the spatial Laplacian on the slices, the Lorentzian (ADM) flow in the Einstein-Hilbert truncation and find a non-Gaussian fixed point whose coordinates differ from the Euclidean ones by regulator-dependent factors arising from the foliation. Hypothesis H$_{\mathrm{rot}}$: on $(U,g^{\mathrm L})$ the Wick rotation $\mathsf t=-i\mathsf t_E$ of the regulated propagators $(\Gamma^{(2)}_{\mathrm L}+R_k)^{-1}$ commutes with the heat-kernel expansion (10.8) term by term, which holds when the $\mathsf t$-dependence of the traces is Gaussian and the regulator is $\mathsf t$-independent. Under H$_{\mathrm W}$ and H$_{\mathrm{rot}}$ the Lorentzian beta functions in the Einstein-Hilbert truncation with the Litim regulator coincide with (11.3)-(11.4), and the fixed point (11.7) and exponents (11.8) are those of the Lorentzian flow on $U$.* *Proof.* Under H$_{\mathrm{rot}}$ each trace in (10.3) evaluated on $(U,g^{\mathrm L})$ equals the corresponding Euclidean trace on $(U,g)$, because the Wick rotation maps the Lorentzian Hessian to the Euclidean one (H$_{\mathrm W}$ makes the rotation an isometry of the static background onto $(U,g)$) and, by H$_{\mathrm{rot}}$, maps the regulated inverse to the Euclidean regulated inverse with the same heat-kernel coefficients. The projection on the two invariants is then identical, and the beta functions coincide. Without H$_{\mathrm{rot}}$, the cited foliated construction gives different beta functions; the identity of fixed points is therefore Conditional and not a theorem about the ADM flow. $\square$ H$_{\mathrm W}$ holds on the static reference families of Chapter 20 (Minkowski, static de Sitter patch, Schwarzschild exterior); H$_{\mathrm{rot}}$ is discharged on Minkowski, where the traces are explicitly Gaussian in $\mathsf t$, and is a hypothesis elsewhere. ### 12.6. Hypothesis discharge ledger, Chapter 12 | Result | Classical inputs | Hypotheses used | Where consumed | |---|---|---|---| | Lem 12.3 | Lichnerowicz formula; Vassilevich $b_0,b_2$; Lem 10.14 | minimal coupling; matter $Z=1$; type Ia Litim | traces | | Prop 12.4, 12.5 | intermediate value theorem | domain $\mathcal D_M$ | reduction, existence | | Thm 12.1 | Thm 11.6 (Reuter) via Thm 11.7 | as above | SM root, digits | | Thm 12.2 | Classical M | $\det M^*\neq0$ | persistence | | Prop 12.6 | Manrique-Rechenberger-Saueressig (cited for the foliated flow) | H$_{\mathrm W}$, H$_{\mathrm{rot}}$ | continuation | --- ## Chapter 13. The Cosmological Constant: Edge Modes and Unimodularity ### 13.1. The volume sector Fix a node $n$ of valence $d(n)\leq D$ (H1) and let $\mathcal H_{\mathrm{geo}}(n)$ be the gauge-invariant intertwiner space of the incident $\mathrm{SU}(2)$ representations with spins $\leq j_{\max}$ (finite-dimensional). The Ashtekar-Lewandowski volume operator $\widehat V_n$ has spectrum $\ell_P^d\sigma_n$ with $\sigma_n\subset[0,\infty)$ finite, depending on $d(n)$, $j_{\max}$ and the Immirzi parameter (Ashtekar, Lewandowski, *Adv. Theor. Math. Phys.* **1** (1998) 388). On the sector of equal spins $j_i=j$ the eigenvalues are $v=\kappa_0\ell_P^dj^{d/2}(1+O(j^{-1}))$, $\kappa_0=\kappa_0(d(n))>0$. **Definition 13.5** (Node energy; stabilising potential). The node-local geometric energy is $$ \mathcal E_n(v,\mathbf j)=\tfrac12\alpha_{\mathrm{geo}}C(\mathbf j)+V(v),\qquad C(\mathbf j)=\sum_{i=1}^{d(n)}j_i(j_i+1),\qquad V(v)=\frac\mu2\ell_P^{-d}(v-\ell_P^d)^2,\tag{13.1} $$ with $\mu>0$ of dimension energy$\cdot$length$^{-d}$ (the constant shift of §10.2 is omitted, being irrelevant to derivatives). On the equal-spin sector $C=d(n)(v/\kappa_0\ell_P^d)^{4/d}(1+O(j^{-1}))$, and $d=4$ gives $C=d(n)\,v/(\kappa_0\ell_P^4)$ to leading order. **Definition 13.6** (Volume-dependence of tensions; hypothesis H$_{\mathrm{vol}}$; scaling generator). Hypothesis H$_{\mathrm{vol}}$: the edge tension of $e=(n,m)$ depends on the node volumes through $$ T(e;v)=\ell(e)^{-1},\qquad\ell(e)=\ell_0(e)\,(v_nv_m)^{1/(2d)},\tag{13.2} $$ with $\ell_0(e)>0$ a dimensionless combinatorial factor independent of $v$. (Axiom L4 defines $d_{\mathrm{eff}}$ from $T$; (13.2) is an additional modelling assumption stating that an edge spans one mean inter-node spacing.) Let $\Phi_{\mathrm{vol}}=\{(v_n)_{n\in N}\colon v_n>0\}$ and $$ E(v,\mathbf j)=\sum_n\mathcal E_n(v_n,\mathbf j_n)+\sum_{e\in E}T(e;v)\mathcal K(e),\tag{13.3} $$ where the second sum is Axiom L5 with (13.2) inserted. The *volume-scaling generator* at $n$ is $\mathcal D_n=v_n\partial/\partial v_n$, acting on functions $C^1$ in $v_n$ through the interpolation $v=\kappa_0\ell_P^dj^{d/2}$ of the AL eigenvalues in $j$. A *stationary point of the volume sector* is $(v^*,\mathbf j^*)$ with $\mathcal D_nE(v^*,\mathbf j^*)=0$ for all $n$; the relation to Axiom L5 is that $E$ is the L5 energy augmented by the node energies of (4.4.1), and $\mathcal D_nE=0$ is the stationarity of the gradient flow in the $v_n$ direction. ### 13.2. The virial identity and volume quantisation **Lemma 13.1** $\boxed{\text{Theorem}}$ (Volume quantisation, $d=4$). *Assume L1-L5, H1, H1', H$_{\mathrm{vol}}$, the finite-spin sector, $d=4$, and* $$ \mu\ell_P^4>A_n:=\frac{2\alpha_{\mathrm{geo}}d(n)}{d\,\kappa_0}=\frac{\alpha_{\mathrm{geo}}d(n)}{2\kappa_0}\quad\text{for all }n\qquad(\text{H}_{\mathrm{stab}}), $$ $$ r_n:=\frac1{2d}\sum_{e\ni n}T(e;v)\mathcal K(e)\ \text{ satisfies }\ 4\mu\ell_P^4|r_n|\leq(\mu\ell_P^4-A_n)^2\qquad(\text{H}_{\mathrm{small}}). $$ *Then at every stationary point of the volume sector with $v^*_n>\tfrac12\kappa_d(n)\ell_P^4$ for every $n$ (the non-collapsed branch), for every node $n$,* $$ v^*_n=\kappa_d(n)\,\ell_P^4\,(1+\epsilon_n),\qquad\kappa_d(n)=1-\frac{A_n}{\mu\ell_P^4}\in(0,1),\qquad|\epsilon_n|\leq\frac{2\mu\ell_P^4}{(\mu\ell_P^4-A_n)^2}\,|r_n|\leq C_{\mathrm{vir}}\,T_{\max}\sup_{e\ni n}|\mathcal K(e)|,\tag{13.4} $$ *with $C_{\mathrm{vir}}=\frac{2D\mu\ell_P^4}{d\,(\mu\ell_P^4-\max_nA_n)^2}$, depending only on $D$, $d$, $\alpha_{\mathrm{geo}}$, $\kappa_0$, $\mu$. The constant $\kappa_d(n)$ depends only on the valence $d(n)$ and on $\alpha_{\mathrm{geo}},\kappa_0,\mu$; it is independent of the matter state and of the tensions. In particular on a QSN with $|\mathcal K(e)|\leq C\ell_P^2/L^2$ (a sprinkling of a manifold with curvature radius $L$, Lemma 5.2), $\epsilon_n=O(\ell_P^2/L^2)$.* *Proof.* *Step 1 (virial identity).* For $d=4$, $\mathcal E_n=\tfrac12\alpha_{\mathrm{geo}}d(n)\,v_n/(\kappa_0\ell_P^4)+\frac\mu2\ell_P^{-4}(v_n-\ell_P^4)^2$ on the equal-spin interpolation. Hence, with $\kappa=v_n/\ell_P^4$, $$ \mathcal D_n\mathcal E_n=v_n\partial_{v_n}\mathcal E_n=A_n\kappa+\mu\ell_P^4\kappa(\kappa-1)=:\Psi_n(\kappa). $$ For the edge term, by H$_{\mathrm{vol}}$ and Axiom L3 ($\mathcal K(e)$ depends only on the graph and $d_G$, not on $v$): $v_n\partial_{v_n}(v_nv_m)^{-1/(2d)}=-\frac1{2d}(v_nv_m)^{-1/(2d)}$, so $$ \mathcal D_n\sum_{e\ni n}T(e;v)\mathcal K(e)=-\frac1{2d}\sum_{e\ni n}T(e;v)\mathcal K(e)=-r_n. $$ Terms of $E$ not involving $v_n$ are annihilated by $\mathcal D_n$ ($H_{\mathrm{mat}},H_{\mathrm{gauge}},H_{\mathrm{ent}}$ act on other factors; edges not incident to $n$ do not involve $v_n$). The stationarity condition is therefore the *virial identity* $$ \Psi_n(\kappa)=r_n,\qquad\text{i.e.}\qquad\mu\ell_P^4\kappa^2+(A_n-\mu\ell_P^4)\kappa-r_n=0.\tag{13.5} $$ *Step 2 (solution).* (13.5) is a quadratic in $\kappa$ with roots $\kappa_\pm=\frac{(\mu\ell_P^4-A_n)\pm\sqrt{(\mu\ell_P^4-A_n)^2+4\mu\ell_P^4r_n}}{2\mu\ell_P^4}$, real by H$_{\mathrm{small}}$. Under H$_{\mathrm{stab}}$, $\mu\ell_P^4-A_n>0$, and for $r_n=0$ the roots are $\kappa=0$ and $\kappa=\kappa_d(n)=1-A_n/(\mu\ell_P^4)\in(0,1)$. The root $\kappa_-$ satisfies $\kappa_-\leq\kappa_d/2$ (it is negative for $r_n>0$ and in $[0,\kappa_d/2]$ for $r_n\leq0$); it is a stationary point of $E$ in $v_n$ whenever $\kappa_-\ell_P^4$ lies in the admissible spectrum, which is why the statement restricts to the non-collapsed branch $\kappa>\kappa_d/2$, on which $\kappa_+$ is the unique root. (The collapsed branch is excluded altogether under the additional hypothesis $\kappa_-<\min\sigma_n$, i.e. $\tfrac12\kappa_d(1-\sqrt{1+y})<\min\sigma_n$, which is not implied by H$_{\mathrm{stab}}$ or H$_{\mathrm{small}}$ and is not assumed.) Writing $y=4\mu\ell_P^4r_n/(\mu\ell_P^4-A_n)^2\in[-1,1]$, $\kappa_+/\kappa_d=\tfrac12(1+\sqrt{1+y})$, so $\epsilon_n=\tfrac12(\sqrt{1+y}-1)$, and the elementary inequalities $|\sqrt{1+y}-1|\leq|y|$ for $y\in[-1,1]$ give $|\epsilon_n|\leq|y|/2=\frac{2\mu\ell_P^4}{(\mu\ell_P^4-A_n)^2}|r_n|$. *Step 3 (bound on $r_n$).* By H1 (in-degree and out-degree each bounded by $D$, §3.2) at most $2D$ edges are incident to $n$, by H1' $T\leq T_{\max}$, so $|r_n|\leq\frac{2D}{2d}T_{\max}\sup_{e\ni n}|\mathcal K(e)|=\frac DdT_{\max}\sup_{e\ni n}|\mathcal K(e)|$, which with Step 2 gives (13.4) and the stated $C_{\mathrm{vir}}$. *Step 4 (curvature scale).* On a sprinkling of a manifold with $|\mathrm{Ric}|\leq L^{-2}$, Lemma 5.2 gives $|\mathcal K(e)|\leq C\ell(e)^2/L^2+O(\rho^{-1/(d+1)})$ with $\ell(e)=O(\ell_P)$ and $T_{\max}=O(\ell_P^{-1})$, so $C_{\mathrm{vir}}T_{\max}\sup|\mathcal K|=O(\ell_P^2/L^2)$ in natural units. $\square$ **Remark 13.8** (Scope; general $d$; dimensions). For $d\neq4$ the node term is $A_n\kappa^{4/d}$ and (13.5) is not quadratic; the inverse function theorem at the zero of $\Psi_n$ in $(\tfrac12,1)$ (where $\Psi_n'>0$ under H$_{\mathrm{stab}}$) gives the same conclusion with $C_{\mathrm{vir}}$ depending on $\Psi_n'$ and $\sup|\Psi_n''|$, and uniqueness only on $(\tfrac12,1)$; the closed form is special to $d=4$. Lemma 13.1 does not assert that the flow reaches a stationary point. Dimensions: $v$ is a length$^4$, $A_n$ and $\mu\ell_P^4$ are energies, $r_n=\frac1{2d}\sum T\mathcal K$ is an inverse length, which in natural units ($\hbar=c=1$) is an energy, so $y$ and $\epsilon_n$ are dimensionless; with units restored, replace $r_n$ by $\hbar c\,r_n$. The Ollivier range $\mathcal K\in[-2,1]$ gives the uniform bound $|r_n|\leq\frac{2D}dT_{\max}$ and makes H$_{\mathrm{small}}$ a condition on $T_{\max}$ alone when curvature information is unavailable. The v2.0 remainder "$O(\ell_P^{d+1})$" is replaced by the dimensionally consistent $O(\ell_P^{d+2}/L^2)$. ### 13.3. Unimodularity and the Henneaux-Teitelboim decomposition **Theorem 13.7** (Classical U: trace-free Einstein equations). *Let $\mathcal M$ be a $d$-manifold with a fixed volume form $\omega$, and $S[g,\psi]=\frac1{16\pi G}\int R\sqrt{-g}\,\mathrm d^dx+S_{\mathrm{mat}}[g,\psi]$ restricted to metrics with $\sqrt{-g}\,\mathrm d^dx=\omega$. Its Euler-Lagrange equations are $R_{\mu\nu}-\frac1dRg_{\mu\nu}=8\pi G(T_{\mu\nu}-\frac1dTg_{\mu\nu})$; if $\nabla^\mu T_{\mu\nu}=0$ there is a constant $\Lambda_0$ with $\frac{d-2}2R+8\pi GT=d\Lambda_0$, and the pair is equivalent to $G_{\mu\nu}+\Lambda_0g_{\mu\nu}=8\pi GT_{\mu\nu}$.* (Trace-free equations and integration constant: Einstein, *Sitzungsber. Preuss. Akad. Wiss.* (1919) 349; Unruh, *Phys. Rev. D* **40** (1989) 1048; Weinberg, *Rev. Mod. Phys.* **61** (1989) 1, §5. Covariant action with auxiliary field and $\Lambda$ as canonical variable: Henneaux, Teitelboim, *Phys. Lett. B* **222** (1989) 195.) *Proof.* Variations preserving $\sqrt{-g}$ satisfy $g^{\mu\nu}\delta g_{\mu\nu}=0$, so stationarity is the vanishing of the trace-free part of $G^{\mu\nu}-8\pi GT^{\mu\nu}$. Taking the divergence of the trace-free equation and using $\nabla^\mu R_{\mu\nu}=\frac12\nabla_\nu R$ and $\nabla^\mu T_{\mu\nu}=0$: $(\frac12-\frac1d)\nabla_\nu R=-\frac{8\pi G}d\nabla_\nu T$, i.e. $\nabla_\nu[\frac{d-2}{2d}R+\frac{8\pi G}dT]=0$; call the constant $\Lambda_0$. Adding $\frac1dg_{\mu\nu}(d\Lambda_0)$ to the trace-free equation reconstructs $G_{\mu\nu}+\Lambda_0g_{\mu\nu}=8\pi GT_{\mu\nu}$, whose trace $-\frac{d-2}2R+d\Lambda_0=8\pi GT$ is consistent. $\square$ **Theorem 13.2** $\boxed{\text{Conditional (on H-cf, H-TCD, C-Lor, H10, E-Lor, H12, H13, H14, H$_{\mathrm{vol}}$, H$_{\mathrm{stab}}$, H$_{\mathrm{small}}$, H$_{\mathrm{LLN}}$) modulo Classical (A-Lor, D, U)}}$, H$_{\mathrm{stab}}$, H$_{\mathrm{small}}$, H$_{\mathrm{LLN}}$) modulo Classical A-Lor, B-Lor, C-Lor (active research), D, E, U}}$ (Henneaux-Teitelboim decomposition on the QSN). *Let $(\mathsf Q_n)$ satisfy the hypotheses of the Riemannian-sector replacement of Theorem 8.1 (errata C21, C22: Conditional on H-cf, H-TCD, C-Lor, H10, E-Lor, H12, H13, H14; the Γ-convergence statement of Theorem 8.1 is withdrawn as a theorem, and "Γ-limit" below denotes the conditional limit functional of Chapter 8), let each $\mathsf Q_n$ be at a stationary point of its volume sector (Lemma 13.1 with H$_{\mathrm{vol}}$, H$_{\mathrm{stab}}$, H$_{\mathrm{small}}$), and assume* *(H$_{\mathrm{LLN}}$) on the probability space $(\Omega,\mathcal F,\mathbb P_\rho)$ of the Poisson sprinkling at density $\rho=n^d$ that generates $\mathsf Q_n$, the weighted empirical measures $\mu_n=\sum_{m\in N_n}\kappa_d(d(m))(1+\epsilon_m)\delta_m$, rescaled as in Theorem 5.6, converge $\mathbb P_\rho$-almost surely in the measured GHL topology to $\bar\kappa_d\,\rho_\infty\,\mathrm{vol}_{g^{\mathrm L}}$, where $\bar\kappa_d=\mathbb E_\rho[\kappa_d(d(m))]$ and $\rho_\infty\mathrm{vol}_{g^{\mathrm L}}$ is the limit of the unweighted rescaled counting measure.* *Then:* *(i) In the Γ-limit of Theorem 8.1 (Lorentzian signature), the volume form $\sqrt{-g}\,\mathrm d^dx=\ell_P^d\bar\kappa_d\rho_\infty\mathrm{vol}_{g^{\mathrm L}}$ is a fixed datum determined by the QSN sequence, not a variable of the action.* *(ii) The Euler-Lagrange equations of the Γ-limit action are* $$ R_{\mu\nu}-\tfrac1dRg_{\mu\nu}=8\pi G\bigl(\langle\widehat T_{\mu\nu}\rangle-\tfrac1d\langle\widehat T\rangle g_{\mu\nu}\bigr),\qquad\tfrac{d-2}2R+8\pi G\langle\widehat T\rangle=d\Lambda_0,\tag{13.6} $$ *with $\Lambda_0$ an integration constant.* *(iii) $\Lambda_0$ is independent of the coefficient of $\int\sqrt g$ in $\Gamma_k$; in particular the matter term $\frac g{4\pi}V_{\mathrm{mat}}$ of (12.1) does not enter the equations of motion.* *Proof.* (i) By Lemma 13.1, $\langle\widehat V_m\rangle=\ell_P^d\kappa_d(d(m))(1+\epsilon_m)$, so the volume measure is $\ell_P^d\mu_n$ with $\mu_n$ as in H$_{\mathrm{LLN}}$. H$_{\mathrm{LLN}}$ is the statement that the node-dependent weights average to $\bar\kappa_d$ against the limit; it is a spatial law of large numbers for the stationary point process generating the sprinkling (for a local functional of the Poisson process with finite mean, the spatial ergodic theorem for stationary point processes, Nguyen, Zessin, *Math. Nachr.* **88** (1979) 105, gives the almost-sure convergence of the empirical average over growing regions; its application to the measured GHL limit requires in addition the uniform control of $\epsilon_m$ from Lemma 13.1 and the convergence of the unweighted measure from Theorem 5.6, and is stated as a hypothesis because the valence of a sprinkled Hasse diagram is bounded only under H1, which is itself an added hypothesis on the sequence). The recovery sequence of Theorem 8.1 is built on sprinklings whose node volumes are at their virial values, so the Γ-limit is defined on metrics with the stated volume form and variations are restricted to that class. (ii) Apply Theorem 13.7 to the Γ-limit action, with $\omega$ from (i) and $\nabla^\mu T_{\mu\nu}=0$ from §8.3. (iii) Under the constraint $\sqrt{-g}\,\mathrm d^dx=\omega$, $\int\sqrt{-g}=\mathrm{Vol}_\omega(\mathcal M)$ is independent of $g$; its coefficient in $\Gamma_k$, whatever its $k$-dependence, drops out of $\delta\Gamma_k/\delta g_{\mu\nu}$ on volume-preserving variations. $\Lambda_0$ arises from integrating the Bianchi identity and is fixed by boundary data of the solution. $\square$ **Remark 13.9** (Corrections to v2.0; what unimodularity buys). The v2.0 constraint $R-8\pi GT=-d\Lambda_0$ is replaced by (13.6), which in $d=4$ reads $R+8\pi G\langle\widehat T\rangle=4\Lambda_0$. The v2.0 tag "unconditional" for Theorem 13.2 is replaced by the inherited tag above: only Lemma 13.1 is a theorem at the QSN level; Theorem 13.2 lives in the continuum limit. Unimodularity removes the radiative instability of the cosmological term: the coefficient of $\int\sqrt g$, which receives the matter contributions (12.1), does not enter the equations of motion. It does not explain the value of $\Lambda_0$ (Weinberg, *loc. cit.*; Smolin, *Phys. Rev. D* **80** (2009) 084003). ### 13.4. The infrared flow of the cosmological term **Proposition 13.3** $\boxed{\text{Theorem modulo Classical (Reuter)}}$ (Infrared behaviour). *Along any trajectory of (11.3) that enters a neighbourhood of the Gaussian fixed point $(0,0)$ as $k\to0$, the stability matrix at $(0,0)$ is $\begin{pmatrix}2&0\\\frac1{2\pi}&-2\end{pmatrix}$ with $\theta=(-2,+2)$; $G_k\to G_0$ and $\Lambda_k\to\Lambda_{\mathrm{IR}}$, finite constants, with $\lambda_k=\Lambda_{\mathrm{IR}}/k^2\,(1+O(g_k))$. The dimensionless $\lambda_k$ tends to $0$ as $k\to0$ if and only if $\Lambda_{\mathrm{IR}}=0$; otherwise it diverges. In the unimodular Γ-limit of Theorem 13.2, $\Lambda_{\mathrm{IR}}$ is the infrared coefficient of $\int\sqrt g$ and by Theorem 13.2(iii) does not enter the equations of motion; the observable cosmological term is the integration constant $\Lambda_0$ of (13.6), which is a different object.* *Proof.* Linearising (11.3) at $(0,0)$: $\eta_N=O(g)$, $\partial_tg=2g+O(g^2)$, $\partial_t\lambda=-2\lambda+\frac g{2\pi}+O(g\lambda,g^2)$; eigenvalues $2,-2$. The linear solution is $g=g_0e^{2t}$, $\lambda=\lambda_0e^{-2t}+\frac{g_0}{8\pi}e^{2t}$, so $G_k=g/k^2\to g_0/k_0^2$ and $\Lambda_k=\lambda k^2=\lambda_0k_0^2+O(k^4)=:\Lambda_{\mathrm{IR}}+O(k^4)$. The last sentence is Theorem 13.2(iii). $\square$ **Remark 13.10** (Correction to v2.0). The v2.0 text asserted $\widetilde\Lambda_k\to0$ in the infrared and identified the infrared coefficient with the integration constant; both are withdrawn. $\Lambda_k$ (dimensionful) tends to a constant, and that constant is not $\Lambda_0$. ### 13.5. The observed value **Phenomenology 13.4** $\boxed{\text{Phenomenological}}$ (Observed cosmological constant as a boundary condition). *The integration constant $\Lambda_0$ of (13.6) is fixed by observation: with $H_0=67.4\ \mathrm{km\,s^{-1}Mpc^{-1}}$ and $\Omega_\Lambda=0.685$ (Planck Collaboration, *Astron. Astrophys.* **641** (2020) A6), $\Lambda_0=3\Omega_\Lambda H_0^2/c^2=1.09\times10^{-52}\ \mathrm m^{-2}$, i.e. $\Lambda_0\ell_P^2=2.8\times10^{-122}$.* This is a retrodiction. The theorem-level content of the chapter is Theorem 13.2(iii): the boundary condition is stable against the radiative corrections (12.1). The hierarchy $\Lambda_0\ell_P^2\sim10^{-122}$, the square of $H_0\ell_P/c$, is not explained. Dimensions: $\Lambda_0$ is an inverse length squared; $\Lambda_0\ell_P^2=\Lambda_0\hbar G/c^3$ is dimensionless. ### 13.6. Hypothesis discharge ledger, Chapter 13 | Result | Classical inputs | Hypotheses used | Where consumed | |---|---|---|---| | Lem 13.1 | Ashtekar-Lewandowski spectrum; quadratic formula | L1-L5, H1, H1', H$_{\mathrm{vol}}$, H$_{\mathrm{stab}}$, H$_{\mathrm{small}}$, finite-spin sector, $d=4$ | Steps 1-3 | | Thm 13.7 | contracted Bianchi identity | none | Classical U | | Thm 13.2 | Classical U; Thms 5.6, 8.1 replacement (A-Lor, D; C-Lor and E-Lor as Conjectures per C22); Nguyen-Zessin (motivating H$_{\mathrm{LLN}}$) | hypotheses of the Thm 8.1 replacement (H-cf, H-TCD, C-Lor, H10, E-Lor, H12, H13, H14); H$_{\mathrm{vol}}$, H$_{\mathrm{stab}}$, H$_{\mathrm{small}}$; H$_{\mathrm{LLN}}$ | (i)-(iii) | | Prop 13.3 | none | EH truncation, Litim | linearisation | | Phen 13.4 | Planck 2018 | none | input | --- ## Referee response (review4.md) Items are numbered as in the report. Every numerical value in this Part was recomputed after the edits (exact rational system, sympy at 30 digits; Jacobians by symbolic differentiation). 1. Bare action redefined on $\mathrm{Herm}(\mathcal H^{\mathrm{phys}})$ as $\mathrm{tr}(\varphi\mathcal S\varphi)$ with $\mathcal S=H_{\mathsf Q}+\sum_e\widehat T\widehat{\mathcal K}$, Definition 10.1. 2. Coercivity stated as $\mathcal S\geq c_1$ and discharged by an explicit constant shift of the stabilising potential (§10.2). 3. Jensen bound written out; Laplace remainder replaced by the exact Gaussian evaluation with explicit error $\tfrac12\mathcal N^2\|\mathcal S_{\mathrm{sym}}\|\|R_k^{-1}\|$ (Theorem 10.4). 4. Remark 10.5 rewritten: Picard-Lindelöf applies to truncations, not to (10.3). 5. $\Delta_{\mathsf Q}$ redefined from the undirected Hasse graph on $\ell^2(N,\deg)$; self-adjointness and $\mathrm{spec}\subset[0,2c_d\ell_P^{-2}]$ proved (Lemma 10.7); H$_{\mathrm{iso}}$ restated for the symmetric walk with the vanishing of the odd term explained; regulator defined by the spectral theorem (Definition 10.8). 6. Lemma 10.14 restated with Reuter's factor $\tfrac12$ and the $+\eta R_k$ numerator; the identity $Q_n[2k^2\theta/(P+w)^p]=2k^{2(n-p+1)}\Phi^p_n$ displayed. 7. Reuter 1998 stated as numbered classical input Theorem 11.6 with hypotheses and discharge; Proposition 11.5, Theorem 11.7, and consequently Theorems 11.1, 11.2, 12.1 retagged "Theorem modulo Classical (Reuter)". 8. Crossing value corrected to $\lambda=0.338$. 9. $g_c'(x^*)=2.229$ and $\max_{[0.6,0.62]}|g_c'|=2.243$ displayed (the referee's $1.87$ is not reproduced; the bound $<3$ holds either way). 10. $\|\mathrm dM/\mathrm dx\|_2=14.0$ displayed. 11. Newton step signs corrected. 12. Decomposition corrected: $c_2^{\mathrm{mix}}=\tfrac{23}3$; sign statement rephrased as cited. 13. Value corrected to $-0.9967$. 14. Folkerts-Litim-Pawlowski stated as classical input Theorem 11.15 with gauge, regulator and background hypotheses; Proposition 11.3 retagged "Theorem modulo Classical (FLP)". 15. Proposition 11.4(a) restricted to flat-background $Q_n$ traces; curvature-weighted traces explicitly excluded. 16. Proposition 11.4(b) retagged Conditional on H$_{\mathrm{iso}}$, Proposition 11.3, and the single-$Q_2$ hypothesis, which is stated. 17. Definition 10.8 states that truncations are written with $\Delta_{\mathsf Q}$ in place of $-\nabla^2$. 18. Scheme made consistent: type Ia throughout; Lemma 12.3 recomputed, $M_{\mathrm{mat}}=N_S-N_D-N_V$; type II discussed in Remark 12.7. 19. Theorem 12.1 recomputed: cubic $4(46x^3-175x^2+59x-24)$, $x^*=3.4787637$, $(g^*,\lambda^*)=(0.8632468,-1.2393818)$, $\theta=(1.9694115,3.8483397)$, $\det=7.5789647$, $1-g^*B_2=1.0216$; $N_D=22.5$ case likewise. The monotonicity argument was re-verified ($q'$ roots $0.18157$, $2.3547$; $q(0.18157)=-18.78$). 20. Proposition 12.5 now covers $11.656\leq M<14$. 21. Theorem 12.2 uses the recomputed $\det M^*_{\mathrm{SM}}$. 22. Proposition 12.6 restates what Manrique-Rechenberger-Saueressig prove, drops "identical", and is Conditional on H$_{\mathrm W}$ and a named hypothesis H$_{\mathrm{rot}}$; the Euclidean continuation is now Definition 10.10 in §10.4, so Chapter 11 no longer forward-references Chapter 12. 23. H$_{\mathrm{vol}}$ stated as a hypothesis (Definition 13.6) and listed in the ledger. 24. Step 1 rewritten for $d=4$ with the closed form $\kappa_d=1-A_n/(\mu\ell_P^4)$; the incorrect limiting-sign claim removed; uniqueness stated for the physical root; general $d$ relegated to Remark 13.8 with uniqueness on $(\tfrac12,1)$. 25. Smallness hypothesis H$_{\mathrm{small}}$ stated with the explicit threshold $4\mu\ell_P^4|r_n|\leq(\mu\ell_P^4-A_n)^2$; no inverse function theorem needed in $d=4$. 26. Display (13.4) corrected to $|\epsilon_n|\leq\frac{2\mu\ell_P^4}{(\mu\ell_P^4-A_n)^2}|r_n|\leq C_{\mathrm{vir}}T_{\max}\sup|\mathcal K|$ with $C_{\mathrm{vir}}$'s dependency list corrected; dimensions in Remark 13.8. 27. H2 dropped from Lemma 13.1; stationarity defined for $E$ of Definition 13.6 with its relation to L5 stated. 28. H$_{\mathrm{LLN}}$ stated with probability space and a.s. qualifier; Nguyen-Zessin cited as motivation; signature of the Γ-limit stated as Lorentzian per Theorem 8.1. 29. Attribution corrected (Einstein 1919, Unruh 1989, Weinberg 1989 for the decomposition; Henneaux-Teitelboim for the action); unverified equation numbers removed. 30. Tag of Theorem 13.2 spelled out with all inherited inputs, including C-Lor flagged and errata C12 applied. 31. Proposition 13.3 rewritten with $\Lambda_{\mathrm{IR}}$; the identification with $\Lambda_0$ deleted; Remark 13.10 corrected. 32. Phenomenology 13.4 quotes only the Planck-scale value; the reference to Proposition 13.3 at $k=H_0/c$ removed. 33. Labels renumbered consecutively after the anchors (convention stated at the head of Part IV); ledgers corrected (no cross-chapter forward dependence; H$_{\mathrm{vol}}$, H$_{\mathrm{stab}}$, H$_{\mathrm{small}}$, H$_{\mathrm{LLN}}$ listed); the anchor "Theorem 11.3" is a Proposition here, flagged for Appendix C. Errata C12 applied (Henneaux-Teitelboim constraint form, Proposition 13.3, tag of Theorem 13.2). Errata C14 applied (Ollivier range $[-2,1]$ used in the coercivity discharge and in Remark 13.8). ### Pass 2 (review4b.md) B1. Pulled-back fields defined in Definition 10.8 ($\iota_n^*\phi\in\ell^2(N,\deg)\otimes\mathcal V_{\mathrm{fib}}$, $\Delta_{\mathsf Q}\otimes\mathbb 1$, traces $\mathrm{Tr}_{N\otimes\mathcal V}$); Definition 10.11 states that the truncated flows live on the continuum theory space with regulators on pulled-back fields and that (10.3) on $\Phi$ is not the equation whose truncations are solved; Proposition 11.4(a) now carries the symbol hypothesis H$_{\mathrm{sym}}$ (H$_{\mathrm{iso}}$ sufficient) and the trace is specified. B2. Lemma 13.1 restricted to the non-collapsed branch $v^*_n>\tfrac12\kappa_d\ell_P^4$; Step 2 states $\kappa_-\leq\kappa_d/2$ with the sign cases, and records the optional exclusion hypothesis without assuming it. B3. $\partial_tg=(d-2+\eta_N)g$ in Theorem 11.6(b). B4. Lemma 11.13: mixed loop contributes to both $c_1$ and $c_2$ according to the line carrying $\partial_tR_k$. B5. Coercivity wording corrected (shift of $\sum_nV$; $\Gamma_k$ shifts by the constant, the flow is unchanged). B6. Uniform tag "Theorem modulo Classical (Reuter)" for Theorem 12.2 (with Classical M) and Proposition 13.3; explanatory sentence at the head of §11.2. B7. Step 3 of Lemma 13.1 cites H1's form (in- and out-degree each $\leq D$), so $2D$ stands. B8. Chapter 10 labels renumbered in text order: 10.8 regulator, 10.9 symbol lemma, 10.10 Euclidean continuation, 10.11 truncation, 10.12 fixed points, 10.13 conjecture, 10.14 Litim lemma, 10.15 dimensions; no primes remain. B9. Lemma 11.14: factor 2 between $\sum_i\mathcal L_2(e^{(i)})$ and $\mathrm{tr}E$ made explicit. No numerical value changed in this pass; the values of Pass 1 were re-run unchanged. # Part V. Operator-Algebraic Holography --- ## Chapter 14. Edge-Mode Algebras and the Type $\mathrm{II}_1$ Factor ### 14.1. Scope and retraction record This chapter constructs the von Neumann algebra of edge modes on a causal boundary of the QSN and identifies its type. The construction is an inductive limit of finite-dimensional matrix algebras with trace-preserving inclusions, and the type identification is a direct application of the Murray-von Neumann-Connes classification. Everything in §14.2 to §14.5 is proved; §14.6 records what is not proved and what was withdrawn from earlier versions. The following statements from v1.0 are withdrawn and do not reappear anywhere in this monograph: (i) The claim that the modular flow of the edge-mode algebra in its natural state generates proper-time translation. In the tracial state constructed below the modular group is the identity (Proposition 14.9), so no nontrivial dynamics can be read off from it. (ii) The claim that the edge-mode algebra is isomorphic, as a theorem, to the Chandrasekaran-Penington-Witten crossed-product algebra of a gravitationally dressed horizon region. This is Conjecture 14.6 below, with the precise obstruction stated in Proposition 14.10. (iii) The claim that the edge-mode algebra exhausts the gauge-invariant observables of a tubular neighbourhood of the boundary, argued via Stone-Weierstrass and Reeh-Schlieder. The hypotheses of Reeh-Schlieder (a cyclic and separating vacuum for a net of local algebras satisfying spectrum condition and locality) are not available on the QSN. This is Conjecture 14.7 below, with the partial result Proposition 14.11. ### 14.2. The finite edge-mode algebra Throughout, $\mathsf{Q}=(N,E)$ satisfies L1 and H1. Let $B\subset N$ be a finite region and $\partial B\subset E$ its edge boundary, the set of edges with exactly one endpoint in $B$. For each edge $e\in E$ fix the edge Hilbert space $\mathcal{H}_e=\mathcal{H}_{\mathrm{grav}}(e)\otimes\mathcal{H}_{\mathrm{gauge}}(e)$ of Chapter 4, and for each integer $D_e\geq 1$ a $D_e$-dimensional subspace $\mathcal{H}_e^{(D_e)}\subset\mathcal{H}_e$ spanned by the first $D_e$ eigenvectors of the edge Hamiltonian $H_e$ of L6. Since $H_e$ has discrete spectrum by L6, this subspace is well defined once an ordering of any degenerate eigenvalues is fixed; the constructions below do not depend on that ordering. **Definition 14.0a** (Finite edge-mode algebra). Let $\Gamma\subset\partial B$ be a finite set of boundary edges and let $D=(D_e)_{e\in\Gamma}$ be a family of positive integers, the *representation cutoffs*. The finite edge-mode algebra is $$ \mathcal{A}_\Gamma^{(D)}:=\bigotimes_{e\in\Gamma}M_{D_e}(\mathbb{C}), \tag{14.1} $$ where $M_{D_e}(\mathbb{C})=B(\mathcal{H}_e^{(D_e)})$. It acts on $\mathcal{H}_\Gamma^{(D)}:=\bigotimes_{e\in\Gamma}\mathcal{H}_e^{(D_e)}$, of dimension $|D|:=\prod_{e\in\Gamma}D_e$. The *normalised trace* on $\mathcal{A}_\Gamma^{(D)}$ is $$ \tau_\Gamma(a):=\frac{1}{|D|}\,\mathrm{Tr}_{\mathcal{H}_\Gamma^{(D)}}(a). \tag{14.2} $$ **Lemma 14.1** $\boxed{\text{Theorem}}$ (Finite type $\mathrm{I}$ factor). *$\mathcal{A}_\Gamma^{(D)}$ is a factor of type $\mathrm{I}_{|D|}$. The functional $\tau_\Gamma$ is the unique tracial state on $\mathcal{A}_\Gamma^{(D)}$, and it is faithful.* *Proof.* The tensor product of full matrix algebras is a full matrix algebra: $\bigotimes_{e\in\Gamma}M_{D_e}(\mathbb{C})\cong M_{|D|}(\mathbb{C})$ via the canonical identification $\bigotimes_e\mathcal{H}_e^{(D_e)}\cong\mathbb{C}^{|D|}$. The centre of $M_n(\mathbb{C})$ is $\mathbb{C}\mathbf{1}$: if $z$ commutes with all matrix units $E_{ij}$ then $E_{ii}zE_{jj}=E_{ii}E_{jj}z=\delta_{ij}E_{ii}z$, so $z$ is diagonal, and $E_{ij}z=zE_{ij}$ forces equal diagonal entries. Hence $\mathcal{A}_\Gamma^{(D)}$ is a factor, and being finite-dimensional it is of type $\mathrm{I}_{|D|}$ (Kadison, Ringrose, *Fundamentals of the Theory of Operator Algebras II*, AMS, 1997, Thm 6.6.1). Uniqueness of the tracial state. Let $\phi$ be a tracial state on $M_n(\mathbb{C})$. For matrix units, $\phi(E_{ij})=\phi(E_{ik}E_{kj})=\phi(E_{kj}E_{ik})=\delta_{ij}\phi(E_{kk})$ for any $k$, so $\phi(E_{ii})=\phi(E_{kk})$ for all $i,k$, and $\sum_i\phi(E_{ii})=\phi(\mathbf{1})=1$ gives $\phi(E_{ii})=1/n$. Linearity then yields $\phi(a)=\frac1n\sum_i a_{ii}=\frac1n\mathrm{Tr}(a)$. Faithfulness: $\tau_\Gamma(a^*a)=\frac1{|D|}\sum_{ij}|a_{ij}|^2$, which vanishes only if $a=0$. $\square$ ### 14.3. The inductive system and its trace Let $\mathcal{I}$ be the set of pairs $(\Gamma,D)$ with $\Gamma\subset\partial B$ finite and $D=(D_e)_{e\in\Gamma}$ a cutoff family. Partially order $\mathcal{I}$ by $(\Gamma,D)\leq(\Gamma',D')$ if $\Gamma\subseteq\Gamma'$ and $D_e\leq D'_e$ for every $e\in\Gamma$. This is a directed set: for $(\Gamma_1,D^1)$ and $(\Gamma_2,D^2)$ take $\Gamma=\Gamma_1\cup\Gamma_2$ and $D_e=\max(D^1_e,D^2_e)$ with $D^i_e:=1$ when $e\notin\Gamma_i$. For $(\Gamma,D)\leq(\Gamma',D')$ define the inclusion $$ \iota_{(\Gamma,D),(\Gamma',D')}\colon\mathcal{A}_\Gamma^{(D)}\to\mathcal{A}_{\Gamma'}^{(D')},\qquad \bigotimes_{e\in\Gamma}a_e\;\longmapsto\;\bigotimes_{e\in\Gamma}\bigl(a_e\oplus 0_{D'_e-D_e}\bigr)\otimes\bigotimes_{e\in\Gamma'\setminus\Gamma}\mathbf{1}_{D'_e}, \tag{14.3} $$ extended linearly. Here $a_e\oplus 0_{D'_e-D_e}$ denotes $a_e$ acting on $\mathcal{H}_e^{(D_e)}\subset\mathcal{H}_e^{(D'_e)}$ and as zero on the orthogonal complement. This is an injective $*$-homomorphism, but it is not unital when some $D'_e>D_e$, since it maps $\mathbf{1}$ to the projection $P=\bigotimes_{e\in\Gamma}P_e^{(D_e)}\otimes\mathbf{1}$ onto $\mathcal{H}_\Gamma^{(D)}\otimes\mathcal{H}_{\Gamma'\setminus\Gamma}^{(D')}$. A non-unital inclusion cannot preserve normalised traces. We therefore make the choice that is forced by the physics of Chapter 4, where the representation cutoff is a regulator on the edge Hamiltonian spectrum and physical states at scale $D_e$ are those supported on $\mathcal{H}_e^{(D_e)}$: we require the inclusions to be unital on the edges already present, and to enlarge the algebra only by adjoining new edges. That is, we restrict $\mathcal{I}$ to sequences along which cutoffs are increased by *refining* the edge set rather than by enlarging a fixed edge's cutoff. Concretely, we use the following directed subsystem. **Definition 14.0b** (Admissible inductive system). Assume $\mathsf{Q}$ is connected as an undirected graph; then $N$, hence $E$ and $\partial B$, are countable (from any node, the set of nodes at undirected distance $\leq r$ is finite by L1 and H1, and $N$ is the union over $r\in\mathbb{N}$). Fix an exhaustion $\partial B=\bigcup_{k\geq 1}\Gamma_k$ by finite sets $\Gamma_1\subset\Gamma_2\subset\cdots$, and an assignment $e\mapsto D_e\in\mathbb{N}$ with $D_e\geq 2$ for all $e$. The admissible inductive system is $\{\mathcal{A}_k\}_{k\geq 1}$ with $\mathcal{A}_k:=\mathcal{A}_{\Gamma_k}^{(D|_{\Gamma_k})}$ and unital inclusions $$ \iota_{k}\colon\mathcal{A}_k\to\mathcal{A}_{k+1},\qquad a\mapsto a\otimes\mathbf{1}_{\Gamma_{k+1}\setminus\Gamma_k}. \tag{14.4} $$ If $\partial B$ is finite the system is eventually constant in $\Gamma$; to obtain an infinite system in that case one passes to a sequence of regions $B_1\subset B_2\subset\cdots$ with $|\partial B_k|\to\infty$, which exists whenever $N$ is infinite and satisfies H4'. For the remainder of the chapter $\partial B$ is infinite or is understood as such a sequence. The condition $D_e\geq 2$ is what makes the limit infinite-dimensional; the role of letting $D_e$ grow, when needed, is played in Chapter 15 by the bond-dimension parameter $\chi$ and in §14.5 by Hypothesis H8. **Proposition 14.2** $\boxed{\text{Theorem}}$ (Inductive-limit trace). *Let $\{\mathcal{A}_k,\iota_k\}$ be an admissible inductive system. Then:* *(a) The maps $\iota_k$ are injective unital $*$-homomorphisms and $\tau_{k+1}\circ\iota_k=\tau_k$ for every $k$.* *(b) The algebraic inductive limit $\mathcal{A}_\infty:=\varinjlim\mathcal{A}_k$ carries a unique state $\tau_\infty$ with $\tau_\infty\circ\iota_{k,\infty}=\tau_k$ for all $k$, where $\iota_{k,\infty}\colon\mathcal{A}_k\to\mathcal{A}_\infty$ are the canonical maps. This $\tau_\infty$ is tracial.* *(c) $\tau_\infty$ extends uniquely to a tracial state on the norm completion $\widetilde{\mathcal{A}}_{\partial B}:=\overline{\mathcal{A}_\infty}^{\|\cdot\|}$, and the extension is faithful.* *Proof.* (a) $\iota_k$ is the map $a\mapsto a\otimes\mathbf{1}$, which is a unital $*$-homomorphism; it is injective because $a\otimes\mathbf{1}=0$ implies $a=0$ in a tensor product of matrix algebras. For the trace, with $m_k:=\dim\mathcal{H}_{\Gamma_k}$ and $r:=\dim\mathcal{H}_{\Gamma_{k+1}\setminus\Gamma_k}$, so $m_{k+1}=m_kr$, $$ \tau_{k+1}(a\otimes\mathbf{1})=\frac{1}{m_kr}\mathrm{Tr}(a\otimes\mathbf{1})=\frac{1}{m_kr}\,\mathrm{Tr}(a)\,\mathrm{Tr}(\mathbf{1}_r)=\frac{1}{m_k}\mathrm{Tr}(a)=\tau_k(a). $$ (b) The algebraic inductive limit is the quotient of the disjoint union $\bigsqcup_k\mathcal{A}_k$ by the relation $a\sim\iota_{k,l}(a)$ for $k\leq l$, where $\iota_{k,l}=\iota_{l-1}\circ\cdots\circ\iota_k$. Since all $\iota_k$ are injective, each $\iota_{k,\infty}$ is injective and $\mathcal{A}_\infty=\bigcup_k\iota_{k,\infty}(\mathcal{A}_k)$ is an increasing union of unital $*$-subalgebras. Define $\tau_\infty$ on $\iota_{k,\infty}(\mathcal{A}_k)$ by $\tau_\infty(\iota_{k,\infty}(a)):=\tau_k(a)$. This is well defined: if $\iota_{k,\infty}(a)=\iota_{l,\infty}(b)$ with $k\leq l$, then $\iota_{k,l}(a)=b$ by injectivity of $\iota_{l,\infty}$, and $\tau_l(b)=\tau_l(\iota_{k,l}(a))=\tau_k(a)$ by iterating (a). The resulting functional is linear, positive (each $\tau_k$ is positive and any finitely many elements lie in a common $\mathcal{A}_k$), unital, and tracial: for $x,y\in\mathcal{A}_\infty$ pick $k$ with $x=\iota_{k,\infty}(a)$, $y=\iota_{k,\infty}(b)$; then $\tau_\infty(xy)=\tau_k(ab)=\tau_k(ba)=\tau_\infty(yx)$. Uniqueness: any state $\tau'$ with $\tau'\circ\iota_{k,\infty}=\tau_k$ for all $k$ agrees with $\tau_\infty$ on each $\iota_{k,\infty}(\mathcal{A}_k)$, hence on their union $\mathcal{A}_\infty$. (c) Each $\tau_k$ is a state on the C$^*$-algebra $\mathcal{A}_k$ and hence has norm one; therefore $|\tau_\infty(x)|\leq\|x\|$ on $\mathcal{A}_\infty$, where the norm on $\mathcal{A}_\infty$ is the unique C$^*$-norm (each $\mathcal{A}_k$ is a C$^*$-algebra and the $\iota_k$ are injective, hence isometric). Thus $\tau_\infty$ is uniformly continuous and extends uniquely to a bounded linear functional on $\widetilde{\mathcal{A}}_{\partial B}$, still positive, unital, and tracial by continuity of $x\mapsto\tau_\infty(xy-yx)$. Faithfulness is not automatic: a state on an inductive limit that is faithful on every $\mathcal{A}_k$ can fail to be faithful on the closure when the $\mathcal{A}_k$ are not simple (take $\mathcal{A}_k=\mathbb{C}^2$ with constant inclusions and $\tau=(1-2^{-k})\mathrm{ev}_1+2^{-k}\mathrm{ev}_2$ on the $k$-th level; the compatible limit state is $\mathrm{ev}_1$, which kills $(0,1)$). The argument must therefore use simplicity of the $\mathcal{A}_k$. This is Lemma 14.2a below, applied with $\mathcal{A}=\widetilde{\mathcal{A}}_{\partial B}$ and $\mathcal{A}_0=\mathcal{A}_\infty$. $\square$ **Lemma 14.2a** $\boxed{\text{Theorem}}$ (Faithfulness on the closure). *Let $\mathcal{A}$ be a unital C$^*$-algebra which is the norm closure of an increasing union $\mathcal{A}_0=\bigcup_k\mathcal{A}_k$ of finite-dimensional factors $\mathcal{A}_k\cong M_{m_k}(\mathbb{C})$, and let $\tau$ be a tracial state on $\mathcal{A}$. Then $\tau$ is faithful.* *Proof.* Let $J:=\{x\in\mathcal{A}\colon\tau(x^*x)=0\}$. For any state, $J$ is a closed left ideal (Cauchy-Schwarz gives $\tau((yx)^*(yx))\leq\|y\|^2\tau(x^*x)$). Traciality makes it a right ideal as well: $\tau((xy)^*(xy))=\tau(y^*x^*xy)=\tau(x^*xyy^*)\leq\|y\|^2\tau(x^*x)$. So $J$ is a closed two-sided ideal. For each $k$, $J\cap\mathcal{A}_k$ is a two-sided ideal of the simple algebra $M_{m_k}(\mathbb{C})$, hence is $0$ or $\mathcal{A}_k$; since $\tau(\mathbf{1})=1$, it is $0$. It remains to show that a closed two-sided ideal $J$ of an AF algebra with $J\cap\mathcal{A}_k=0$ for all $k$ is zero (Bratteli, *Trans. AMS* **171** (1972) 195, Thm 3.3). Let $q\colon\mathcal{A}\to\mathcal{A}/J$ be the quotient map. Take $x\in J$ and $\varepsilon>0$, and choose $k$ and $a\in\mathcal{A}_k$ with $\|x-a\|<\varepsilon$; then $\|q(a)\|=\|q(a-x)\|<\varepsilon$. The restriction $q|_{\mathcal{A}_k}$ has kernel $J\cap\mathcal{A}_k=0$, so it is an injective $*$-homomorphism of C$^*$-algebras and therefore isometric: $\|a\|=\|q(a)\|<\varepsilon$. Hence $\|x\|<2\varepsilon$ for every $\varepsilon$, so $x=0$. Thus $J=0$ and $\tau$ is faithful. $\square$ **Remark 14.2b.** The Hilbert-Schmidt comparison $\|a\|^2\leq m_k\tau_k(a^*a)$ degrades like $1/m_k$ and cannot give faithfulness on the closure; the statement is ideal-theoretic, and the argument above uses only simplicity of the $\mathcal{A}_k$ and the isometry of injective $*$-homomorphisms. Replacing the unital inclusions (14.4) by the non-unital ones of (14.3) breaks trace preservation at the first line of Proposition 14.2 and would require a different state. ### 14.4. The von Neumann algebra and hyperfiniteness Let $(\pi_{\tau_\infty},\mathcal{H}_{\tau_\infty},\Omega)$ be the GNS triple of $(\widetilde{\mathcal{A}}_{\partial B},\tau_\infty)$: $\mathcal{H}_{\tau_\infty}$ is the completion of $\widetilde{\mathcal{A}}_{\partial B}$ in the inner product $\langle x,y\rangle:=\tau_\infty(x^*y)$, $\pi_{\tau_\infty}(x)$ acts by left multiplication, and $\Omega$ is the class of $\mathbf{1}$. Faithfulness of $\tau_\infty$ (Proposition 14.2) means the inner product is definite on $\widetilde{\mathcal{A}}_{\partial B}$ itself, so no quotient is taken. **Definition 14.0c.** The edge-mode von Neumann algebra of $\partial B$ is $$ \mathcal{M}_{\partial B}:=\pi_{\tau_\infty}\bigl(\widetilde{\mathcal{A}}_{\partial B}\bigr)''\subset B(\mathcal{H}_{\tau_\infty}). \tag{14.5} $$ The state $\tau_\infty$ extends to $\mathcal{M}_{\partial B}$ as the vector state $\tau_\infty(m):=\langle\Omega,m\Omega\rangle$, which is normal, and remains tracial and faithful: tracial because $\langle\Omega,xy\Omega\rangle=\tau_\infty(xy)$ on the weakly dense subalgebra $\pi(\widetilde{\mathcal{A}})$ and both sides are separately weakly continuous; faithful because $\Omega$ is separating for $\mathcal{M}_{\partial B}$ (it is cyclic for the commutant, as $\pi(\widetilde{\mathcal{A}})'\supset$ right multiplications, which are well defined and bounded since $\tau_\infty$ is tracial, and right multiplications applied to $\Omega$ reproduce the dense set $\widetilde{\mathcal{A}}\Omega$). **Proposition 14.3** $\boxed{\text{Theorem}}$ (Hyperfiniteness). *$\mathcal{M}_{\partial B}$ is hyperfinite: it is the weak-operator closure of an increasing sequence of finite-dimensional $*$-subalgebras.* *Proof.* Set $\mathcal{M}_k:=\pi_{\tau_\infty}(\iota_{k,\infty}(\mathcal{A}_k))$. Each $\mathcal{M}_k$ is a finite-dimensional $*$-subalgebra of $B(\mathcal{H}_{\tau_\infty})$ containing $\mathbf{1}$, isomorphic to $M_{m_k}(\mathbb{C})$ since $\pi_{\tau_\infty}$ is injective (the GNS representation of a faithful state is faithful). The union $\bigcup_k\mathcal{M}_k=\pi_{\tau_\infty}(\mathcal{A}_\infty)$ is norm dense in $\pi_{\tau_\infty}(\widetilde{\mathcal{A}}_{\partial B})$, hence weakly dense in $\mathcal{M}_{\partial B}$. By the double commutant theorem, $\mathcal{M}_{\partial B}=\overline{\bigcup_k\mathcal{M}_k}^{\,\mathrm{wo}}$. This is the definition of hyperfiniteness (Murray, von Neumann, *Ann. Math.* **44** (1943) 716, Def. 4.1.1; equivalently approximate finite-dimensionality). $\square$ ### 14.5. Factoriality and type Hyperfiniteness alone does not give a factor: the tensor product of matrix algebras along an infinite sequence is a factor, but a general AF inductive limit may have nontrivial centre if the inclusions fail to mix central elements. In the admissible system (14.4) each inclusion is a tensor-product-with-identity map, and this suffices. We state the hypothesis in the form used throughout Parts V and VI, which also covers non-tensor-product admissible systems arising from gauge-invariant restrictions. **Hypothesis H8** (Density of central elements). *For every $k$ and every $z$ in the centre $Z(\mathcal{M}_{\partial B})$ of the edge-mode algebra, the conditional expectation $E_k\colon\mathcal{M}_{\partial B}\to\mathcal{M}_k$ (the unique $\tau_\infty$-preserving one) satisfies $\lim_{k\to\infty}\|E_k(z)-\tau_\infty(z)\mathbf{1}\|_2=0$, where $\|x\|_2^2:=\tau_\infty(x^*x)$.* For the tensor-product system (14.4), H8 is a theorem (Lemma 14.4a below); H8 is retained as a named hypothesis because Chapter 17 uses gauge-invariant subsystems of (14.4) for which it must be verified separately. **Theorem 14.4U** (Classical, Umegaki; Umegaki, *Tôhoku Math. J.* **6** (1954) 177, for finite-dimensional $\mathcal{N}$, which is the case used; Takesaki, *Theory of Operator Algebras II*, Springer, 2003, Thm IX.4.2, in general). *Let $\mathcal{M}$ be a von Neumann algebra with a faithful normal tracial state $\tau$ and $\mathcal{N}\subseteq\mathcal{M}$ a von Neumann subalgebra. There is a unique normal conditional expectation $E\colon\mathcal{M}\to\mathcal{N}$ with $\tau\circ E=\tau$. It is the restriction to $\mathcal{M}\subset L^2(\mathcal{M},\tau)$ of the orthogonal projection $e_{\mathcal{N}}$ of $L^2(\mathcal{M},\tau)$ onto $\overline{\mathcal{N}\Omega}$, i.e. $E(x)\Omega=e_{\mathcal{N}}(x\Omega)$, and $E$ is $\mathcal{N}$-bimodular, positive, and $\|E(x)\|_2\leq\|x\|_2$.* *Hypotheses discharged for $\mathcal{M}=\mathcal{M}_{\partial B}$, $\mathcal{N}=\mathcal{M}_k$:* $\tau_\infty$ is a faithful normal tracial state on $\mathcal{M}_{\partial B}$ (§14.4); $\mathcal{M}_k$ is a finite-dimensional, hence weakly closed, unital $*$-subalgebra. **Lemma 14.4a** $\boxed{\text{Theorem}}$. *The admissible inductive system (14.4) satisfies H8.* *Proof.* We first show $Z(\mathcal{M}_{\partial B})=\mathbb{C}\mathbf{1}$ directly; H8 then holds, since $E_k(\lambda\mathbf{1})=\lambda\mathbf{1}=\tau_\infty(\lambda\mathbf{1})\mathbf{1}$. Write $\mathcal{M}_{\partial B}$ as the weak closure of $\bigcup_k\mathcal{M}_k$ with $\mathcal{M}_{k+1}=\mathcal{M}_k\otimes\mathcal{N}_k$, $\mathcal{N}_k\cong M_{r_k}(\mathbb{C})$, $r_k=\dim\mathcal{H}_{\Gamma_{k+1}\setminus\Gamma_k}\geq 2$. Let $E_k\colon\mathcal{M}_{\partial B}\to\mathcal{M}_k$ be the $\tau_\infty$-preserving conditional expectation; it exists and is unique by Theorem 14.4U, and it is given on $\mathcal{M}_l$, $l\geq k$, by the partial trace $E_k(a\otimes b)=a\,\tau(b)$ for $a\in\mathcal{M}_k$, $b\in\bigotimes_{j=k}^{l-1}\mathcal{N}_j$. Let $z\in Z(\mathcal{M}_{\partial B})$. For every $k$, $z$ commutes with $\mathcal{M}_k$, hence $E_k(z)\in Z(\mathcal{M}_k)=\mathbb{C}\mathbf{1}$ by Lemma 14.1, and since $E_k$ preserves $\tau_\infty$, $E_k(z)=\tau_\infty(z)\mathbf{1}$. On the other hand the $E_k$ converge strongly to the identity on $\mathcal{H}_{\tau_\infty}$: $E_k$ extends to the orthogonal projection of $\mathcal{H}_{\tau_\infty}=L^2(\mathcal{M}_{\partial B},\tau_\infty)$ onto the closed subspace $\overline{\mathcal{M}_k\Omega}$, these subspaces increase to the dense subspace $\bigcup_k\mathcal{M}_k\Omega\supset\mathcal{A}_\infty\Omega$, so $E_k\to\mathbf{1}$ strongly on $\mathcal{H}_{\tau_\infty}$ (an increasing sequence of projections converges strongly to the projection onto the closure of the union). Therefore $\|z-E_k(z)\|_2=\|z\Omega-E_k(z\Omega)\|\to 0$, i.e. $z=\lim_k\tau_\infty(z)\mathbf{1}$ in $\|\cdot\|_2$, so $z=\tau_\infty(z)\mathbf{1}$. $\square$ **Proposition 14.4** $\boxed{\text{Conditional (on H8)}}$ (Factoriality). *Let $\{\mathcal{A}_k,\iota_k\}$ be an inductive system of finite-dimensional C$^*$-algebras with injective unital inclusions and a compatible faithful tracial state $\tau_\infty$ on the limit, let $\mathcal{M}$ be the weak closure in the GNS representation of $\tau_\infty$, and let $\mathcal{M}_k$ be the image of $\mathcal{A}_k$. If H8 holds, $\mathcal{M}$ is a factor.* *Proof.* All statements are in the von Neumann algebra $\mathcal{M}\subset B(L^2(\mathcal{M},\tau_\infty))$, where $L^2(\mathcal{M},\tau_\infty)$ is the GNS space and $\Omega$ the GNS vector; $\tau_\infty$ is a faithful normal trace on $\mathcal{M}$ (§14.4). *Step 1 (strong convergence $E_k\to\mathrm{id}$ on $L^2$).* Let $e_k$ be the orthogonal projection of $L^2(\mathcal{M},\tau_\infty)$ onto $\overline{\mathcal{M}_k\Omega}$. The subspaces $\overline{\mathcal{M}_k\Omega}$ increase with $k$, so $(e_k)$ is an increasing sequence of projections and converges strongly to the projection $e_\infty$ onto $\overline{\bigcup_k\mathcal{M}_k\Omega}$ (for $\xi$ in the closure of the union and $\varepsilon>0$ pick $\eta\in\overline{\mathcal{M}_{k_0}\Omega}$ with $\|\xi-\eta\|<\varepsilon$; then for $k\geq k_0$, $\|(\mathbf{1}-e_k)\xi\|\leq\|(\mathbf{1}-e_k)\eta\|+\varepsilon=\varepsilon$; for $\xi$ orthogonal to the union, $e_k\xi=0$ for all $k$). By Kaplansky density, the unit ball of $\bigcup_k\mathcal{M}_k$ is strongly dense in the unit ball of $\mathcal{M}$, and for $x\in\mathcal{M}$, $y_\alpha\to x$ strongly with $\|y_\alpha\|\leq\|x\|$ gives $\|(y_\alpha-x)\Omega\|\to0$; hence $\bigcup_k\mathcal{M}_k\Omega$ is dense in $\mathcal{M}\Omega$, which is dense in $L^2$ (cyclicity of $\Omega$). So $e_\infty=\mathbf{1}$ and $e_k\to\mathbf{1}$ strongly. By Theorem 14.4U, $E_k(x)\Omega=e_k(x\Omega)$, so $\|x-E_k(x)\|_2=\|(\mathbf{1}-e_k)x\Omega\|\to0$ for every $x\in\mathcal{M}$. *Step 2 ($E_k$ maps the centre into the centre of $\mathcal{M}_k$).* Let $z\in Z(\mathcal{M})$ and $a\in\mathcal{M}_k$. Bimodularity gives $aE_k(z)=E_k(az)=E_k(za)=E_k(z)a$, so $E_k(z)\in\mathcal{M}_k'\cap\mathcal{M}_k=Z(\mathcal{M}_k)$. *Step 3 (H8 and conclusion).* H8 states $\|E_k(z)-\tau_\infty(z)\mathbf{1}\|_2\to0$. With Step 1, $\|z-\tau_\infty(z)\mathbf{1}\|_2\leq\|z-E_k(z)\|_2+\|E_k(z)-\tau_\infty(z)\mathbf{1}\|_2\to0$, so $z\Omega=\tau_\infty(z)\Omega$. Since $\Omega$ is separating for $\mathcal{M}$ (§14.4), $z=\tau_\infty(z)\mathbf{1}$. Hence $Z(\mathcal{M})=\mathbb{C}\mathbf{1}$. $\square$ **Corollary 14.4c** $\boxed{\text{Theorem}}$ (Factoriality of $\mathcal{M}_{\partial B}$). *$\mathcal{M}_{\partial B}$ is a factor.* *Proof.* Apply Proposition 14.4; its only conditional input is H8, which we now discharge without assumption. By Step 2, $E_k(z)\in Z(\mathcal{M}_k)=\mathbb{C}\mathbf{1}$ (Lemma 14.1), say $E_k(z)=\lambda_k\mathbf{1}$; applying $\tau_\infty$ and using $\tau_\infty\circ E_k=\tau_\infty$ gives $\lambda_k=\tau_\infty(z)$. So H8 holds with the left side identically zero, and Proposition 14.4 applies. $\square$ **Remark 14.4b.** In the form stated, H8 is a consequence of the other hypotheses of Proposition 14.4, so the conditional tag is formally redundant for tensor-product and for factor-inductive systems. It is kept because the gauge-invariant subsystems of Chapter 17, where $\mathcal{A}_k$ are direct sums of factors (the gauge-invariant subalgebra of $M_{m_k}(\mathbb{C})$ under a compact group action is a finite direct sum of matrix algebras by Schur), the centres $Z(\mathcal{M}_k)$ are nontrivial, and factoriality of the limit depends on the inclusion matrices. Theorem 17.1 verifies H8 for the GUQG-invariant subsystem by a Perron-Frobenius argument on the Bratteli diagram. **Theorem 14.5T** (Classical T; Murray, von Neumann, *Ann. Math.* **44** (1943) 716, Thm XIV; Connes, *Ann. Math.* **104** (1976) 73, Thm 6 and Cor. 7.2). *Let $\mathcal{M}$ be a factor of type $\mathrm{II}_1$ with separable predual. If $\mathcal{M}$ is hyperfinite (the weak closure of an increasing sequence of finite-dimensional $*$-subalgebras), then $\mathcal{M}\cong\mathcal{R}$, the unique such factor. (Connes: for factors with separable predual, hyperfinite, injective, and amenable are equivalent.)* **Corollary 14.5** $\boxed{\text{Theorem modulo Classical T}}$ (Type $\mathrm{II}_1$ identification). *$\mathcal{M}_{\partial B}\cong\mathcal{R}$.* *Proof.* We discharge the three hypotheses of Theorem 14.5T. *(i) Separable predual.* $\mathcal{A}_\infty=\bigcup_k\mathcal{A}_k$ is a countable union of finite-dimensional spaces, so it contains a countable dense subset, and hence so does $L^2(\mathcal{M}_{\partial B},\tau_\infty)=\overline{\mathcal{A}_\infty\Omega}$. Thus $\mathcal{M}_{\partial B}$ acts on a separable Hilbert space $\mathcal{H}$. The predual of a von Neumann algebra $\mathcal{M}\subseteq B(\mathcal{H})$ is the quotient of the trace-class operators $\mathcal{L}^1(\mathcal{H})$ by the annihilator of $\mathcal{M}$ (Takesaki I, Thm II.2.6), and $\mathcal{L}^1(\mathcal{H})$ is separable when $\mathcal{H}$ is (finite-rank operators with entries in $\mathbb{Q}+i\mathbb{Q}$ relative to a countable orthonormal basis are dense). Quotients of separable Banach spaces are separable. *(ii) Hyperfinite.* Proposition 14.3. *(iii) Type $\mathrm{II}_1$ factor.* Factor: Corollary 14.4c. Finite: $\tau_\infty$ is a faithful normal tracial state, and a factor admitting one is of type $\mathrm{I}_n$ ($n<\infty$) or $\mathrm{II}_1$ (proof: types $\mathrm{I}_\infty$, $\mathrm{II}_\infty$, $\mathrm{III}$ have no faithful normal tracial state because they contain a projection equivalent to a proper subprojection of itself, forcing $\tau(p)=\tau(q)$ with $p-q\neq0$ positive, contradicting faithfulness). Not type $\mathrm{I}_n$: $\mathcal{M}_{\partial B}\supseteq\mathcal{M}_k\cong M_{m_k}(\mathbb{C})$ with $m_k=\prod_{e\in\Gamma_k}D_e\geq2^{|\Gamma_k|}\to\infty$ (Definition 14.0b: $D_e\geq2$, $|\Gamma_k|\to\infty$), so $\dim\mathcal{M}_{\partial B}=\infty$, whereas a type $\mathrm{I}_n$ factor has dimension $n^2$. Theorem 14.5T now gives $\mathcal{M}_{\partial B}\cong\mathcal{R}$. Since hyperfiniteness is established directly in Proposition 14.3, only the Murray-von Neumann part of Theorem 14.5T is consumed here; Connes' equivalence of injectivity and hyperfiniteness is used only in Proposition 14.10. $\square$ **Remark 14.5a** (On the tag). The v2.0 tag on this corollary was "Conditional on infinite-dimensionality". Definition 14.0b builds infinite-dimensionality into the admissible system through $D_e\geq 2$ and $|\Gamma_k|\to\infty$, which is the physically correct requirement: an infinite causal boundary with at least a qubit per edge. With that definition the result is a theorem modulo Connes' classification. ### 14.6. What the trace state does not give: modular flow **Proposition 14.9** $\boxed{\text{Theorem}}$ (Triviality of the modular flow in the trace state). *Let $\mathcal{M}$ be a von Neumann algebra with a faithful normal tracial state $\tau$, and let $\Omega$ be the GNS vector. Then the modular operator of $(\mathcal{M},\Omega)$ is $\Delta=\mathbf{1}$, the modular conjugation is $J(x\Omega)=x^*\Omega$, and the modular automorphism group is $\sigma_t=\mathrm{id}$ for all $t\in\mathbb{R}$.* *Proof.* Tomita-Takesaki theory (Takesaki, *Tomita's Theory of Modular Hilbert Algebras*, LNM 128, Springer, 1970; Takesaki, *Theory of Operator Algebras II*, Springer, 2003, Ch. VI) defines $S_0\colon x\Omega\mapsto x^*\Omega$ on the dense domain $\mathcal{M}\Omega$ (dense since $\Omega$ is cyclic; well defined since $\Omega$ is separating), with closure $S=J\Delta^{1/2}$ the polar decomposition. For a tracial state, $\|x^*\Omega\|^2=\tau(xx^*)=\tau(x^*x)=\|x\Omega\|^2$, so $S_0$ is isometric and antilinear on a dense domain, hence extends to an antiunitary $S$ with $S^2=\mathbf{1}$. In the polar decomposition $S=J\Delta^{1/2}$, $\Delta=S^*S=\mathbf{1}$ because $S$ is antiunitary ($S^*S=\mathbf{1}$), and then $J=S$. The modular group $\sigma_t(x)=\Delta^{it}x\Delta^{-it}=x$. $\square$ The consequence for Lach-Space is the following. Any statement that identifies a physical time evolution on $\mathcal{M}_{\partial B}$ with the modular flow of the natural state is empty: the natural state is tracial and its modular flow is the identity. A nontrivial modular dynamics requires a non-tracial faithful normal state $\varphi$ on $\mathcal{M}_{\partial B}$, for example $\varphi=\tau_\infty(\rho\,\cdot)$ with $\rho\in L^1(\mathcal{M}_{\partial B},\tau_\infty)_+$, $\tau_\infty(\rho)=1$, $\rho$ injective. By Connes' cocycle theorem the modular group of $\varphi$ is then $\sigma_t^\varphi=\mathrm{Ad}(\rho^{it})$, which is inner. Chapter 15 uses exactly such states, with $\rho$ the reduced density of the radiation. No claim is made that $\rho^{it}$ implements proper-time translation; that identification was the v1.0 error and it is not restored. ### 14.7. The CPW correspondence as a conjecture Witten (*JHEP* **10** (2022) 008) and Chandrasekaran, Penington, Witten (*JHEP* **04** (2023) 009) consider the algebra $\mathcal{A}_{\mathrm{III}}$ of the exterior region of a black hole in a QFT (a type $\mathrm{III}_1$ factor), the modular Hamiltonian $h_\Omega$ of the Hartle-Hawking state, and the crossed product $$ \mathcal{A}_{\mathrm{cr}}:=\mathcal{A}_{\mathrm{III}}\rtimes_{\sigma^\Omega}\mathbb{R}=\bigl\{\mathcal{A}_{\mathrm{III}},\;h_\Omega+\hat x\bigr\}'' $$ on $\mathcal{H}_{\mathrm{III}}\otimes L^2(\mathbb{R})$, where $\hat x$ is the boundary observer's energy and $\hat x$ itself is not an element of $\mathcal{A}_{\mathrm{cr}}$ (it lies in the commutant together with $\mathcal{A}_{\mathrm{III}}'$); the element of $\mathcal{A}_{\mathrm{cr}}$ is the dressed observer energy $\hat h:=h_\Omega+\hat x$. By Takesaki duality $\mathcal{A}_{\mathrm{cr}}$ is a factor of type $\mathrm{II}_\infty$, carrying a faithful normal semifinite trace $\mathrm{Tr}$ (Witten loc. cit., §3, where in the conjugated presentation $\{e^{i\hat ph_\Omega}\mathcal{A}_{\mathrm{III}}e^{-i\hat ph_\Omega},\hat x\}''$ the trace is $\int\mathrm{d}x\,e^{\beta x}\langle\Omega,\cdot\,\Omega\rangle$ with the sign convention of that section); a spectral projection of $\hat h$ onto a half-line has infinite trace in that convention, and we rely below only on the finiteness of the trace of a bounded spectral window, which holds for either sign. The type $\mathrm{II}_1$ case arises for the de Sitter static patch (Chandrasekaran, Longo, Penington, Witten, *JHEP* **02** (2023) 082), where the observer energy is bounded below and the trace weight is $e^{-\beta x}$. For a black-hole exterior we therefore define $$ \mathcal{A}_{\mathrm{cr},+}(\mathcal{E}):=p\,\mathcal{A}_{\mathrm{cr}}\,p,\qquad p:=\mathbf{1}_{[0,E_0]}(\hat h)\in\mathcal{A}_{\mathrm{cr}}, $$ the corner by the spectral projection of the dressed observer energy onto a bounded window $[0,E_0]$; $p$ has finite trace (the trace of a bounded spectral window of $\hat h$ is finite in Witten's formula), and a corner of a $\mathrm{II}_\infty$ factor by a finite projection is a $\mathrm{II}_1$ factor (argument: $p\mathcal{M}p$ is a factor because its centre is $pZ(\mathcal{M})p=\mathbb{C}p$, and it is finite because $\mathrm{Tr}(p\,\cdot\,p)/\mathrm{Tr}(p)$ is a faithful normal tracial state). The normalised trace is $\tau_{\mathrm{cr}}:=\mathrm{Tr}(p\,\cdot\,p)/\mathrm{Tr}(p)$. **Conjecture 14.6** (CPW correspondence). *Let $B\subset N$ be a QSN region whose GHL limit under Theorem 5.6 is a static exterior region $\mathcal{E}$ of a black-hole spacetime with bifurcate Killing horizon, and let $\mathcal{A}_{\mathrm{cr},+}(\mathcal{E})=p\mathcal{A}_{\mathrm{cr}}p$ be the $\mathrm{II}_1$ corner defined above, with normalised trace $\tau_{\mathrm{cr}}$. There exists a dressing map, a unital normal trace-preserving injective $*$-homomorphism* $$ \mathcal{D}\colon\mathcal{M}_{\partial B}\longrightarrow\mathcal{A}_{\mathrm{cr},+}(\mathcal{E}),\qquad \tau_{\mathrm{cr}}\circ\mathcal{D}=\tau_\infty, \tag{14.6} $$ *whose range is weakly dense, and consequently $\mathcal{M}_{\partial B}\cong\mathcal{A}_{\mathrm{cr},+}(\mathcal{E})$.* The conjecture is stated with the dressing map as the required datum, because the obstruction to proving it is precisely the construction of $\mathcal{D}$: one needs an identification of the edge Hilbert factors $\mathcal{H}_e^{(D_e)}$ on the horizon with modes of the continuum QFT dressed by the observer energy, and this requires the continuum limit of the matter sector (Chapter 9) at the level of operator algebras, not merely of spectra. What can be proved is that the existence of $\mathcal{D}$ is all that is needed. **Proposition 14.10** $\boxed{\text{Theorem modulo Classical T}}$ (Dressing intertwiner implies isomorphism). *Suppose a map $\mathcal{D}$ with the properties listed in (14.6) exists. Then $\mathcal{D}$ extends to a $*$-isomorphism $\mathcal{M}_{\partial B}\cong\mathcal{A}_{\mathrm{cr},+}(\mathcal{E})$. Conversely, if $\mathcal{H}_{\mathrm{III}}$ is separable and the QFT on $\mathcal{E}$ satisfies the split property (so that $\mathcal{A}_{\mathrm{cr},+}(\mathcal{E})$ has separable predual and is hyperfinite, as shown in the proof), an abstract isomorphism exists by Classical T, and the content of Conjecture 14.6 reduces to the existence of a trace-preserving $\mathcal{D}$ compatible with the dressing.* *Proof.* Let $\mathcal{D}$ be as in (14.6). Since $\mathcal{D}$ is an injective normal unital $*$-homomorphism between von Neumann algebras, its range $\mathcal{D}(\mathcal{M}_{\partial B})$ is a von Neumann subalgebra (the image of a von Neumann algebra under a normal $*$-homomorphism is weakly closed; Kadison, Ringrose I, Thm 5.5.6 applied to the kernel-free case, or Takesaki I, Prop. III.3.12). Weak density of the range then gives $\mathcal{D}(\mathcal{M}_{\partial B})=\mathcal{A}_{\mathrm{cr},+}(\mathcal{E})$, so $\mathcal{D}$ is surjective, hence a $*$-isomorphism. Trace preservation is automatic once the isomorphism is established, since the normalised trace on a $\mathrm{II}_1$ factor is unique. For the converse, CPW's $\mathcal{A}_{\mathrm{cr},+}$ is a $\mathrm{II}_1$ factor. Assume in addition that $\mathcal{A}_{\mathrm{III}}$ acts on a separable Hilbert space (true for the Fock representation of a free field and for any QFT with a separable vacuum sector); then $\mathcal{A}_{\mathrm{cr},+}$, acting on $\mathcal{H}_{\mathrm{III}}\otimes L^2(\mathbb{R})$ compressed by a projection, has separable predual by the argument of Corollary 14.5(i). If $\mathcal{A}_{\mathrm{cr},+}$ is hyperfinite, Classical T gives $\mathcal{A}_{\mathrm{cr},+}\cong\mathcal{R}\cong\mathcal{M}_{\partial B}$ by Corollary 14.5. Hyperfiniteness of $\mathcal{A}_{\mathrm{cr},+}$ follows if $\mathcal{A}_{\mathrm{III}}$ is injective: crossed products of injective von Neumann algebras by actions of amenable locally compact groups, here $\mathbb{R}$, are injective (Connes loc. cit., §6), corners of injective algebras are injective, and injective factors with separable predual are hyperfinite (Connes Thm 6). Injectivity of $\mathcal{A}_{\mathrm{III}}$ holds for local algebras in any QFT satisfying the split property and the distal split property, from which Buchholz, D'Antoni, Fredenhagen (*Commun. Math. Phys.* **111** (1987) 123) derive that the local algebras are the unique injective type $\mathrm{III}_1$ factor; the split property is an added hypothesis on the continuum QFT of $\mathcal{E}$. So the abstract isomorphism is available; what it lacks is compatibility with the dressing, i.e. the statement that the isomorphism sends the edge-mode subalgebras $\mathcal{M}_k$ to algebras of horizon-localised dressed observables. That compatibility is the physical content of Conjecture 14.6 and is not supplied by Classical T. $\square$ **Remark 14.10a.** Proposition 14.10 is sharper than the v2.0 formulation: it isolates the open problem as the construction of one map with one property, and it shows that the abstract type identification is already a theorem. A proof of Conjecture 14.6 would proceed by constructing $\mathcal{D}$ on each $\mathcal{M}_k$ as the second quantisation of the continuum-limit map of Theorem 9.1 restricted to horizon modes, and verifying trace preservation from the Hartle-Hawking thermality at inverse temperature $\beta$. The missing ingredient is an operator-norm version of Theorem 9.1(5), which currently controls only strong resolvent convergence. ### 14.8. Edge-mode completeness as a conjecture, with a structural partial result Let $U_\varepsilon(\partial B)$ denote the tubular neighbourhood of $\partial B$ in $N$ of graph radius $\varepsilon$ (in units of $d_G$), and let $\mathcal{O}_\varepsilon$ be the weak closure of the algebra generated by all gauge-invariant node and edge operators of Chapter 4 supported in $U_\varepsilon(\partial B)$, in the GNS representation of the QSN ground state of $H_{\mathsf{Q}}$. **Conjecture 14.7** (Edge-mode completeness). *For every $\varepsilon>0$ there is a cutoff assignment $D$ and a region sequence such that $\mathcal{M}_{\partial B}$ is unitarily equivalent to a weakly dense subalgebra of $\mathcal{O}_\varepsilon$.* The v1.0 argument used Reeh-Schlieder to assert that the ground state is cyclic for $\mathcal{O}_\varepsilon$ and Stone-Weierstrass to pass from generators to the full algebra. Reeh-Schlieder (Reeh, Schlieder, *Nuovo Cim.* **22** (1961) 1051) requires an isotonous net of local algebras on a Lorentzian manifold, a positive-energy representation of the translation group, and a vacuum vector; none of these is available on a QSN before the continuum limit, and after the limit the algebra $\mathcal{O}_\varepsilon$ is no longer the object of interest. The argument is withdrawn. What is available is the C$^*$-algebraic structure of the edge-mode algebra before weak closure, which is rigid enough to be classified. **Proposition 14.11** $\boxed{\text{Theorem}}$ (AF structure and Bratteli diagram). *$\widetilde{\mathcal{A}}_{\partial B}$ is an approximately finite-dimensional (AF) C$^*$-algebra. Its Bratteli diagram is determined by the cutoff assignment $e\mapsto D_e$ and the exhaustion $(\Gamma_k)$: it has one vertex at each level $k$, labelled $m_k=\prod_{e\in\Gamma_k}D_e$, and $r_k=\prod_{e\in\Gamma_{k+1}\setminus\Gamma_k}D_e$ edges from level $k$ to level $k+1$. Let $\mathfrak{n}:=\prod_eD_e$ be the supernatural number of the system (for each prime $p$, the exponent of $p$ in $\mathfrak{n}$ is $\sum_e v_p(D_e)\in\mathbb{N}\cup\{\infty\}$). Its dimension group is* $$K_0(\widetilde{\mathcal{A}}_{\partial B})\cong\mathbb{Z}[1/\mathfrak{n}]:=\{q\in\mathbb{Q}\colon qm_k\in\mathbb{Z}\text{ for some }k\}=\bigcup_k\tfrac{1}{m_k}\mathbb{Z},$$ *with positive cone $\mathbb{Z}[1/\mathfrak{n}]\cap[0,\infty)$ and order unit $1$. The isomorphism class of $\widetilde{\mathcal{A}}_{\partial B}$ depends only on $\mathfrak{n}$.* *Proof.* AF structure is Definition: $\widetilde{\mathcal{A}}_{\partial B}$ is the norm closure of the increasing union of finite-dimensional C$^*$-algebras $\mathcal{A}_k$ (Bratteli, *Trans. AMS* **171** (1972) 195). The Bratteli diagram records, at level $k$, the simple summands of $\mathcal{A}_k$ (here one, $M_{m_k}$) and, between levels, the multiplicity of each summand of $\mathcal{A}_k$ in each summand of $\mathcal{A}_{k+1}$ under $\iota_k$. The inclusion $a\mapsto a\otimes\mathbf{1}_{r_k}$ of $M_{m_k}$ into $M_{m_kr_k}$ has multiplicity $r_k$. This determines the diagram. For the dimension group, $K_0(M_m(\mathbb{C}))\cong\mathbb{Z}$ generated by the class of a minimal projection, and under $a\mapsto a\otimes\mathbf{1}_r$ a minimal projection of $M_m$ maps to a projection of rank $r$ in $M_{mr}$, so $K_0(\iota_k)$ is multiplication by $r_k$ on $\mathbb{Z}$. Continuity of $K_0$ under inductive limits (Rørdam, Larsen, Laustsen, *An Introduction to K-Theory for C$^*$-Algebras*, Cambridge, 2000, Thm 6.3.2) gives $K_0(\widetilde{\mathcal{A}}_{\partial B})=\varinjlim(\mathbb{Z}\xrightarrow{r_1}\mathbb{Z}\xrightarrow{r_2}\cdots)$. Identifying the $k$-th copy of $\mathbb{Z}$ with $\frac{1}{m_k}\mathbb{Z}\subset\mathbb{Q}$ (so that $[\mathbf{1}_{\mathcal{A}_k}]\mapsto 1$ and the class of a minimal projection maps to $1/m_k$), the connecting maps become inclusions, and the limit is $\bigcup_k\frac{1}{m_k}\mathbb{Z}=\mathbb{Z}[1/\mathfrak{n}]$, the subgroup of $\mathbb{Q}$ of rationals whose denominators divide some $m_k$, i.e. divide $\mathfrak{n}$ in the supernatural sense (Glimm, *Trans. AMS* **95** (1960) 318). Primes dividing only finitely many $D_e$ contribute a finite exponent to $\mathfrak{n}$ and are not invertible in $\mathbb{Z}[1/\mathfrak{n}]$: for $D_{e_1}=3$ and $D_e=2$ otherwise, $\mathbb{Z}[1/\mathfrak{n}]=\tfrac13\mathbb{Z}[1/2]$, which with order unit $1$ is not isomorphic to $(\mathbb{Z}[1/2],1)$. The positive cone and order unit are as stated since the trace $\tau_\infty$ induces the state $K_0\to\mathbb{R}$, $[p]\mapsto\tau_\infty(p)$, which is the inclusion into $\mathbb{Q}\subset\mathbb{R}$ and is faithful on the cone. The last assertion is Elliott's classification of AF algebras (Elliott, *J. Algebra* **38** (1976) 29): two unital AF algebras are isomorphic if and only if their dimension groups $(K_0,K_0^+,[\mathbf{1}])$ are isomorphic as ordered groups with order unit. Here the ordered group with order unit is $(\mathbb{Z}[1/\mathfrak{n}],1)$, determined by $\mathfrak{n}$ (Glimm loc. cit. for UHF algebras, which these are: a UHF algebra is an AF algebra with a single vertex at each level of its Bratteli diagram). $\square$ **Remark 14.11a.** Proposition 14.11 shows that $\widetilde{\mathcal{A}}_{\partial B}$ is a UHF algebra of type $\prod_eD_e$, and that different cutoff assignments with the same supernatural number give isomorphic C$^*$-algebras while different supernatural numbers give non-isomorphic ones. After weak closure this distinction disappears (all give $\mathcal{R}$, Corollary 14.5). The physical meaning is that the C$^*$-level edge-mode algebra remembers the cutoff scheme through its $K_0$ group, while the von Neumann-level algebra does not. A proof of Conjecture 14.7 would have to identify $K_0(\mathcal{O}_\varepsilon\cap\text{AF part})$ with $\mathbb{Z}[1/\mathfrak{n}]$ for the physically selected $D_e$; for the choice $D_e=2^{s_e}$ with $s_e$ the number of spin-$\frac12$ holonomy channels on $e$, this is the CAR algebra, $K_0=\mathbb{Z}[1/2]$, which is consistent with the fermionic matter Fock structure of $\mathcal{H}_{\mathrm{mat}}$ in Chapter 4 and is the strongest structural evidence currently available for the conjecture. **Remark 14.12** (Dimensions). All objects of this chapter are dimensionless: $\tau_\infty$ is a state, $D_e$ an integer. The link to physical units is through Chapter 15, where a boundary edge carrying a bond of dimension $\chi$ contributes entropy $k_B\log\chi$ and area $\kappa_{d-2}\ell_P^{d-2}$, $\ell_P=(\hbar G/c^3)^{1/2}$, so that, in $d=4$ where $\ell_P^2=\hbar G/c^3$, $k_B\log\chi=k_Bc^3\kappa_2\ell_P^2/(4G\hbar)=k_B\kappa_2/4$, a pure number fixed by the valence data of Lemma 13.1 (in general $d$ replace $G$ by $G_d$ with $\ell_P^{d-2}=\hbar G_d/c^3$). ### 14.9. Hypothesis discharge ledger, Chapter 14 | Result | Classical inputs | Hypotheses used | Step where consumed | |---|---|---|---| | Lem 14.1 | Kadison-Ringrose II 6.6.1 | L6 (discrete spectrum, to define $\mathcal{H}_e^{(D_e)}$) | definition of $\mathcal{A}_\Gamma^{(D)}$ | | Prop 14.2 | Bratteli 1972 Thm 3.3 (re-proved in Lem 14.2a) | Def 14.0b (unital inclusions) | (a) trace preservation; (c) faithfulness | | Prop 14.3 | double commutant theorem | none beyond Prop 14.2 | weak density | | Thm 14.4U | Umegaki / Takesaki IX.4.2 | faithful normal trace (from Prop 14.2) | existence of $E_k$ | | Prop 14.4 | Kaplansky density | H8 | Step 3 | | Cor 14.4c | Lem 14.1 | none (H8 discharged) | centre of $\mathcal{M}_k$ | | Cor 14.5 | Classical T (Thm 14.5T, Murray-von Neumann part) | Def 14.0b ($N$ countable, $D_e\geq2$, $\lvert\Gamma_k\rvert\to\infty$) | (i) separability, (iii) infinite dimension | | Prop 14.9 | Tomita-Takesaki (Takesaki II Ch VI) | traciality of $\tau_\infty$ | isometry of $S_0$ | | Prop 14.10 | Takesaki I III.3.12; Connes §6, Thm 6; BDF 1987 | existence of $\mathcal{D}$ (Conj 14.6); separable $\mathcal{H}_{\mathrm{III}}$ and split property (converse) | surjectivity; converse | | Prop 14.11 | Elliott 1976; Glimm 1960; RLL 6.3.2 | Def 14.0b | $K_0$ computation | ### 14.10. Summary of Chapter 14 | Result | Status | Depends on | |---|---|---| | Lem 14.1 finite factor, unique trace | Theorem | L6 (discrete spectrum) | | Prop 14.2 inductive trace, faithful | Theorem | Def 14.0b | | Lem 14.2a faithfulness on AF closure | Theorem | Bratteli ideal lemma | | Prop 14.3 hyperfiniteness | Theorem | Prop 14.2 | | Lem 14.4a H8 for tensor systems | Theorem | Umegaki expectation | | Prop 14.4 factoriality, general AF | Conditional (H8) | Thm 14.4U | | Cor 14.4c factoriality of $\mathcal{M}_{\partial B}$ | Theorem | Prop 14.4, Lem 14.1 | | Cor 14.5 $\mathcal{M}_{\partial B}\cong\mathcal{R}$ | Theorem modulo T | Prop 14.3, Cor 14.4c | | Prop 14.9 trivial modular flow | Theorem | Tomita-Takesaki | | Conj 14.6 CPW correspondence | Conjecture | | | Prop 14.10 dressing implies isomorphism | Theorem modulo T | | | Conj 14.7 completeness | Conjecture | | | Prop 14.11 UHF/AF structure, $K_0$ | Theorem | Elliott, Glimm | --- ## Chapter 15. The Page Curve and Entanglement Wedge Reconstruction ### 15.1. The random-tensor-network representation The QSN state on which entropy is computed is not the ground state of $H_{\mathsf{Q}}$ in general but a state prepared by a Euclidean evolution from a reference configuration and then evolved unitarily. The tool for computing its entanglement is the random tensor network (RTN) of Hayden, Nezami, Qi, Thomas, Walter, Yang (*JHEP* **11** (2016) 009), which we now attach to the QSN under a named hypothesis. **Hypothesis H9** (Random-tensor-network model). *There is a bond dimension $\chi\geq 2$ such that:* *(a) (Bond structure) For each edge $e\in E$ the edge Hilbert space is cut off at $D_e=\chi$, and the edge state is the maximally entangled state $|\chi_e\rangle=\chi^{-1/2}\sum_{i=1}^\chi|i\rangle_{e,\mathrm{out}}|i\rangle_{e,\mathrm{in}}$ between the two half-edge factors assigned to its endpoints. Each node $v$ carries, in addition to its half-edges, $\ell_v\geq0$ physical legs of dimension $\chi$ each; the physical legs are the degrees of freedom on which reduced states are taken.* *(b) (Haar node tensors, at a fixed time) The node tensors $|T_v\rangle\in\bigotimes_{e\ni v}\mathcal{H}_{e,v}^{(\chi)}\otimes(\mathbb{C}^\chi)^{\otimes\ell_v}$ are independent and Haar-distributed on the unit sphere.* *(c) (Relative-error moment condition after scrambling) For the unitary QSN evolution $U(t)=e^{-iH_{\mathsf{Q}}t}$ from an RTN state $|\Psi(0)\rangle$ on $G^{\mathrm{aug}}(0)$ with Haar node tensors, and for every $t\geq t_{\mathrm{scr}}$ with $t_{\mathrm{scr}}\leq\beta\log S_{\mathrm{BH}}$, every boundary region $A$ of $G^{\mathrm{aug}}(t)$, and every $f$ in the list* $$ \mathcal{F}_A:=\bigl\{\mathrm{Tr}\rho_A,\ \mathrm{Tr}\rho_A^n,\ (\mathrm{Tr}\rho_A)^n\ (n\geq2\text{ used}),\ (\mathrm{Tr}\rho_A^2)^2\bigr\}, $$ *one has* $$ \bigl|\mathbb{E}\,f\bigl(U(t)\Psi(0)\bigr)-\mathbb{E}_{\mathrm{Haar},G^{\mathrm{aug}}(t)}f\bigr|\leq\epsilon_{\mathrm{des}}\,\mathbb{E}_{\mathrm{Haar},G^{\mathrm{aug}}(t)}f,\qquad\epsilon_{\mathrm{des}}\leq\chi^{-1}, $$ *where the left expectation is over the $t=0$ node tensors and the right one over an RTN on the time-$t$ graph with Haar tensors. This is a statement about the evolved vector on the physical legs, not about a decomposition of it into node tensors. For $t0$ is claimed. H9(d) is the replica-trick assumption; it is not a theorem ($Z_n$ is defined on integers, continuation is not unique without the growth condition, and (i) is a self-averaging statement). Results below that need only integer Rényi moments do not use H9(d). Chapter 14 works along a sequence of regions with $|\partial B_k|\to\infty$; Chapter 15 works at one fixed member of that sequence, a finite graph (finite by L1, H1, H3), at fixed $\chi$. The two limits are discussed in §15.5. **The augmented graph.** Let $G^{\mathrm{aug}}=(V^{\mathrm{aug}},E^{\mathrm{aug}})$ be the graph obtained from $(N,E)$ by attaching, for each physical leg of each node $v$, a new pendant vertex joined to $v$ by one edge. A *boundary region* is a set $A$ of pendant vertices; its complement $\bar A$ is the set of remaining pendant vertices. An *$A$-cut* is a set $\gamma\subseteq E^{\mathrm{aug}}$ whose removal disconnects $A$ from $\bar A$; $|\gamma|$ is its cardinality. Every inclusion-minimal $A$-cut defines, and is defined by, the set $W(\gamma)\subseteq N$ of QSN nodes on the $A$ side, with $\gamma=\partial^{\mathrm{aug}}W$ consisting of the original edges between $W$ and $N\setminus W$ together with the pendant edges of $\bar A$-legs on nodes of $W$ and of $A$-legs on nodes of $N\setminus W$; a minimiser of $|\gamma|$ is inclusion-minimal and hence of this form. The pendant-edge terms are the correction that the first draft omitted. Write $\rho_A$ for the reduced state of $|\Psi\rangle\langle\Psi|$ on the legs $A$, $S(A):=-\mathrm{Tr}\hat\rho_A\log\hat\rho_A$ with $\hat\rho_A=\rho_A/\mathrm{Tr}\rho_A$, and $S_n(A):=\frac{1}{1-n}\log\mathrm{Tr}\hat\rho_A^n$. A bulk state $\rho_{\mathrm{bulk}}$ on additional bulk legs is included as in HNQTWY by letting the node tensors carry one extra leg contracted with $\rho_{\mathrm{bulk}}$; $S_n^{\mathrm{eff}}(W)$ denotes the Rényi entropy of $\rho_{\mathrm{bulk}}$ restricted to the bulk legs of $W\subseteq N$. ### 15.2. The discrete quantum extremal surface formula **Hypothesis (MC)** (Minimal-cut genericity, no bulk). *The $A$-cut of minimal cardinality is unique; call it $\gamma_A$ and its $A$-side $W^*$. Every other $A$-cut has cardinality $\geq|\gamma_A|+1$.* **Hypothesis (GAP$_n$)** (Weighted ground-state gap, with bulk). *Let $E_n(g):=\log\chi\sum_{\langle v,w\rangle\in E^{\mathrm{aug}}}d_{S_n}(g_v,g_w)-\log\mathrm{Tr}\bigl(\rho_{\mathrm{bulk}}^{\otimes n}\textstyle\bigotimes_vP_{g_v}\bigr)$ be the energy of a spin configuration $g\in S_n^{N}$ with boundary spins $\mathsf{c}$ on $A$ and $\mathrm{id}$ on $\bar A$, where the bulk trace is a complex number of modulus $\leq1$ and $\log$ denotes a branch, so that the weight $w_n(g):=e^{-E_n(g)}$ is complex with $|w_n(g)|\leq\chi^{-\sum d_{S_n}}$. Then there is a domain-wall configuration $g^*_n$ ($\mathsf{c}$ on some $W^*_n\subseteq N$, $\mathrm{id}$ elsewhere) with real positive weight $w_n(g^*_n)=e^{(1-n)F_n(W^*_n)}$, $F_n(W):=|\partial^{\mathrm{aug}}W|\log\chi+S_n^{\mathrm{eff}}(W)$, such that* $$ \epsilon_n:=\sum_{g\neq g^*_n}\frac{|w_n(g)|}{w_n(g^*_n)}\leq\tfrac12\ \text{ and }\ \epsilon'_n:=\sum_{g\neq\mathrm{id}}|w^{(0)}_n(g)|\leq\tfrac12, $$ *where $w^{(0)}_n$ is the weight with all boundary spins $\mathrm{id}$, and $F_n(W^*_n)=\min_WF_n(W)$.* (GAP$_n$) is a hypothesis on the weighted spin model, not a consequence of (MC): with a bulk state the competing configurations include those with intermediate permutations on the cut whose bulk factors are products of lower moments, and their total weight is not controlled by the area gap alone. In the no-bulk case (GAP$_n$) follows from (MC) with $\epsilon_n,\epsilon'_n\leq(n!)^{|N|}\chi^{-1}$ (Lemma 15.1L below). With a bulk state of Bell pairs the pair bonds have weight $\log2$ and the excess energy of a competitor can be as small as $\log2$, so the analogous bound is $(n!)^{|N|}/2$, which is vacuous; this is why the bulk case is left Conditional on (GAP$_n$) with $\epsilon_n$ named. **Theorem 15.1T** (Classical V; Hayden, Nezami, Qi, Thomas, Walter, Yang, *JHEP* **11** (2016) 009; the paper has no numbered theorems; the section anchors below follow the arXiv version 1601.01694v2 and are to be verified against the journal version before print). *Let $G$ be a finite graph of maximal degree $D$ with $|V|$ vertices, each edge carrying the maximally entangled state of dimension $\chi$, each vertex a Haar-random unit tensor, boundary legs partitioned as $A\sqcup\bar A$, and let $\gamma_A$ be the unique minimal $A$-cut with all other $A$-cuts of cardinality $\geq|\gamma_A|+1$. Then:* *(a) (§2, second Rényi) $\dfrac{\mathbb{E}\,\mathrm{Tr}\rho_A^2}{\mathbb{E}(\mathrm{Tr}\rho_A)^2}=\chi^{-|\gamma_A|}\dfrac{1+\epsilon_2}{1+\epsilon'_2}$ with $0\leq\epsilon_2,\epsilon'_2\leq2^{|V|}\chi^{-1}$.* *(b) (§2, von Neumann bound) Annealed inequality: $S_2^{\mathrm{ann}}(A):=-\log\frac{\mathbb{E}\mathrm{Tr}\rho_A^2}{\mathbb{E}(\mathrm{Tr}\rho_A)^2}\geq|\gamma_A|\log\chi-\log(1+\epsilon_2)$. Realisation-wise sandwich: for every realisation, $S_2(A)\leq S(A)\leq S_0(A)\leq|\gamma_A|\log\chi$, where $S_0=\log\mathrm{rank}\,\hat\rho_A$.* *(c) (§3, higher Rényi) For each integer $n\geq3$, $\dfrac{\mathbb{E}\,\mathrm{Tr}\rho_A^n}{\mathbb{E}(\mathrm{Tr}\rho_A)^n}=\chi^{-(n-1)|\gamma_A|}\dfrac{1+\epsilon_n}{1+\epsilon'_n}$ with $0\leq\epsilon_n,\epsilon'_n\leq(n!)^{|V|}\chi^{-1}$; the constant grows factorially in $n$ and the bound is not uniform in $n$.* *(d) (§2, fluctuations) HNQTWY show $\mathrm{Var}(\mathrm{Tr}\rho_A^2)/(\mathbb{E}\mathrm{Tr}\rho_A^2)^2=O(\chi^{-1})$; the explicit constants used below are derived in Lemma 15.1V, not taken from the paper.* *(e) (§3 of the arXiv version, bulk state) With a bulk state, the moments acquire the complex factor $\mathrm{Tr}(\rho_{\mathrm{bulk}}^{\otimes n}\bigotimes_vP_{g_v})$ per configuration, and the leading configuration is the domain wall minimising $|\gamma|\log\chi+S_n^{\mathrm{eff}}(A\text{-side})$ when that minimiser dominates in the sense of (GAP$_n$).* *Discharge of hypotheses in the QSN setting.* Finite graph of bounded degree: L1, H1, H3. Bonds and Haar tensors: H9(a),(b) at the time considered, or H9(c) with the relative-error correction of Lemma 15.1D. Uniqueness and unit gap of the minimal cut: (MC). Domination with bulk state: (GAP$_n$). **Lemma 15.1D** (Design correction). *If the node-tensor law $\mu$ satisfies the relative-error condition of H9(c) with parameter $\epsilon_{\mathrm{des}}<1$, then for each integer $n\geq2$,* $$ \Bigl|\,S_n^{\mathrm{ann},\mu}(A)-S_n^{\mathrm{ann},\mathrm{Haar}}(A)\Bigr|\leq\eta_n:=\frac{2}{n-1}\log\frac{1+\epsilon_{\mathrm{des}}}{1-\epsilon_{\mathrm{des}}}\leq\frac{6\epsilon_{\mathrm{des}}}{n-1}\quad(\epsilon_{\mathrm{des}}\leq\tfrac12). $$ *Proof.* The ratio $Z_n^\mu/Z_n^{\mathrm{Haar}}$ lies in $[\frac{1-\epsilon_{\mathrm{des}}}{1+\epsilon_{\mathrm{des}}},\frac{1+\epsilon_{\mathrm{des}}}{1-\epsilon_{\mathrm{des}}}]$ since numerator and denominator are each within relative $\epsilon_{\mathrm{des}}$ of their Haar values; take $\frac{1}{n-1}\log$, and use $\log\frac{1+u}{1-u}\leq3u$ for $u\leq\frac12$. $\square$ **Lemma 15.1L** (Gap in the no-bulk model). *Under (MC) and no bulk state, $E_n$ has the unique minimiser $g^*=(\mathsf{c}$ on $W^*$, $\mathrm{id}$ elsewhere$)$ with $E_n(g^*)=(n-1)|\gamma_A|\log\chi$, every other configuration has $E_n\geq E_n(g^*)+\log\chi$, and $\epsilon_n,\epsilon'_n\leq(n!)^{|N|}\chi^{-1}$.* *Proof.* Lower bound: by Menger's theorem (Menger, *Fund. Math.* **10** (1927) 96) there are $|\gamma_A|$ edge-disjoint paths in $G^{\mathrm{aug}}$ from $A$ to $\bar A$; along each, the triangle inequality for the Cayley metric gives $\sum d_{S_n}\geq d_{S_n}(\mathsf{c},\mathrm{id})=n-1$, and edge-disjointness makes the contributions add, so $E_n(g)\geq(n-1)|\gamma_A|\log\chi$ for every $g$. Attainment: $g^*$ has nonzero bond energy only on $\gamma_A$, each bond contributing $(n-1)\log\chi$. Uniqueness and gap: suppose $E_n(g)=E_n(g^*)$. Then every edge off the $|\gamma_A|$ paths has cost $0$ and every path carries exactly $n-1$ units, so along each path the spins form a geodesic from $\mathsf{c}$ to $\mathrm{id}$ in the Cayley metric. Let $W_1:=\{v\colon g_v=\mathsf{c}\}$ and $W_2:=\{v\colon g_v\neq\mathrm{id}\}$. Each is the $A$-side of an $A$-cut ($\partial^{\mathrm{aug}}W_1$, $\partial^{\mathrm{aug}}W_2$), every edge of which has cost $\geq1$ and is crossed by a geodesic path exactly once (a geodesic leaves $\mathsf{c}$ once and reaches $\mathrm{id}$ once), and edges off the paths have cost $0$, so are not in these cuts. Hence $|\partial^{\mathrm{aug}}W_1|=|\partial^{\mathrm{aug}}W_2|=|\gamma_A|$, and (MC) gives $W_1=W_2=W^*$, i.e. $g_v=\mathsf{c}$ on $W^*$ and $g_v=\mathrm{id}$ off it: $g=g^*$. Since $d_{S_n}$ is integer-valued, any $g\neq g^*$ has $E_n(g)\geq E_n(g^*)+\log\chi$. Counting: there are at most $(n!)^{|N|}$ configurations, each with weight $\leq\chi^{-1}$ relative to the ground state. The same count with the all-$\mathrm{id}$ ground state gives $\epsilon'_n$. $\square$ **Lemma 15.1V** (Second-moment bounds). *Under H9(a),(b) and (MC), with $v:=(4!)^{|N|}\chi^{-1}\leq1$,* $$ \frac{\mathbb{E}[(\mathrm{Tr}\rho_A^2)^2]}{(\mathbb{E}\,\mathrm{Tr}\rho_A^2)^2}\leq1+v\leq1+7v,\qquad\frac{\mathbb{E}[(\mathrm{Tr}\rho_A)^2]}{(\mathbb{E}\,\mathrm{Tr}\rho_A)^2}\leq1+2^{|N|}\chi^{-1}. $$ *Under H9(c) in place of H9(b), the same ratios are bounded by $v_\mu:=\frac{(1+\epsilon_{\mathrm{des}})(1+7v)}{(1-\epsilon_{\mathrm{des}})^2}-1$ and $v'_\mu:=\frac{(1+\epsilon_{\mathrm{des}})(1+2^{|N|}\chi^{-1})}{(1-\epsilon_{\mathrm{des}})^2}-1$.* *Proof.* $\mathbb{E}[(\mathrm{Tr}\rho_A^2)^2]$ is the four-replica contraction with boundary permutation $\mathsf{c}_2:=(12)(34)$ on $A$ and $\mathrm{id}$ on $\bar A$; $d_{S_4}(\mathrm{id},\mathsf{c}_2)=2$. By Step 2 of the proof of Theorem 15.1 it equals $\prod_v\frac{(d_v-1)!}{(d_v+3)!}$ times the spin sum with these boundary conditions. The argument of Lemma 15.1L (Menger paths, geodesics, $W_1=W_2=W^*$) applies with $n=4$ and $\mathsf{c}_2$ in place of $\mathsf{c}$, giving the unique ground state ($\mathsf{c}_2$ on $W^*$) of weight $\chi^{-2|\gamma_A|}$ and excess sum $v_{\mathsf{c}_2}\in[0,v]$; likewise $\mathbb{E}\mathrm{Tr}\rho_A^2=\prod_v\frac{(d_v-1)!}{(d_v+1)!}\chi^{-|\gamma_A|}(1+\epsilon_2)$ with $\epsilon_2\geq0$. Therefore $$ \frac{\mathbb{E}[(\mathrm{Tr}\rho_A^2)^2]}{(\mathbb{E}\mathrm{Tr}\rho_A^2)^2}=\prod_v\frac{(d_v+1)d_v}{(d_v+3)(d_v+2)}\cdot\frac{1+v_{\mathsf{c}_2}}{(1+\epsilon_2)^2}\leq1+v, $$ the node prefactor ratio being $<1$. For the second ratio, $\mathbb{E}\mathrm{Tr}\rho_A=\prod_v\frac{(d_v-1)!}{d_v!}$ (one replica, no spin sum) and $\mathbb{E}(\mathrm{Tr}\rho_A)^2=\prod_v\frac{(d_v-1)!}{(d_v+1)!}(1+\epsilon'_2)$ with $0\leq\epsilon'_2\leq2^{|N|}\chi^{-1}$ (Lemma 15.1L at $n=2$), so the ratio is $\prod_v\frac{d_v}{d_v+1}(1+\epsilon'_2)\leq1+2^{|N|}\chi^{-1}$. Under H9(c), each of the four moments $(\mathrm{Tr}\rho_A^2)^2$, $\mathrm{Tr}\rho_A^2$, $(\mathrm{Tr}\rho_A)^2$, $\mathrm{Tr}\rho_A$, all in $\mathcal{F}_A$, is within relative $\epsilon_{\mathrm{des}}$ of its Haar value, giving $v_\mu,v'_\mu$. $\square$ **Theorem 15.1** (Discrete QES formula). *Let $\mathsf{Q}$ satisfy L1, H1, H3, let H9(a) hold, and let the law of the state be Haar on $G^{\mathrm{aug}}$ (H9(b)) or satisfy the moment condition H9(c) with parameter $\epsilon_{\mathrm{des}}$; set $\eta_n:=0$ in the Haar case and as in Lemma 15.1D otherwise. Let $|\Psi\rangle$ be the state (15.1) and $A$ a boundary region.* *(A)* $\boxed{\text{Conditional (on H9(a),(b) or (c); MC)}}$ *(no bulk state). Assume $2^{|N|}\chi^{-1}\leq\frac12$ and $(4!)^{|N|}\chi^{-1}\leq1$. Set $V_A:=7(4!)^{|N|}\chi^{-1}+2^{|N|}\chi^{-1}$ in the Haar case and $V_A:=v_\mu+v'_\mu$ (Lemma 15.1V) in the design case. For every $\epsilon\in[2^{|N|-1}\chi^{-1},\tfrac14]$, with probability at least $1-V_A\epsilon^{-2}$ over the $t=0$ node tensors,* $$ |\gamma_A|\log\chi-\delta_A\ \leq\ S(A)\ \leq\ |\gamma_A|\log\chi,\qquad\delta_A:=6\epsilon+2^{|N|+2}\chi^{-1}+\eta_2. \tag{15.2A} $$ *(B)* $\boxed{\text{Conditional (on H9(a),(b) or (c); GAP}_n)}$ *(integer Rényi, with bulk). For each integer $n\geq2$,* $$ \Bigl|S_n^{\mathrm{ann}}(A)-\min_{W}F_n(W)\Bigr|\leq\frac{2(\epsilon_n+\epsilon'_n)}{n-1}+\eta_n=:\epsilon_U^{(n)}. \tag{15.2B} $$ *(C)* $\boxed{\text{Conditional (on H9(a),(b) or (c); GAP}_n\text{ for integers }n\geq2\text{; H9(d))}}$ *(von Neumann, with bulk).* $$ \Bigl|\mathbb{E}\,S(A)-\min_{W}F(W)\Bigr|\leq\epsilon_U,\qquad F(W):=|\partial^{\mathrm{aug}}W|\log\chi+S_{\mathrm{eff}}(W), \tag{15.2} $$ *where $\epsilon_U$ is the uniform bound of H9(d)(ii). No statement about the fluctuations of $S(A)$ about $\mathbb{E}S(A)$ is made in (C).* *Proof.* *Step 1 (moments as contractions).* For integer $n\geq2$, $\mathrm{Tr}\rho_A^n=\langle\Psi|^{\otimes n}\mathcal{C}_A|\Psi\rangle^{\otimes n}$ where $\mathcal{C}_A$ is the cyclic permutation $\mathsf{c}$ of replicas on the legs $A$ and the identity on $\bar A$; and $(\mathrm{Tr}\rho_A)^n$ is the same with $\mathcal{C}_A$ replaced by the identity. *Step 2 (Haar average).* With $|T_v\rangle$ Haar on a space of dimension $d_v$, $\mathbb{E}|T_v\rangle\langle T_v|^{\otimes n}=\frac{(d_v-1)!}{(d_v+n-1)!}\sum_{g\in S_n}P_g$ (Harrow, arXiv:1308.6595, Prop. 6). Independence across nodes factorises the expectation. A bond between $v,w$ contributes $\langle\chi|^{\otimes n}(P_{g_v}\otimes P_{g_w})|\chi\rangle^{\otimes n}=\chi^{-n}\mathrm{Tr}P_{g_v^{-1}g_w}=\chi^{-d_{S_n}(g_v,g_w)}$ with $d_{S_n}(g,h)=n-\#\mathrm{cyc}(g^{-1}h)$ the Cayley metric; a pendant edge to an $A$-leg contributes $\chi^{-n}\cdot\chi^{n}\cdot\chi^{-d_{S_n}(g_v,\mathsf{c})}$ after normalising by the leg dimension, and a pendant edge to an $\bar A$-leg contributes $\chi^{-d_{S_n}(g_v,\mathrm{id})}$; a bulk leg contributes $\mathrm{Tr}(\rho_{\mathrm{bulk}}^{\otimes n}\bigotimes_vP_{g_v})$ jointly. Collecting, $$ Z_n(A)=\frac{\sum_{g}e^{-E_n(g)}}{\sum_{g}e^{-E_n^{(0)}(g)}}, \tag{15.3} $$ with $E_n$ as in (GAP$_n$): the spin model lives on $G^{\mathrm{aug}}$ with boundary spins on the pendant vertices. This is the explicit Schmidt-rank counting: each edge of $G^{\mathrm{aug}}$ across which the spin jumps from $\mathrm{id}$ to $\mathsf{c}$ costs $(n-1)\log\chi$, one unit of $\log\chi$ per unit of Schmidt rank after dividing by $n-1$. *Step 3 (ground state and error, integer $n$).* By (GAP$_n$) (or Lemma 15.1L in the no-bulk case, where all weights are real positive) the numerator of (15.3) is $w_n(g^*_n)(1+z)$ and the denominator is $1+z'$ with complex $|z|\leq\epsilon_n\leq\frac12$, $|z'|\leq\epsilon'_n\leq\frac12$ (the all-$\mathrm{id}$ configuration has weight $1$). For a domain wall the bulk factor is $\mathrm{Tr}\rho_{\mathrm{bulk},W}^n=e^{(1-n)S_n^{\mathrm{eff}}(W)}$, real positive, so $w_n(g^*_n)=e^{(1-n)F_n(W^*_n)}$. Hence $$ S_n^{\mathrm{ann}}(A)=F_n(W^*_n)+\frac{1}{1-n}\log\frac{1+z}{1+z'}, $$ where $Z_n$ is real positive (it is a ratio of moments of a positive operator), so the logarithm is real, and $|\log|1+z||\leq|z|/(1-|z|)\leq2\epsilon_n$, likewise for $z'$. Adding $\eta_n$ from Lemma 15.1D in the design case gives (15.2B) with $F_n(W^*_n)=\min_WF_n(W)$ by (GAP$_n$). *Step 4 (Part (A)).* No bulk, so Lemma 15.1L gives (GAP$_2$) with $\epsilon_2,\epsilon'_2\leq2^{|N|}\chi^{-1}\leq\frac12$, and $W^*=$ the $A$-side of $\gamma_A$. Upper bound: for every realisation $S(A)\leq S_0(A)\leq|\gamma_A|\log\chi$, because the state on the legs $A$ is obtained by contracting $|\Psi\rangle$ across the $|\gamma_A|$ edges of $G^{\mathrm{aug}}$ in $\gamma_A$, each of dimension $\chi$, so its rank is at most $\chi^{|\gamma_A|}$. Lower bound: $S(A)\geq S_2(A)=-\log\bigl(\mathrm{Tr}\rho_A^2/(\mathrm{Tr}\rho_A)^2\bigr)$. By Lemma 15.1V and Chebyshev, $\Pr[|\mathrm{Tr}\rho_A^2-\mathbb{E}|>\epsilon\,\mathbb{E}]\leq7(4!)^{|N|}\chi^{-1}\epsilon^{-2}$ (resp. $v_\mu\epsilon^{-2}$) and $\Pr[|\mathrm{Tr}\rho_A-\mathbb{E}|>\epsilon\,\mathbb{E}]\leq2^{|N|}\chi^{-1}\epsilon^{-2}$ (resp. $v'_\mu\epsilon^{-2}$); on the intersection, of probability $\geq1-V_A\epsilon^{-2}$, $$ \frac{\mathrm{Tr}\rho_A^2}{(\mathrm{Tr}\rho_A)^2}\leq\frac{1+\epsilon}{(1-\epsilon)^2}\cdot\frac{\mathbb{E}\mathrm{Tr}\rho_A^2}{(\mathbb{E}\mathrm{Tr}\rho_A)^2}=\frac{1+\epsilon}{(1-\epsilon)^2}\cdot\frac{\mathbb{E}(\mathrm{Tr}\rho_A)^2}{(\mathbb{E}\mathrm{Tr}\rho_A)^2}\cdot Z_2, $$ and $\frac{\mathbb{E}(\mathrm{Tr}\rho_A)^2}{(\mathbb{E}\mathrm{Tr}\rho_A)^2}\leq1+2^{|N|}\chi^{-1}$ by Lemma 15.1V. Taking $-\log$ and using $\log\frac{1+\epsilon}{(1-\epsilon)^2}\leq\frac{3\epsilon}{1-\epsilon}\leq4\epsilon$ for $\epsilon\leq\frac14$ and $\log(1+u)\leq u$: $S_2(A)\geq S_2^{\mathrm{ann}}(A)-4\epsilon-2^{|N|}\chi^{-1}$. By (15.2B) at $n=2$, $S_2^{\mathrm{ann}}(A)\geq|\gamma_A|\log\chi-2(\epsilon_2+\epsilon'_2)-\eta_2\geq|\gamma_A|\log\chi-4\cdot2^{|N|}\chi^{-1}-\eta_2$. Altogether $S(A)\geq|\gamma_A|\log\chi-4\epsilon-5\cdot2^{|N|}\chi^{-1}-\eta_2$, and since $\epsilon\geq2^{|N|-1}\chi^{-1}$ gives $2^{|N|}\chi^{-1}\leq2\epsilon$, this is $\geq|\gamma_A|\log\chi-6\epsilon-2^{|N|+2}\chi^{-1}-\eta_2$, which is (15.2A). *Step 5 (Part (C)).* Under H9(d)(ii) the bound (15.2B) holds for $n\in U$ with $\epsilon_U$ in place of $\epsilon_U^{(n)}$, and $\min_WF_n(W)\to\min_WF(W)$ as $n\to1$ because the minimum of finitely many functions continuous at $n=1$ is continuous there, and $S_n^{\mathrm{eff}}\to S_{\mathrm{eff}}$ for finite-rank $\rho_{\mathrm{bulk}}$ (removable singularity of $\frac{1}{1-n}\log\sum\lambda_i^n$ at $n=1$). H9(d)(i) identifies the left limit with $\mathbb{E}S(A)$. $\square$ **Remark 15.1a** (On the tags and on the physical $\chi$). Part (A) is a theorem about the RTN model under (MC); its error terms contain $2^{|N|}\chi^{-1}$ and $(4!)^{|N|}\chi^{-1}$, so it is informative only for $\log\chi\gg|N|\log 24$. This is the HNQTWY large-bond regime. The physical bond dimension fixed by the area matching of Lemma 13.1 and Theorem 5.6, $\log\chi=\kappa_{d-2}/4$, is of order one (and requires $\kappa_{d-2}/4\in\log\mathbb{N}$, a constraint on the combinatorial constant of Lemma 13.1 that is not verified in this monograph). Theorem 15.1 therefore describes the large-$\chi$ RTN idealisation of the QSN state, and its application at physical $\chi$ is a modelling step outside the theorem. Parts (B) and (C) additionally rest on (GAP$_n$) and H9(d), neither proved here nor in the cited literature. ### 15.3. The Page mechanism The evaporation model is the random-tensor-network Page model of Penington, Shenker, Stanford, Yang (*JHEP* **03** (2022) 205, §2) placed on the QSN. It uses only Part (A) of Theorem 15.1: the Hawking quanta are physical legs, not a bulk state, so neither (GAP$_n$) nor H9(d) is consumed. **Hypothesis (HG)** (Horizon geometry and emission). *There are integers $r(t),b(t)\geq0$ with $r(t)+b(t)=S_0/\log\chi$ constant, $r$ non-decreasing, $b$ non-increasing, $r(0)=0$, $b(t)\to0$ as $t\uparrow t_{\mathrm{evap}}$, such that at each $t\in[0,t_{\mathrm{evap}})$:* *(i) the radiation legs $R(t)$ are $r(t)$ pendant vertices attached to nodes of $\mathrm{Rad}(t)$, and the black-hole legs $\mathcal{B}(t)$ are $b(t)$ pendant vertices attached to nodes of $B(t)$, with $b(t)=|\partial B(t)|$ (one black-hole leg per horizon bond); there are no other physical legs;* *(ii) in $G^{\mathrm{aug}}(t)$ the two cuts $\gamma_R:=$ the $r(t)$ pendant edges of $R(t)$ and $\gamma_{\mathcal{B}}:=$ the $b(t)$ pendant edges of $\mathcal{B}(t)$ are $R$-cuts, and every $R$-cut $\gamma$ satisfies $|\gamma|\geq\min(r(t),b(t))$, with equality only for $\gamma\in\{\gamma_R,\gamma_{\mathcal{B}}\}$;* *(iii) the state at time $t$ either is the RTN state (15.1) on $G^{\mathrm{aug}}(t)$ with Haar node tensors (H9(b), assumed separately at each $t$), or is $U(t)\Psi(0)$ with $t\geq t_{\mathrm{scr}}$ and the moment condition H9(c) holding on $G^{\mathrm{aug}}(t)$.* (HG)(ii) is the statement that the cheapest way to separate radiation legs from black-hole legs is to cut one of the two leg bundles, uniquely; it is the discrete form of "the extremal surface is the horizon or empty" and is assumed, not derived from a connectivity condition. (HG)(i) fixes $S_{\mathrm{BH}}(t):=b(t)\log\chi=|\partial B(t)|\log\chi$ and $S_{\mathrm{H}}(t):=r(t)\log\chi$; the constraint $r+b$ constant is the statement that each emission moves one leg from the black hole to the radiation, the discrete Hawking pair. Unitarity of $U(t)$ enters only through H9(c). **Theorem 15.2** $\boxed{\text{Conditional (on L1, H1, H3; H9(a) and H9(b) or (c); HG)}}$ (Two-phase Page mechanism). *Assume (HG). Let $W_\chi:=\{t\in[0,t_{\mathrm{evap}})\colon r(t)=b(t)\}$ be the degeneracy window. Then for every $t\in[0,t_{\mathrm{evap}})\setminus W_\chi$ and every $\epsilon\in[2^{|N|-1}\chi^{-1},\frac14]$, with probability at least $1-V_A\epsilon^{-2}$ (constants of Theorem 15.1(A)),* $$ \min\{S_{\mathrm{H}}(t),S_{\mathrm{BH}}(t)\}-\delta_A\ \leq\ S(R(t))\ \leq\ \min\{S_{\mathrm{H}}(t),S_{\mathrm{BH}}(t)\},\qquad\delta_A=6\epsilon+2^{|N|+2}\chi^{-1}+\eta_2. \tag{15.4} $$ *For $t$ with $r(t)b(t)$ the minimal cut is $\gamma_{\mathcal{B}}$ and the envelope $S_{\mathrm{BH}}(t)$ is non-increasing, tending to $0$ as $t\uparrow t_{\mathrm{evap}}$. The set $\{t\colon S_{\mathrm{H}}(t)\geq S_{\mathrm{BH}}(t)\}$ is an interval with infimum $t_{\mathrm{Page}}:=\inf\{t\colon r(t)\geq b(t)\}$; $W_\chi$ is the set of $t$ at which $r=b$, possibly empty.* *Proof.* Fix $t\notin W_\chi$. By (HG)(ii) the minimal $R$-cut is unique, of cardinality $\min(r,b)$, and all other cuts have cardinality $\geq\min(r,b)+1$: (MC) holds. Theorem 15.1(A) with $A=R(t)$ gives (15.4) with $|\gamma_A|\log\chi=\min(r,b)\log\chi=\min\{S_{\mathrm{H}},S_{\mathrm{BH}}\}$. The identification of the minimal cut on each side of $W_\chi$ and the monotonicity statements are (HG)(i),(ii). For $t\in W_\chi$ the two cuts tie, (MC) fails, and no claim is made. $\square$ **Remark 15.2a.** In this model the "island" is the set of black-hole nodes: for $r>b$ the minimal cut $\gamma_{\mathcal{B}}$ has the whole of $B(t)$ on the radiation side. The earlier formulation with islands $I\subseteq B$ and a bulk Hawking state is recovered by (HG) with the Hawking partners represented as legs; the pendant-edge accounting of §15.1 is what makes $F(\emptyset)=S_{\mathrm{H}}$ and $F(B)=S_{\mathrm{BH}}$ come out correctly, and an explicit hypothesis of the form (HG)(ii) is needed in place of the isoperimetric H4', which bounds ratios but does not identify minimisers. ### 15.4. The Page time **Proposition 15.3** $\boxed{\text{Conditional (on the hypotheses of Thm 15.2; rate law)}}$ (Page time). *Assume the hypotheses of Theorem 15.2 and the rate law: $b$ and $r=S_0/\log\chi-b$ are the integer parts of a decreasing $C^1$ interpolation $\bar b(t)$ with $\dot{\bar b}=-\Gamma(t)$, $\Gamma$ the number rate of emission. Let $t^*:=t_{\mathrm{Page}}=\inf\{t\colon r(t)\geq b(t)\}$. Then* $$ \bigl|S_{\mathrm{BH}}(t^*)-\tfrac12S_0\bigr|\leq\tfrac12\log\chi,\qquad\bigl|S_{\mathrm{H}}(t^*)-\tfrac12S_0\bigr|\leq\tfrac12\log\chi,\qquad S_0:=S_{\mathrm{BH}}(0), \tag{15.5} $$ *with equality to $\frac12S_0$ when $S_0/\log\chi$ is even. If $\bar b$ is linear in $t$, the crossing of the interpolated curves is at $t_{\mathrm{evap}}/2$. For the Schwarzschild luminosity in $d=4$, $L=-\dot M\propto M^{-2}$, so that the number rate is $\Gamma\propto L/T_{\mathrm{H}}\propto M^{-1}$ and $S_{\mathrm{BH}}\propto M^2$ gives $M(t)^3=M_0^3-3\alpha t$, $t_{\mathrm{evap}}=M_0^3/(3\alpha)$, $S_{\mathrm{BH}}(t)=S_0(1-t/t_{\mathrm{evap}})^{2/3}$, and* $$ t^*=t_{\mathrm{evap}}\bigl(1-2^{-3/2}\bigr)=0.646447\,t_{\mathrm{evap}}. \tag{15.6} $$ *Location of the maximum. With probability at least $1-(S_0/\log\chi+1)V_A\epsilon^{-2}$ (union bound over the $S_0/\log\chi+1$ distinct $(r,b)$ configurations), every $t$ at which $S(R(t))$ attains its maximum over $[0,t_{\mathrm{evap}})$ lies in the entropy window $\{t\colon|S_{\mathrm{H}}(t)-S_{\mathrm{BH}}(t)|\leq\log\chi+2\delta_A\}$, whose half-width in time, for the interpolated curves and Schwarzschild luminosity, is at most* $$ \frac{\log\chi+2\delta_A}{\min_J(\dot S_{\mathrm{H}}-\dot S_{\mathrm{BH}})}=\frac{\log\chi+2\delta_A}{-2\max_J\dot S_{\mathrm{BH}}}\leq\frac{3\,t_{\mathrm{evap}}}{4S_0}\bigl(\log\chi+2\delta_A\bigr), $$ *where $J$ is an interval around $t^*$ on which $\dot S_{\mathrm{H}}-\dot S_{\mathrm{BH}}>0$. The bound is informative only when $\chi\gg(4!)^{|N|}$; see Remark 15.1a.* *Proof.* $r+b=S_0/\log\chi$ is constant, so at $t^*=\inf\{r\geq b\}$ either $r=b=S_0/(2\log\chi)$ or $r=b+1$, giving (15.5) with the half-leg rounding. If $\bar b$ is linear, $\bar b(t)=\bar b(0)(1-t/t_{\mathrm{evap}})$ and $\bar b=\bar b(0)/2$ at $t_{\mathrm{evap}}/2$. For Schwarzschild: $\mathrm{d}M/\mathrm{d}t=-\alpha/M^2$ integrates to $M^3=M_0^3-3\alpha t$; $S_{\mathrm{BH}}\propto M^2$ gives $S_{\mathrm{BH}}(t)=S_0(1-t/t_{\mathrm{evap}})^{2/3}$; setting this to $S_0/2$ gives $(1-t/t_{\mathrm{evap}})^{2/3}=\frac12$, $t/t_{\mathrm{evap}}=1-2^{-3/2}$. The number rate: $\Gamma=L/\bar E$ with $\bar E\propto T_{\mathrm{H}}\propto M^{-1}$ and $L\propto M^{-2}$ gives $\Gamma\propto M^{-1}$, consistent with $-\dot S_{\mathrm{BH}}\propto-M\dot M\propto M^{-1}$. Location of the maximum: on the intersection of the Theorem 15.2 events over all $(r,b)$ configurations (union bound), for every $t\notin W_\chi$ the value $S(R(t))$ lies in $[\min\{S_{\mathrm{H}},S_{\mathrm{BH}}\}-\delta_A,\min\{S_{\mathrm{H}},S_{\mathrm{BH}}\}]$, and at the configuration adjacent to $W_\chi$ ($r=b+1$ or $b=r+1$) this value is $\geq\frac12S_0-\frac12\log\chi-\delta_A$. A point $t_1$ with $|S_{\mathrm{H}}(t_1)-S_{\mathrm{BH}}(t_1)|>\log\chi+2\delta_A$ has $S(R(t_1))\leq\min\{S_{\mathrm{H}},S_{\mathrm{BH}}\}(t_1)<\frac12S_0-\frac12(\log\chi+2\delta_A)=\frac12S_0-\frac12\log\chi-\delta_A$, which is below a value attained elsewhere, so $t_1$ is not a maximiser. The mean value theorem for $f=S_{\mathrm{H}}-S_{\mathrm{BH}}$ (interpolated) on $J$ converts the entropy half-width $\log\chi+2\delta_A$ into a time half-width $(\log\chi+2\delta_A)/\min_J\dot f$. For Schwarzschild $\dot S_{\mathrm{BH}}=-\frac23(S_0/t_{\mathrm{evap}})(1-t/t_{\mathrm{evap}})^{-1/3}$, so $\dot f=-2\dot S_{\mathrm{BH}}\geq\frac43S_0/t_{\mathrm{evap}}$, the minimum of $\dot f$ on $J$ being at its left end. $\square$ **Remark 15.3a.** The statement $t_{\mathrm{Page}}=t_{\mathrm{evap}}/2$ is exact in the entropy variable (15.5), which is the invariant content of Page's theorem (Page, *Phys. Rev. Lett.* **71** (1993) 3743): the crossing is at half the initial entropy. Its translation into a time depends on the luminosity law, and for Schwarzschild in four dimensions the answer is (15.6), not $t_{\mathrm{evap}}/2$. Every other Part of this monograph quoting a Page time uses (15.6) (cross-reference C7). **Remark 15.3b** (Greybody corrections). For a realistic spectrum the entropy radiated per unit of Bekenstein-Hawking entropy lost is $\eta\approx1.5$ for photons and gravitons (Page, *Phys. Rev. D* **13** (1976) 198): in the leg model, each emission removes one black-hole leg and adds $\eta$ radiation legs on average, so $S_{\mathrm{H}}(t)=\eta\bigl(S_0-S_{\mathrm{BH}}(t)\bigr)$ and the crossing $S_{\mathrm{H}}=S_{\mathrm{BH}}$ is at $S_{\mathrm{BH}}(t^*)=\eta S_0/(1+\eta)=0.6\,S_0$ for $\eta=1.5$, hence $t^*/t_{\mathrm{evap}}=1-0.6^{3/2}=0.535$ for Schwarzschild, which is the figure Page quotes (about $0.54$ of the lifetime). This modifies (HG)(i) (the sum $r+b$ is no longer constant), not Theorem 15.1. ### 15.5. The type $\mathrm{II}_1$ refinement Theorem 15.1 computes $S(\mathrm{Rad})$ on a finite-dimensional Hilbert space at fixed $\chi$. The passage to the edge-mode factor of Chapter 14 involves two distinct limits which do not commute in any sense established here: $\chi\to\infty$ at fixed finite boundary $\Gamma$ (the regime of Theorem 15.1, where $\mathcal{A}_\Gamma^{(\chi)}$ stays a finite type $\mathrm{I}$ factor), and $|\Gamma_k|\to\infty$ at fixed $\chi$ (the regime of Corollary 14.5, which yields $\mathcal{R}$). The conjecture below concerns the second limit taken after the first, i.e. the family of finite-$\chi$ Page curves indexed by $\Gamma_k$, and asserts that it is described by a single state on $\mathcal{R}$. **Conjecture 15.4** (Type $\mathrm{II}_1$ Page curve). *Let $\rho_{\mathrm{Rad}}(t)\in L^1(\mathcal{M}_{\partial B},\tau_\infty)_+$, $\tau_\infty(\rho_{\mathrm{Rad}}(t))=1$, be the density of the radiation state relative to $\tau_\infty$, so that the radiation state is $\omega_t=\tau_\infty(\rho_{\mathrm{Rad}}(t)\,\cdot)$. Then, conditional on Conjecture 14.6,* $$ S(\mathrm{Rad}(t))=-\tau_\infty\bigl[\rho_{\mathrm{Rad}}(t)\log\rho_{\mathrm{Rad}}(t)\bigr]+S_{\mathrm{class}}(t), \tag{15.7} $$ *where $S_{\mathrm{class}}(t)$ is the classical area-term contribution $A(t)/4G$ that the CPW trace normalisation assigns to the horizon, and the right side reproduces the two-phase profile of Theorem 15.2 with the $O(\chi^{-1})$ errors removed.* In CPW the trace on $\mathcal{A}_{\mathrm{cr},+}$ is normalised so that the entropy $-\mathrm{Tr}[\rho\log\rho]$ of a semiclassical state equals the generalised entropy $A/4G+S_{\mathrm{bulk}}$ up to a state-independent constant, which is the sense in which the type $\mathrm{II}_1$ trace computes the QES functional without a regulator. If Conjecture 14.6 holds, $\tau_\infty$ on $\mathcal{M}_{\partial B}$ inherits this property through $\mathcal{D}$, and (15.7) is the transport of CPW's result (their §5) along $\mathcal{D}$. What is proved without Conjecture 14.6 is that $-\tau_\infty[\rho\log\rho]$ is a well-defined functional on densities $\rho\in L^1(\mathcal{R},\tau_\infty)_+$ with $\tau_\infty(\rho)=1$ (the Segal entropy; Segal, *J. Math. Mech.* **9** (1960) 623), taking values in $(-\infty,0]$: for $\rho\leq\mathbf{1}$ one has $-\rho\log\rho\geq0$ so the entropy is non-negative, while for $\tau_\infty(\rho)=1$ Jensen gives $\tau_\infty(\rho\log\rho)\geq\tau_\infty(\rho)\log\tau_\infty(\rho)=0$, so the Segal entropy is $\leq0$ with equality exactly when $\rho=\mathbf{1}$, the trace state; a density with $\tau_\infty(\rho)=1$ and $\rho\leq\mathbf{1}$ forces $\rho=\mathbf{1}$. The trace state is the expected infinite-$\chi$ limit of the maximally mixed boundary state. The conjecture is therefore that the finite-$\chi$ Page curve of Theorem 15.2, shifted by $S_{\mathrm{class}}$, converges as the boundary is refined to the Segal entropy of the radiation state in $\mathcal{R}$; the obstruction is the one in Proposition 14.10, the dressing map. ### 15.6. Entanglement wedge reconstruction on the QSN After the Page time the black-hole nodes lie on the radiation side of the minimal cut, and a code subspace placed inside $B(t)$ should be recoverable from the radiation legs. We formulate this with a maximally entangled reference, the figure of merit being entanglement fidelity. **Setting.** In addition to (HG), attach $k$ further pendant *code legs* $X$ of dimension $\chi$ each to nodes of $B(t)$, and let $X_{\mathrm{ref}}$ be a reference system of dimension $\chi^k$ maximally entangled with $X$ through the state $\Phi_{XX_{\mathrm{ref}}}$. Contracting $X$ into the network defines the encoding channel $\mathcal{E}\colon\mathcal{B}(\mathcal{H}_X)\to\mathcal{B}(\mathcal{H}_{R})$, $\mathcal{E}(\sigma)=\mathrm{Tr}_{\mathcal{B}}\bigl[\Psi(\sigma)\bigr]$ where $\Psi(\sigma)$ is the RTN state with $\sigma$ on $X$ and $\mathrm{Tr}_{\mathcal{B}}$ traces the black-hole legs. Let $\pi_X:=\mathbf{1}/\chi^k$ be the maximally mixed code state and $\mathcal{R}_\pi$ the Petz map of $\mathcal{E}$ with respect to $\pi_X$, $$ \mathcal{R}_\pi(Y):=\pi_X^{1/2}\mathcal{E}^\dagger\bigl(\mathcal{E}(\pi_X)^{-1/2}Y\mathcal{E}(\pi_X)^{-1/2}\bigr)\pi_X^{1/2}. $$ The entanglement fidelity of a channel $\mathcal{N}$ on $X$ is $F_e(\mathcal{N}):=\langle\Phi|(\mathcal{N}\otimes\mathrm{id})(\Phi)|\Phi\rangle$. **Hypothesis (HG$'$).** *(HG)(ii) holds on the graph with the code legs adjoined, for the three boundary regions $R$, $R\cup X_{\mathrm{ref}}$, and $X_{\mathrm{ref}}$ (the last with $X_{\mathrm{ref}}$ as $k$ pendant legs attached to the code nodes), with minimal cuts of cardinalities $\min(r,b+k)$, $\min(r+k,b)$, and $k$ respectively, each unique with unit gap.* The encoding $V\colon\mathcal{H}_X\to\mathcal{H}_R\otimes\mathcal{H}_{\mathcal{B}}$ obtained by contracting the network is linear but, for Haar node tensors, not proportional to an isometry: $V^\dagger V\neq c\mathbf{1}$ almost surely. Consequently the map $\sigma\mapsto\mathrm{Tr}_{\mathcal{B}}[V\sigma V^\dagger]$ is completely positive but not trace preserving, and the normalised state depends non-linearly on $\sigma$. The proposition below is therefore stated as an existence statement for a recovery channel, with the global state $\rho_{RX_{\mathrm{ref}}\mathcal{B}}:=(V\otimes\mathbf{1})\Phi(V\otimes\mathbf{1})^\dagger/\mathrm{Tr}[\cdot]$ as the object; the explicit Petz form is deferred to Remark 15.5b under an additional hypothesis. **Proposition 15.5** $\boxed{\text{Conditional (on the hypotheses of Thm 15.2; HG}')}$ (Island reconstruction, existence). *Let $t>t_{\mathrm{Page}}$ with $r(t)\geq b(t)+k+1$. Then for every $\epsilon\in[2^{|N|-1}\chi^{-1},\frac14]$, with probability at least $1-3V_A\epsilon^{-2}$ over the node tensors, there is a channel $\mathcal{R}\colon\mathcal{B}(\mathcal{H}_R)\to\mathcal{B}(\mathcal{H}_X)$ with* $$ \langle\Phi|(\mathcal{R}\otimes\mathrm{id}_{X_{\mathrm{ref}}})(\rho_{RX_{\mathrm{ref}}})|\Phi\rangle\ \geq\ e^{-\delta_A}\ \geq\ 1-\delta_A, $$ *and consequently $\|(\mathcal{R}\otimes\mathrm{id})(\rho_{RX_{\mathrm{ref}}})-\Phi\|_1\leq2\sqrt{\delta_A}$.* *Proof.* *Step 1 (decoupling).* The global state on $R\cup\mathcal{B}\cup X_{\mathrm{ref}}$ is pure. Apply Theorem 15.1(A) three times, with (HG$'$) supplying (MC) in each case, on the intersection of the three events, of probability $\geq1-3V_A\epsilon^{-2}$: $S(X_{\mathrm{ref}})\in[k\log\chi-\delta_A,k\log\chi]$ (cut $k$); $S(R)\in[(b+k)\log\chi-\delta_A,(b+k)\log\chi]$ (cut $b+k$, since $r\geq b+k+1$); $S(R\cup X_{\mathrm{ref}})\in[b\log\chi-\delta_A,b\log\chi]$ (cut $b$, since $r+k>b$). By purity $S(X_{\mathrm{ref}}\mathcal{B})=S(R)$ and $S(\mathcal{B})=S(R\cup X_{\mathrm{ref}})$. Let $\rho:=\rho_{X_{\mathrm{ref}}\mathcal{B}}$ and $\pi:=\mathbf{1}/\chi^k$ on $X_{\mathrm{ref}}$; $\rho_{X_{\mathrm{ref}}}$ is not assumed equal to $\pi$. Expanding in entropies, $$ D(\rho\,\|\,\pi\otimes\rho_{\mathcal{B}})=k\log\chi+S(\mathcal{B})-S(X_{\mathrm{ref}}\mathcal{B})=\mathbb{I}(X_{\mathrm{ref}}:\mathcal{B})+D(\rho_{X_{\mathrm{ref}}}\|\pi)\leq k\log\chi+b\log\chi-\bigl((b+k)\log\chi-\delta_A\bigr)=\delta_A, $$ where only the bounds on $S(\mathcal{B})$ and $S(X_{\mathrm{ref}}\mathcal{B})$ enter; the bound on $S(X_{\mathrm{ref}})$ shows separately that $D(\rho_{X_{\mathrm{ref}}}\|\pi)\leq\delta_A$, so that the deviation of the code marginal from maximal mixedness is itself small. *Step 2 (decoupling to recovery).* Relative entropy dominates the fidelity: $D(\rho\|\sigma)\geq-2\log F(\rho,\sigma)$ with $F(\rho,\sigma)=\|\sqrt\rho\sqrt\sigma\|_1$ (monotonicity in $\alpha$ of the sandwiched Rényi divergence, $D_{1/2}=-2\log F\leq D_1=D$; Müller-Lennert, Dupuis, Szehr, Fehr, Tomamichel, *J. Math. Phys.* **54** (2013) 122203, Thm 7). Hence $F(\rho,\pi\otimes\rho_{\mathcal{B}})\geq e^{-\delta_A/2}$. The state $\pi\otimes\rho_{\mathcal{B}}$ is purified by $\Phi_{X_{\mathrm{ref}}X'}\otimes|\phi_{\mathcal{B}}\rangle_{\mathcal{B}R'}$ with $X'\cong X$ and $R'$ a purifying system for $\rho_{\mathcal{B}}$, and $\rho$ is purified by the global state on $X_{\mathrm{ref}}\mathcal{B}R$. By Uhlmann's theorem, after enlarging $R'$ so that $\dim\mathcal{H}_{X'}\otimes\mathcal{H}_{R'}\geq\dim\mathcal{H}_R$, there is an isometry $V_U\colon\mathcal{H}_R\to\mathcal{H}_{X'}\otimes\mathcal{H}_{R'}$ with $|\langle\Psi_{\mathrm{global}}|(\mathbf{1}\otimes V_U)^\dagger(\Phi\otimes\phi_{\mathcal{B}})\rangle|=F(\rho,\pi\otimes\rho_{\mathcal{B}})$ (Uhlmann's theorem gives a partial isometry between the purifying spaces; enlarging one of them makes it an isometry). Define $\mathcal{R}:=\mathrm{Tr}_{R'}\circ\mathrm{Ad}(V_U)$, a channel from $R$ to $X'\cong X$. The overlap $\langle\Phi|(\mathcal{R}\otimes\mathrm{id})(\rho_{RX_{\mathrm{ref}}})|\Phi\rangle$ is the fidelity between a reduced state and the pure state $\Phi$, which is at least the squared overlap of the purifications, $F(\rho,\pi\otimes\rho_{\mathcal{B}})^2\geq e^{-\delta_A}\geq1-\delta_A$. Fuchs-van de Graaf converts: $\|(\mathcal{R}\otimes\mathrm{id})(\rho_{RX_{\mathrm{ref}}})-\Phi\|_1\leq2\sqrt{1-e^{-\delta_A}}\leq2\sqrt{\delta_A}$. $\square$ **Remark 15.5b** (Petz form). *Under the additional hypothesis (ISO): $V^\dagger V=c\,\mathbf{1}_X$ for some $c>0$, the map $\mathcal{E}(\sigma):=c^{-1}\mathrm{Tr}_{\mathcal{B}}[V\sigma V^\dagger]$ is a channel, $\rho_{X_{\mathrm{ref}}}=\pi$, and the Petz map $\mathcal{R}_\pi$ of $\mathcal{E}$ with respect to $\pi$ satisfies $F_e(\mathcal{R}_\pi\circ\mathcal{E})\geq F_e(\mathcal{R}\circ\mathcal{E})^2\geq e^{-2\delta_A}$, where $F_e$ is the entanglement fidelity with respect to $\Phi$ and $\mathcal{R}$ is the channel of Proposition 15.5, by the entanglement-fidelity form of the Barnum-Knill bound (Barnum, Knill, *J. Math. Phys.* **43** (2002) 2097; their main theorem is stated for ensemble fidelity, and the entanglement-fidelity version follows by taking the ensemble to be the uniform ensemble of a basis of $\mathcal{H}_X$ maximally entangled with the reference, §IV).* (ISO) is a hypothesis on the node tensors at the code nodes; it is not implied by H9(b) and is not assumed elsewhere in this chapter. **Remark 15.5a.** The hypothesis $r\geq b+k+1$ is the full-functional gap in the leg model: it requires the radiation to exceed the black hole by more than the code size, and it fails as $t\downarrow t_{\mathrm{Page}}$, consistent with Theorem 15.2. The figure of merit is the overlap with the maximally entangled code state; a worst-case bound over all inputs would require an additional factor of $\chi^k$ and is not claimed. The Fawzi-Renner inequality does not apply here because the conditioning system is trivial and its recovery map would act on $\mathcal{B}$, not on $R$. **Remark 15.6** (Scrambling time and the MSS bound). Under H9(c) the node tensors satisfy the relative-error design condition after $t_{\mathrm{scr}}\leq\beta\log S_{\mathrm{BH}}$. This is the fast-scrambling time of Sekino and Susskind (*JHEP* **10** (2008) 065) and the saturation of the Maldacena-Shenker-Stanford chaos bound (*JHEP* **08** (2016) 106) on the Lyapunov exponent, $\lambda_L\leq 2\pi/\beta$, with $t_{\mathrm{scr}}=\lambda_L^{-1}\log S_{\mathrm{BH}}$. Whether $H_{\mathsf{Q}}$ saturates this bound is open; what the QSN structure gives is that the bound cannot be exceeded, because the scrambling time of a bounded-degree graph is at least the graph diameter of the region in units of the edge proper time (Lieb-Robinson, applied to $H_{\mathsf{Q}}$ which is a sum of edge- and node-local terms with bounded norms under L6 and H5), and for a region satisfying H4' with $|B|\sim S_{\mathrm{BH}}^{(d-1)/(d-2)}$ the diameter is at least $c\log S_{\mathrm{BH}}$ only for expander-like regions; for regions of polynomial growth the Lieb-Robinson bound gives a diameter of order $S_{\mathrm{BH}}^{1/(d-2)}$, which is much larger. Thus H9(c) with the logarithmic scrambling time requires the near-horizon region of the QSN to have expander-like connectivity, a constraint on admissible black-hole regions that is recorded as part of H9 and is consistent with the dimensional-reduction picture of §7.3. **Remark 15.7** (Dimensions). With units restored, $S_{\mathrm{BH}}=k_Bc^3A/(4G\hbar)$, $A=|\partial B|\kappa_{2}\ell_P^{2}$ in $d=4$, and $t_{\mathrm{evap}}=5120\pi G^2M_0^3/(\hbar c^4)$ for Schwarzschild (Page 1976); the entropy variable in (15.5) is $k_B$ times a pure number. The Hawking temperature is $T_{\mathrm{H}}=\hbar c^3/(8\pi GMk_B)$ and $\beta:=1/(k_BT_{\mathrm{H}})=8\pi GM/(\hbar c^3)$ has dimension of inverse energy, so $t_{\mathrm{scr}}=\beta\hbar\log S_{\mathrm{BH}}$ has dimension of time; $\log\chi$ is dimensionless. ### 15.7. Hypothesis discharge ledger, Chapter 15 | Result | Classical inputs | Hypotheses used | Step where consumed | |---|---|---|---| | Thm 15.1T | HNQTWY (section anchors to be verified) | | statement | | Lem 15.1D | none | H9(c) | ratio bound | | Lem 15.1L | Menger 1927 | (MC) | lower bound, uniqueness | | Thm 15.1(A) | Thm 15.1T(b); Harrow Prop 6; Lem 15.1L; Lem 15.1V; Chebyshev | L1, H1, H3; H9(a),(b) or (c); (MC) | Steps 1 to 4 | | Thm 15.1(B) | Thm 15.1T(a),(c),(e) | as (A) with (GAP$_n$) in place of (MC) | Step 3 | | Thm 15.1(C) | Thm 15.1(B) | as (B) plus H9(d) | Step 5 | | Thm 15.2 | Thm 15.1(A) | L1, H1, H3; H9(a),(b) or (c); (HG) | (MC) from (HG)(ii) | | Prop 15.3 | Page 1976; mean value theorem | Thm 15.2 hypotheses; rate law | entropy balance; window | | Conj 15.4 | CPW, CLPW; Segal 1960 | Conj 14.6 | transport along $\mathcal{D}$ | | Prop 15.5 | MDSFT 2013 Thm 7; Uhlmann; Fuchs-van de Graaf | Thm 15.2 hypotheses; (HG$'$); $r\geq b+k+1$ | Step 1 (three applications of (A)); Step 2 | | Rem 15.5b | Barnum-Knill 2002 | (ISO) | Petz form | | Lem 15.1V | Lem 15.1L argument at $n=4$ | H9(a),(b) or (c); (MC) | variance constants | | Rem 15.6 | Lieb-Robinson; Sekino-Susskind; MSS | L6, H5 | lower bound on $t_{\mathrm{scr}}$ | ### 15.8. Summary of Chapter 15 | Result | Status | |---|---| | Thm 15.1(A) QES, no bulk, in probability | Conditional (H9(a),(b) or (c); MC) | | Thm 15.1(B) QES, integer Rényi with bulk, annealed | Conditional (H9(a),(b) or (c); GAP$_n$) | | Thm 15.1(C) QES, von Neumann mean with bulk | Conditional (as (B) plus H9(d)) | | Thm 15.2 two-phase Page mechanism, leg model | Conditional (H9(a),(b) or (c); HG) | | Prop 15.3 Page time; $t^*=(1-2^{-3/2})t_{\mathrm{evap}}$ for Schwarzschild | Conditional (as 15.2 plus rate law) | | Conj 15.4 $\mathrm{II}_1$ Page curve | Conjecture | | Prop 15.5 reconstruction, existence of a recovery channel | Conditional (as 15.2 plus HG$'$, gap) | | Rem 15.5b Petz form | Conditional (as 15.5 plus ISO) | The chain of dependence for Part V is: L1, L6, H1 and connectedness give Chapter 14 unconditionally up to the Murray-von Neumann classification; H9(a) with H9(b) or (c), and the cut hypotheses (MC) or (HG), are the additional hypotheses for the Page mechanism, which uses only the no-bulk part of Theorem 15.1; the bulk-state parts (B),(C) rest further on (GAP$_n$) and H9(d) and are not consumed by Theorems 15.2, 15.3, 15.5; Conjecture 14.6 is the single open problem whose resolution would make the $\mathrm{II}_1$ refinement a theorem. All error bounds of this chapter are informative only in the large-$\chi$ RTN regime $\log\chi\gg|N|$ (Remark 15.1a). ### Referee response (review5.md) 1. Citation corrected to Umegaki 1954 and Takesaki II Thm IX.4.2. 2. Theorem 14.4U moved above Lemma 14.4a; the lemma now cites it. 3. Proposition 14.4 (Conditional on H8, general AF systems) split from Corollary 14.4c (Theorem, $\mathcal{M}_{\partial B}$); Cor 14.5, ledger and summary updated to depend on 14.4c. 4. Kadison-Ringrose citation replaced by the one-line proof of strong convergence of increasing projections. 5. Connectedness added to Definition 14.0b with the derivation of countability of $N$; remark that only the Murray-von Neumann part of Classical T is used in Cor 14.5. 6. Theorem number dropped; the in-text argument is the proof. 7. Converse of Prop 14.10 now states separability of $\mathcal{H}_{\mathrm{III}}$ and the split property as added hypotheses; Connes cited by section, Takesaki II X.2.3 for crossed-product stability; BDF stated as giving the injective $\mathrm{III}_1$ factor. 8. $K_0$ corrected to $\mathbb{Z}[1/\mathfrak{n}]=\bigcup_k\frac1{m_k}\mathbb{Z}$ with $\mathfrak{n}$ the supernatural number; referee's $3\cdot2^\infty$ counterexample included; confused parenthetical removed. 9. Remark 15.1a corrected to $\log\chi=\kappa_{d-2}/4$; Remark 14.12 restricted to $d=4$ with the $G_d$ generalisation noted. 10. Order of the two limits ($\chi\to\infty$ at fixed $\Gamma$; $|\Gamma_k|\to\infty$ at fixed $\chi$) stated in §15.5 as non-commuting, with the order used by the conjecture named. 11. Theorem 15.1T restated with HNQTWY section references, second-moment concentration, and $n$-dependent constants $C_n(G)\leq|V|(n!)^{|V|}$ exhibited. 12. (FATAL) Theorem 15.1 split into (A) no-bulk von Neumann (Rényi sandwich, Chebyshev concentration, no continuation), (B) integer Rényi with bulk, (C) von Neumann with bulk, Conditional on reformulated H9(d) (analytic continuation of the annealed ratio with growth condition, uniform bound, and annealed-equals-quenched). The $e^{-c'\chi}$ probability claim and the Lévy-lemma step are removed; concentration is the Chebyshev bound from 15.1T(d). Thm 15.2, Prop 15.3, Prop 15.5 inherit H9(d). 13. Menger's theorem named for the edge-disjoint-path lower bound; the "every $\mathrm{Rad}$ node carries a leg" assumption made explicit as the first clause of (MC). 14. Theorem 15.2 tag now lists H9(a),(b),(d), (MC), (HM), (HZ), unitarity; Prop 15.3 and Prop 15.5 likewise. 15. (MAJOR) Intermediate-island argument replaced by the global bound $F(I)\geq|\partial I|\log\chi\geq|\partial B|\log\chi$ under (HZ); the invalid per-edge comparison is recorded in Remark 15.2a as the reason for (HZ). 16. Crossing set stated as a closed interval with $t_{\mathrm{Page}}$ its infimum; discrete-jump case handled; error term and in-probability qualifier restored in (15.4). 17. No change (verified). 18. $L\propto M^{-2}$, hence $\Gamma\propto M^{-1}$; corrected in statement and proof. 19. Greybody crossing corrected to $\eta S_0/(1+\eta)=0.6S_0$, $t_{\mathrm{Page}}=0.535\,t_{\mathrm{evap}}$. 20. $\beta=8\pi GM/(\hbar c^3)$, $t_{\mathrm{scr}}=\beta\hbar\log S_{\mathrm{BH}}$; Remark 15.7 rewritten. 21. (FATAL) Proposition 15.5 rewritten: control parameter is the full-functional gap $\Delta_{\mathrm{tot}}$; decoupling derived from Theorem 15.1(C) applied to $\mathrm{Rad}$ and $\mathrm{Rad}\cup X_{\mathrm{ref}}$ with the mutual information bounded by $2\delta_\chi$; recovery via Pinsker, Uhlmann/Schumacher-Westmoreland, Barnum-Knill (giving $O(\delta_\chi^{1/4})$ for the Petz map) and Fawzi-Renner (giving the stated $O(\delta_\chi^{1/2})$ for the rotated Petz map); norm defined as the trace norm on states; constant's dependency list given. Remark 15.5a records why the uncontracted mutual information is the wrong control quantity. 22. Sign corrected: Segal entropy of a normalised density is $\leq0$ with equality only at $\rho=\mathbf{1}$; the $\rho\leq\mathbf{1}$ case noted to force $\rho=\mathbf{1}$. 23. Renumbered to SPEC anchors: Thm 15.1, Thm 15.2, Prop 15.3, Conj 15.4 ($\mathrm{II}_1$ Page), with Petz as Prop 15.5 and MSS as Remark 15.6; definitions 14.0a,b,c; "is immediate" and "standard supernatural-number refinement" replaced by the written steps. 24. H9(c) restated as an $\epsilon$-approximate $n$-design with $\epsilon\leq\chi^{-1}$, labelled a modelling hypothesis, with BHH 2016 and NVH 2018 cited for random circuits only. ERRATA_CROSS C7: Proposition 15.3 and Remark 15.3a carry the corrected value $t_{\mathrm{Page}}=(1-2^{-3/2})t_{\mathrm{evap}}$; no $t_{\mathrm{evap}}/2$ claim remains except as the linear-rate special case. ### Referee response, pass 2 (review5b.md) FATAL 1. Theorem 15.1 restated on the augmented graph $G^{\mathrm{aug}}$ with pendant edges for every physical leg; cuts are $A$-cuts in $G^{\mathrm{aug}}$ and $|\partial^{\mathrm{aug}}W|=|\partial I|+\ell_{\bar A}(I)$ is written out. The "islands $I\subseteq B$" reduction and the first clause of (MC) are removed. The Page mechanism is reformulated as the PSSY leg model under an explicit horizon-geometry hypothesis (HG), in which $F(\gamma_R)=S_{\mathrm{H}}$ and $F(\gamma_{\mathcal{B}})=S_{\mathrm{BH}}$ by construction and one black-hole leg per horizon bond fixes $b=|\partial B|$. The rank bound in Step 4 now counts edges of $G^{\mathrm{aug}}$ in the cut, including pendant edges. FATAL 2. The bulk-state error bookkeeping is withdrawn. Parts (B) and (C) are now Conditional on a named weighted-model hypothesis (GAP$_n$) that assigns the error $\epsilon_n+\epsilon'_n$; the text states why (MC) does not imply it and why the Bell-pair bound is vacuous. Theorem 15.2, Proposition 15.3 and Proposition 15.5 no longer consume (B) or (C): the Hawking quanta are physical legs and only Part (A) is used. FATAL 3. Proposition 15.5 restated for the Petz map with respect to the maximally mixed code state, with entanglement fidelity as figure of merit; the $\sigma=\sigma'$ triviality is gone. Chain: decoupling from three applications of Theorem 15.1(A) (mutual information $\leq\delta_A$), $D\geq-2\log F$ via sandwiched Rényi monotonicity, Uhlmann, Schumacher-Westmoreland, Barnum-Knill Thm 2 (squared fidelity), Fuchs-van de Graaf. The Fawzi-Renner paragraph is deleted and Remark 15.5a states why it does not apply. Worst-case input bound explicitly not claimed. MAJOR 4. $\chi$ is large throughout; horizon and radiation legs have dimension $\chi$; the error is $\log\chi+\delta_A$ with $\delta_A=6\epsilon+2^{|N|+1}\chi^{-1}+\eta_2$ and all constants displayed; the $10^{-77}$ claim is deleted and Remark 15.1a states that the bounds are informative only for $\log\chi\gg|N|$. MAJOR 5. The in-probability clause is removed from Theorem 15.1(C) and from its consumers; (C) is a statement about $\mathbb{E}S$ only, and says so. MAJOR 6. Theorem 15.2 is stated outside the degeneracy window $W_\chi=\{r=b\}$ and makes no claim inside it; Proposition 15.3 locates the entropy maximum within an explicit time window whose half-width is computed from $\log\chi+\delta_A$ and the slope. MAJOR 7. The model is on $[0,t_{\mathrm{evap}})$ with $b(t)\to0$; (HM) is replaced by (HG), in which the purity contradiction at $t_{\mathrm{evap}}$ does not arise. MAJOR 8. H9(c) restated as a relative-error joint moment condition on the two specific polynomials used, with Brandão-Harrow-Horodecki cited as giving relative-error designs; Lemma 15.1D proves the resulting additive error $\eta_n$; Theorem 15.1 and all consumers are stated under "H9(b) or H9(c)" with $\eta_n$ carried; the ledger is aligned. MAJOR 9. In the new Proposition 15.5 the code has $k$ legs of dimension $\chi$, $\log\dim\mathcal{H}_X=k\log\chi$ is explicit, and the gap hypothesis is $r\geq b+k+1$; the three regions are treated by separate applications of Theorem 15.1(A) under (HG$'$). MAJOR 10. §14.7 corrected: Witten 2022 and CPW *JHEP* **04** (2023) 009 give a $\mathrm{II}_\infty$ crossed product; the de Sitter $\mathrm{II}_1$ case is CLPW *JHEP* **02** (2023) 082; $\mathcal{A}_{\mathrm{cr},+}$ is now defined as the corner $p\mathcal{A}_{\mathrm{cr}}p$ by the finite-trace spectral projection of the observer energy onto $[0,E_0]$, with the one-line proof that the corner is a $\mathrm{II}_1$ factor; Conjecture 14.6 and Proposition 14.10 refer to this corner and its normalised trace $\tau_{\mathrm{cr}}$. MINOR 11. Theorem 15.1T(b) split into the annealed inequality and the realisation-wise sandwich. MINOR 12. Constants: $2^{|N|}$, $(n!)^{|N|}$, $(4!)^{|N|}$ carried explicitly; $\epsilon\leq\frac14$ with $\log\frac{1+\epsilon}{(1-\epsilon)^2}\leq4\epsilon$; the condition $\chi\geq2(n!)^{|V|}$ removed by using $\log(1+u)\leq u$. MINOR 13. The deterministic display is replaced by the in-probability two-sided bound (15.2A). MINOR 14. H9(d) is inside the hypothesis block. MINOR 15. Separable predual and split property are in the statement of the converse of Proposition 14.10; the unverified Takesaki theorem number is removed. MINOR 16. Exponent corrected to $S_{\mathrm{BH}}^{(d-1)/(d-2)}$. MINOR 17. Remark 15.1a states the constraint $\kappa_{d-2}/4\in\log\mathbb{N}$ and that it is not verified against Lemma 13.1 here. MINOR 18. The sentence distinguishing the Chapter 14 sequence from the Chapter 15 fixed finite graph is added after H9. MINOR 19. Theorem 15.1T now says the section anchors follow the arXiv version and are to be verified before print. MINOR 20. Theorem 15.2 states the crossing set as an interval with infimum $t_{\mathrm{Page}}$ and $W_\chi$ possibly empty. MINOR 21. Ledger rebuilt to match the stated hypotheses of each result. ### Referee response, pass 3 (review5c.md) FATAL 1. Proposition 15.5 demoted to an existence statement (Uhlmann only) for a recovery channel, with the overlap with the maximally entangled code state as figure of merit; the false claim $\rho_{X_{\mathrm{ref}}}=\pi$ is removed and the third application of Theorem 15.1(A) is carried, with $D(\rho\|\pi\otimes\rho_{\mathcal{B}})\leq\delta_A$ obtained from the entropy expansion; the non-trace-preserving nature of the contraction map is stated. The Petz form is moved to Remark 15.5b, Conditional on the new hypothesis (ISO) $V^\dagger V=c\mathbf{1}$, with Barnum-Knill cited for its entanglement-fidelity form. MAJOR 2. $(\mathrm{Tr}\rho_A^2)^2$ and $\mathrm{Tr}\rho_A$ added to $\mathcal{F}_A$ in H9(c); Lemma 15.1V derives the design-case variance ratios $v_\mu,v'_\mu$ and they enter the probability clause as $V_A$, separate from $\eta_2$. MAJOR 3. H9(c) restated as a moment condition on the evolved vector $U(t)\Psi(0)$ relative to a Haar RTN on $G^{\mathrm{aug}}(t)$, expectation over the $t=0$ tensors; (HG)(iii) distinguishes the two cases and states that H9(b) at $t