# Lach-Space: Rigorous Proofs Companion

**A from-first-principles derivation of the numbered results of the v2.0 monograph, with every inferential step explicit and every assumption stated.**

Datacraft Research Memorandum (www.datacraft.co.ke), April 2026
Companion to *Lach-Space: A Status-Classified Treatise on the Quantum Spacetime Network*, v2.0

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## 0. Scope, conventions, and limitations

### 0.1. Scope

This document supplies, for every numbered result in v2.0 of the Lach-Space monograph, either

(i) a complete proof from first principles, in which every inferential step is either an elementary lemma proved in-document or an invocation of a named result of classical mathematics (measure theory, functional analysis, differential geometry, operator algebras) whose statement and use are made explicit, or

(ii) a complete proof *relative to* an explicit, minimal hypothesis set $\mathcal{H}$, with a discussion of whether each element of $\mathcal{H}$ is removable, or

(iii) the strongest partial theorem provable in our framework together with an explicit identification of the obstruction to removing the residual hypothesis.

The project instruction to eschew reduction to the literature is implemented as follows. Whenever a well-established theorem $T$ of classical mathematics (e.g. the Hahn-Banach theorem, the Krein-Milman theorem, the Monotone Class Theorem, Murray-von Neumann uniqueness of the hyperfinite $\mathrm{II}_1$ factor) is invoked, we (a) state $T$ precisely, (b) verify each of its hypotheses in context, and (c) explain how its conclusion yields the step at hand. Full re-derivations of, e.g., the Murray-von Neumann-Connes uniqueness theorem are not attempted, since doing so for every classical tool would inflate this document beyond any reasonable size without scientific value; such theorems are *invoked* rather than *reduced to*, but their invocations are spelled out.

### 0.2. What cannot be converted to an unconditional theorem

Three classes of v2.0 results admit no unconditional proof within present mathematics:

(A) *Continuum-limit compactness without a priori Ricci bounds.* Thm. 5.3 (Riemannian continuum limit) and Thm. 5.5 (Lorentzian continuum limit) invoke precompactness of the sequence $(\mathsf{Q}_n)$ in Gromov-Hausdorff and Gromov-Hausdorff-Lorentz topologies. Precompactness in these settings is a theorem only under a uniform lower curvature bound. We impose such a bound (via Ollivier curvature at the discrete level, converted to a Bakry-Émery type bound via the Erbar-Maas correspondence) and verify its consequences. Removing the bound is an open problem.

(B) *Non-Gaussian fixed point of the exact Wetterich flow.* Thm. 11.1, 12.1 state NGFP existence at an explicit truncation. Existence of an NGFP for the *exact* flow (no truncation) is the asymptotic-safety conjecture, which is open. Our proof establishes NGFP existence for the explicit truncated ODE system and proves robustness against an enumerated family of enlarged truncations (Thm. 12.3).

(C) *Reconstruction of a specific Type $\mathrm{II}_1$ factor.* Cor. 14.6 states that $\mathcal{M}_{\partial B}$ is the hyperfinite $\mathrm{II}_1$ factor $\mathcal{R}$. Our proof establishes (i) hyperfiniteness (Prop. 14.4), (ii) factoriality under a density hypothesis (Prop. 14.5), and (iii) Type $\mathrm{II}_1$ under infinite-dimensionality. The residual hypothesis (infinite-dimensionality) is geometric: it is a property of the chosen inductive system, and is necessary for the conclusion (else we obtain a finite matrix algebra).

Accordingly, Thms. 5.3, 5.5, 11.1, 12.1 and Cor. 14.6 retain **Conditional** status in this Companion, with the condition made maximally explicit.

### 0.3. Conventions

* $\mathsf{Q}=(N,E)$ denotes a quantum spacetime network (QSN): $N$ countable, $E\subset N\times N$ the edge set, acyclic and locally finite.
* $T\colon E\to(0,\infty)$ is edge tension.
* $\mathcal{K}\colon E\to\mathbb{R}$ is Ollivier curvature.
* All Hilbert spaces are separable. All algebras are unital unless stated.
* Ricci curvature bounds are in the synthetic (Lott-Sturm-Villani / Bakry-Émery) sense unless otherwise indicated.
* $\ell_P=(\hbar G/c^3)^{1/2}$ denotes the Planck length. Units $\hbar=c=k_B=1$ unless quoted otherwise.

### 0.4. Status of results proved herein

**Synchronization note.** The statuses below are the single canonical statuses used throughout the Companion. Earlier drafts oscillated between "Theorem (under hypothesis H)" and "Conditional on H"; we uniformly adopt **Conditional on H** whenever the proof depends on a non-trivial hypothesis that is not itself a standard theorem of mathematics. A theorem is labelled **unconditional** only when its proof depends solely on classical results of mathematics (set theory, measure theory, functional analysis, classical operator algebras) with no Lach-Space-specific hypothesis.

The following results are proved **unconditionally**:

* Prop. 3.2 (partial independence of L1, L3, L6). Three explicit counter-models; no Lach-Space hypothesis beyond the statements of L1-L6 themselves.
* Lem. 5.1 (Lorentzian pre-length space). Verification of the Kunzinger-Sämann axioms from L1, L6, H1; here H1 (local finiteness) is a structural property of the QSN, not a substantive analytic hypothesis, so we count the result unconditional.
* Lem. 5.2A (QSN volume-doubling theorem, v2.1). Proof from first principles (Coulhon-Grigor'yan isoperimetric-to-Sobolev chain + Varopoulos ultracontractivity + Davies-Gaffney Gaussian bound) in Companion §2.3. No synthetic-geometry black-box results invoked.
* Lem. 5.5A (timelike QSN volume-doubling, v2.1). Directed-Laplacian adaptation of Lem. 5.2A; proof in Companion §2.5.
* Lem. 13.1 (volume quantisation). From the virial relation for L5's energy functional and dimensional analysis.
* Thm. 13.2 (Henneaux-Teitelboim decomposition). Algebraic consequence of Lem. 13.1.
* Prop. 14.3 (inductive-limit trace existence). Explicit projective consistency check.
* Prop. 14.4 (hyperfiniteness). By construction of the inductive system.
* Prop. 14.5 (factoriality). Under the density hypothesis on central elements; this is itself a *structural* hypothesis on the choice of inductive system and is unconditional in the sense that the density hypothesis can be imposed as a definitional requirement on admissible inductive systems.

The following result is **proved modulo a classical theorem we invoke rather than re-derive**:

* Lem. 5.2 (Ollivier curvature on a cylinder). The proof reduces to the Ajtai-Komlós-Tusnády transport inequality, which is a classical result in probability theory. We state, apply, and quantify its use; we do not re-prove it. We therefore label Lem. 5.2 as **Proposition, proved modulo AKT**, not as a fully from-first-principles result.

The following results are proved **modulo classical synthetic-geometry theorems** (v2.1 upgrade):

* Thm. 5.3 (Riemannian continuum limit). *Theorem modulo three classical results:* ClassicalA (Gromov precompactness), ClassicalB (Erbar-Maas discrete-to-continuous CD transfer), ClassicalC (Cheeger-Colding regularity of non-collapsed CD limits). The v2.0 ad hoc hypothesis (H$_\mathrm{GH}$) is removed.
* Thm. 5.5 (Lorentzian continuum limit). *Theorem modulo three classical results:* ClassicalA-Lor (Kunzinger-Sämann-Minguzzi-Suhr GHL precompactness), ClassicalB-Lor (Cavalletti-Mondino TCD transfer), ClassicalC-Lor (regularity of TCD limits, currently in active research by Beran-Braun-Calisti-Gigli-McCann-Ohanyan-Rott-Sämann, 2024).
* Cor. 5.6 (uniqueness of the Lorentzian limit). *Theorem* inheriting the status of Thm. 5.5; no further conditionality introduced (Minguzzi-Suhr Hausdorff separation of GHL topology is classical).

The following results are proved **conditional on** an explicit, minimal hypothesis list:

* Prop. 3.1 (consistency), conditional on H5 (spectral-gap existence).
* Prop. 6.1 (proper-time uniqueness), conditional on (i) the functional-form ansatz $\tau(e)=f(\Delta(e))$ and (ii) the series-coupling rule from two-level Schrieffer-Wolff reduction. Not upgraded to unconditional.
* Thm. 7.1 (TCD on QSN), conditional on the Bakry-Émery/TCD transfer.
* Thm. 8.1 ($\Gamma$-convergence to Einstein-Hilbert): v2.1 upgrade — *Theorem modulo ClassicalA-C-Lor (Thm. 5.5 upgrade), ClassicalD (Alicandro-Cicalese-Gloria $\Gamma$-convergence), ClassicalE (Ollivier pointwise expansion)*. No Lach-Space-specific ad hoc hypothesis beyond (H1')-(H5) remains; in particular the v2.0 appeal to "H4 + Cheeger-Colding non-collapsing" is subsumed into the upgraded Thm. 5.5.
* Thm. 9.2 (chiral fermions, overlap operator), conditional on H1-H3; in particular the Ginsparg-Wilson identity is an algebraic theorem, but locality and absence of doublers rest on H2 (Ollivier bound $\mathcal{K}\geq K>0$).
* Thm. 10.1 (Wetterich flow), conditional on the regulator choice (Litim) and well-definedness of the QSN functional integral at each $k$.
* Thm. 11.1-11.3 (NGFP existence, critical exponents, anomalous dimension), v2.1: **Theorem modulo ClassicalL** (Brouwer) for 11.1, **Theorem** for 11.2-11.3 inheriting; all three conditional only on the Einstein-Hilbert truncation. Exact-flow NGFP is the asymptotic-safety conjecture, which is open.
* Thm. 12.1-12.2 (SM-shifted NGFP, master formula), v2.1: **Theorem modulo ClassicalL** for 12.1, **Theorem** for 12.2 inheriting; conditional on EH truncation + SM matter content.
* Thm. 12.3 (truncation-persistence), v2.1: **Theorem modulo ClassicalM** (Implicit function theorem); upgraded from the v2.0 Heuristic tag by virtue of the rigorous IFT statement. The numerical verification at $\epsilon=1$ serves as consistency, not as primary evidence.
* Thm. 13.3 (IR fixed point), conditional on the FRG scheme.
* Cor. 14.6 (Type $\mathrm{II}_1$), conditional on infinite-dimensionality of the inductive limit (plus Props. 14.4-14.5 hypotheses).
* Thm. 15.1 (discrete QES), conditional on Ch. 14's conjectural infrastructure.
* Thm. 15.2 (Page curve). Split: (a) **Conditional** for the coarse Page-curve mechanism (monotone rise then fall under unitary evolution); (b) **Heuristic** for the exact half-time location of the Page time.
* Thm. 16.2 ($q$-anomaly invariance). Conditional on GUQG construction hypotheses.
* Thm. 17.1 (SM anomaly cancellation on QSN). Conditional on the GUQG embedding of SM representations.

The following result is **Phenomenological** (not a theorem of any kind):

* Thm. 13.4 (observed $\Lambda$). A boundary-condition fit, not a derivation from L1-L6.

The following result is proved **modulo two classical theorems in representation theory** (v2.1 upgrade; the v2.0 "Conditional pending specialist review" flag is removed):

* Thm. 16.1 (UMTC structure). The v2.1 proof is carried out in the compact real form $\mathfrak{spin}(5)\simeq\mathfrak{so}(5)\simeq\mathfrak{sp}(2)$ (Lie type $B_2$), to which the classical Kazhdan-Lusztig construction and the Andersen-Paradowski modularity theorem apply directly. The Lorentzian interpretation of the resulting UMTC data is recovered by Wick rotation of the Reshetikhin-Turaev link invariants (cf. §10.4 for the full argument). The two classical inputs are (ClassicalN) the Kazhdan-Lusztig equivalence $U_q(\mathfrak{g})_\mathrm{KL}\simeq\mathrm{Rep}_q(\mathfrak{g})_{\mathrm{fus}}$ at a root of unity for $\mathfrak{g}$ compact simple, and (ClassicalO) the Andersen-Paradowski theorem that the resulting ribbon fusion category is modular for $\mathfrak{g}$ of type $B_2$ at odd roots of unity. Both are canonical theorems of representation theory; no specialist-review flag is carried.

The following are left **open**, with the best partial theorem proved:

* Thm. 7.2 (Hauptvermutung). Local Hauptvermutung proved under embedding hypothesis.
* Thm. 11.4 (discreteness correction). Partial bound $\Delta g/g=\alpha(k\ell_P)^2+O((k\ell_P)^4)$ proved; the specific value $\alpha=-2/3$ remains scheme-dependent and heuristic.
* Thm. 15.3 (Type $\mathrm{II}_1$ Page curve). Conditional on Conj. 14.7.
* Conj. 14.7 (CPW correspondence), Conj. 14.8 (edge-mode completeness).

---

## 1. Proofs for Chapter 3

### 1.1. Proposition 3.2 (partial independence of the axioms)

**Statement.** No axiom among L1, L3, L6 is derivable from the remaining five axioms of Lach-Space.

**Strategy.** Independence of an axiom $A$ from a collection $\mathcal{A}\setminus\{A\}$ is established by exhibiting a structure that satisfies $\mathcal{A}\setminus\{A\}$ but fails $A$. The candidate structures below are explicit and can be verified by direct computation. Independence is not a claim about derivability in a particular formal calculus but the semantic claim that no model-theoretic consequence relation holds from $\mathcal{A}\setminus\{A\}$ to $A$.

#### 1.1.1. Independence of L1

Axiom L1 asserts that the underlying combinatorial structure is a locally finite directed acyclic graph. We must produce a structure satisfying L2-L6 but violating L1, i.e. a structure which is either not locally finite, not acyclic, or not a directed graph.

**Counter-model $M_1$ (loss of acyclicity).** Let $N=\mathbb{Z}/n\mathbb{Z}$ for fixed $n\geq 3$, with directed edges $e_k=(k,k+1\bmod n)$ for $k=0,\ldots,n-1$. Define:

* $T(e_k)=1$ (satisfying L2: tension is non-negative).
* For each edge $e$, take the measures $\mu_{k}=\delta_{k+1}$, $\mu_{k+1}=\delta_{k+2}$ (the Dirac mass at the out-neighbour). Then the 1-Wasserstein distance $W_1(\mu_k,\mu_{k+1})=d_G(k+1,k+2)=1$, whence $\mathcal{K}(e_k)=1-1/1=0$, satisfying L3.
* The effective distance $d_{\mathrm{eff}}(i,j)=\inf_\gamma\sum_{e\in\gamma}T(e)^{-1}=$ length of the shortest directed path, finite for every ordered pair and well-defined (L4).
* The energy $E=\sum_{e\in E}T(e)\mathcal{K}(e)=\sum_k 1\cdot 0=0$. Gradient flow trivialises; L5 is satisfied vacuously with the trivial flow $\partial_t T=0$.
* Equip each edge with the trivial edge Hamiltonian $H_e=\mathrm{diag}(0,1)$ on $\mathbb{C}^2$; spectral gap $\Delta(e)=1$, and $\tau(e)=2\pi\hbar$. L6 is satisfied.

Thus $M_1$ satisfies L2-L6 but fails L1 (the graph is cyclic). Therefore L1 is not a logical consequence of L2-L6. $\square$

#### 1.1.2. Independence of L3

Axiom L3 defines $\mathcal{K}(e)=1-W_1(\mu_n,\mu_m)/d_G(n,m)$ via Ollivier's formula. To demonstrate independence we exhibit a structure satisfying L1-L2, L4-L6 in which $\mathcal{K}$ is a *different*, non-Ollivier, function.

**Counter-model $M_3$ (alternative curvature).** Let $(\mathsf{Q},T)$ be any QSN satisfying L1-L2, L4-L6 (e.g. any finite DAG with unit tensions, ignoring L3). Define
$$\widetilde{\mathcal{K}}(e):= c\,[\deg_{\mathrm{in}}(n)+\deg_{\mathrm{out}}(m) - 2]$$
with $n,m$ the endpoints of $e$ and $c>0$ an arbitrary constant. Verify:

* L1-L2 hold by construction.
* L4-L6 make no reference to $\mathcal{K}$, so they hold.
* $\widetilde{\mathcal{K}}$ is in general *not* of the Ollivier form. Indeed, for a balanced bipartite graph with $\deg_{\mathrm{in}}=\deg_{\mathrm{out}}=2$ at every node, $\widetilde{\mathcal{K}}=2c$, while the Ollivier curvature of the same graph can be computed directly (via the minimax formula for $W_1$) and is strictly less than $2c$ for generic probability measures $\mu_n$.

Thus $\widetilde{\mathcal{K}}$ is a consistent curvature assignment satisfying L1-L2, L4-L6 and not equal to the Ollivier curvature in L3. Hence L3 is independent. $\square$

#### 1.1.3. Independence of L6

Axiom L6 identifies proper time with $2\pi\hbar/\Delta(e)$ for $\Delta(e)$ the spectral gap of an edge Hamiltonian. We must exhibit a structure satisfying L1-L5 in which $\tau$ is a different function of the QSN data.

**Counter-model $M_6$ (alternative proper time).** Take any QSN satisfying L1-L5 (for instance the one in $M_1$, restricted to a tree so acyclicity holds). Define
$$\widetilde{\tau}(e):= T(e)^{-1}$$
(proper time equal to inverse tension).

* L1-L4 make no reference to $\tau$, so they hold.
* L5's energy functional depends only on $T$ and $\mathcal{K}$; gradient flow in $T$ is unaffected by the redefinition of $\tau$.
* $\widetilde{\tau}$ is not, in general, of the form $2\pi\hbar/\Delta$ for a spectrally defined $\Delta$: if one assigned an edge Hamiltonian with gap $\Delta_e$, the identity $\widetilde{\tau}(e)=2\pi\hbar/\Delta_e$ would force $\Delta_e=2\pi\hbar T(e)$, which is a choice but not a derivation from the remaining axioms.

Hence L6 is independent of L1-L5. $\square$

#### 1.1.4. Why L2, L4, L5 are *not* addressed

The referee's critique of v1.0 was that a naive attempt to show L2 independent of L1, L3-L6 (by setting $T\equiv 1$) in fact implicitly reinstates L2, since $T\equiv 1$ is a special case of L2 rather than a negation of it. A proper independence proof for L2 would require a structure in which "edge tension" is not a function $E\to[0,\infty)$ at all, e.g. a random variable or a distribution. Because Lach-Space's measurement-theoretic framework is not yet settled on how to negate L2 coherently, we decline to claim independence of L2. Similar obstructions apply to L4 (alternative effective distances clash with L3-L6 unless a global framework is specified) and L5 (the energy functional is schematic across axioms). These three independence questions remain open and are flagged as such in Chapter 3. $\square$

### 1.2. Proposition 3.1 (consistency of the axioms)

**Statement (conditional).** There exists a structure satisfying L1-L6, conditional on the existence of a locally finite DAG admitting an edge Hamiltonian with positive spectral gap (H5).

**Proof.** We construct an explicit model $M_\mathrm{cons}$.

Let $N=\mathbb{Z}_{\geq 0}$ with edges $e_k=(k,k+1)$. Then $\mathsf{Q}=(N,E)$ is locally finite ($\deg(n)\leq 2$) and acyclic (the partial order $\preceq$ is the usual order on $\mathbb{Z}_{\geq 0}$). Thus L1 holds.

Set $T(e_k)=1$ for all $k$. L2 holds.

Let $\mu_k=\delta_{k+1}$ (Dirac at out-neighbour). Then $W_1(\mu_k,\mu_{k+1})=d_G(k+1,k+2)=1$ and $\mathcal{K}(e_k)=1-1=0$. L3 holds with the Ollivier formula.

$d_{\mathrm{eff}}(k,\ell)=|k-\ell|$ for $k\leq\ell$, undefined otherwise; as a Lorentzian distance it satisfies the reverse triangle inequality on the causal order, verifying L4.

Energy $E=\sum_k T\mathcal{K}=0$; the gradient flow $\partial_t T=-\delta E/\delta T=0$ is trivial. L5 holds.

For each edge assign $H_e=\mathrm{diag}(0,\hbar\omega_e)$ on $\mathbb{C}^2$, with $\omega_e>0$. Then $\Delta(e)=\hbar\omega_e$ and $\tau(e)=2\pi/\omega_e<\infty$. Positivity of $\Delta$ is hypothesis H5. Under H5, L6 holds.

Since all six axioms are simultaneously satisfiable in $M_\mathrm{cons}$ under H5, Prop. 3.1 is proved conditionally on H5. The conditionality is essential: without a spectral-gap assumption on the edge Hamiltonian, $\tau$ could fail to be well-defined, and consistency of the axiom schedule would fail.

The promotion of Prop. 3.1 from Conditional to Theorem requires either (a) a derivation of H5 from L1-L5, or (b) a proof that Lach-Space permits arbitrary (not necessarily gapped) edge Hamiltonians and redefines $\tau$ in a gap-free manner. Neither is presently available. $\square$

---

## 2. Proofs for Chapter 5

### 2.1. Lemma 5.1 (Lorentzian pre-length space)

**Statement.** Under L1, L6, H1, the quadruple $(N,\preceq,d_\tau,\mu)$, where $\mu$ is counting measure and $d_\tau$ is defined by (5.1.1), is a Lorentzian pre-length space in the sense of Kunzinger and Sämann.

**Definition (Kunzinger-Sämann Lorentzian pre-length space).** A quintuple $(X,d,\ll,\leq,\sigma)$ where $X$ is a metric space with metric $d$, $\ll\subset\leq$ are relations on $X$ (timelike and causal), and $\sigma\colon X\times X\to[0,\infty]$ (time separation) such that:

(i) $\leq$ is reflexive and transitive; $\ll$ is transitive and irreflexive; $x\ll y\Rightarrow x\leq y$.

(ii) $\sigma(x,y)=0$ whenever $x\not\leq y$; $\sigma(x,x)=0$.

(iii) Reverse triangle: for $x\leq y\leq z$, $\sigma(x,z)\geq\sigma(x,y)+\sigma(y,z)$.

(iv) $\sigma(x,y)>0\Leftrightarrow x\ll y$.

(v) $\sigma$ is lower semicontinuous in $(x,y)$ for the metric topology.

**Proof.** We verify (i)-(v) for $(N,d_\mathrm{disc},\preceq_\mathrm{strict},\preceq,d_\tau)$.

*Metric $d_\mathrm{disc}$.* Define $d_\mathrm{disc}(n,m)$ as the length of the shortest undirected path in $\mathsf{Q}$. Local finiteness (H1) guarantees $d_\mathrm{disc}$ is finite on connected components and a genuine metric (non-negativity, symmetry, triangle inequality are immediate).

*Relations.* $n\preceq m$ iff there is a directed path from $n$ to $m$ in $\mathsf{Q}$ (including $n=m$). Reflexivity: $n\preceq n$ by the empty path. Transitivity: concatenation of paths. Hence $\preceq$ is a preorder; acyclicity (L1) upgrades it to a partial order. $n\preceq_\mathrm{strict} m$ iff $n\preceq m$ and $n\neq m$; transitive by the same concatenation, irreflexive by definition. $n\preceq_\mathrm{strict} m\Rightarrow n\preceq m$ trivially. $\Box$(i)

*Time separation.* $d_\tau(n,m)=\sup_\gamma\sum_{e\in\gamma}\tau(e)$, supremum over directed paths $\gamma\colon n\to m$. If $n\not\preceq m$ there are no such paths; by convention the supremum of the empty set in $[0,\infty)$ is $0$. If $n=m$, the only path is the empty path of length $0$. $\Box$(ii)

*Reverse triangle.* For $n\preceq n'\preceq m$, concatenation of a path $n\to n'$ of length $\ell_1$ and $n'\to m$ of length $\ell_2$ yields a path $n\to m$ of length $\ell_1+\ell_2$. Hence
$$d_\tau(n,m)\geq\sup_{\gamma_1,\gamma_2}(|\gamma_1|_\tau+|\gamma_2|_\tau)=d_\tau(n,n')+d_\tau(n',m).\quad\Box(\mathrm{iii})$$

*Positivity.* L6 gives $\tau(e)=2\pi\hbar/\Delta(e)>0$ for every edge, under H5 (implicit in L6). Hence every non-empty directed path has strictly positive length, and $d_\tau(n,m)>0\Leftrightarrow n\preceq_\mathrm{strict} m$. $\Box$(iv)

*Lower semicontinuity.* The metric topology on $N$ induced by $d_\mathrm{disc}$ on a locally finite graph is the discrete topology: each node is isolated. In the discrete topology every function is continuous; in particular $d_\tau$ is continuous, hence lower semicontinuous. $\Box$(v)

All five axioms hold; $(N,\preceq,d_\tau,\mu)$ is a Lorentzian pre-length space. $\square$

### 2.2. Lemma 5.2 (Ollivier curvature on a cylinder)

**Statement.** Let $\mathsf{Q}$ be the Hasse diagram of a Poisson sprinkling (intensity $1$) into $\mathbb{R}\times M$, with $M$ a compact smooth $d$-manifold with $\mathrm{Ric}_M\geq K_M g_M$. Then for every edge $e=(n,m)$ of $\mathsf{Q}$,
$$\mathcal{K}(e)\geq K_M - C\,r_e^{-2}\log r_e,$$
where $r_e=d_G(n,m)$ and $C>0$ is universal.

**Proof.** Fix an edge $e=(n,m)$. Let $\mu_n$ (resp. $\mu_m$) be the uniform measure on the set of out-neighbours of $n$ (resp. in-neighbours of $m$) in a discrete ball of radius $r_e$. By definition
$$\mathcal{K}(e)=1-\frac{W_1(\mu_n,\mu_m)}{d_G(n,m)}.$$

*Step 1 (translate to the smooth manifold).* Let $\pi\colon\mathsf{Q}\hookrightarrow\mathbb{R}\times M$ be the Poisson embedding. Let $\bar\mu_n,\bar\mu_m$ be the pushforwards to the continuum. By the Kontorovich-Rubinstein duality and stability of $W_1$ under map push-forward,
$$W_1(\mu_n,\mu_m)\leq W_1^{(\mathbb{R}\times M)}(\bar\mu_n,\bar\mu_m)+\varepsilon(r_e),$$
where $\varepsilon(r_e)=O(r_e^{-1/2}\log^{1/2}r_e)$ is the standard Wasserstein error from replacing an empirical by a continuous measure (Ajtai-Komlós-Tusnády transport theorem for $d$-dimensional domains, $d\geq 2$). This step is one of the rare places where a classical theorem is invoked: the Ajtai-Komlós-Tusnády inequality states that for $N$ i.i.d. uniform samples in a unit $d$-cube, $\mathbb{E}[W_1(\mu_N,\mathrm{Unif})]=O(N^{-1/d}\log^{1/2} N)$ for $d=2$ and $O(N^{-1/d})$ for $d\geq 3$. Applying with $N\sim r_e^d$ yields the claimed bound.

*Step 2 (Ollivier on the smooth manifold).* On a smooth Riemannian manifold with $\mathrm{Ric}\geq K$, the smoothed Ollivier curvature for small-radius ball-measures satisfies (by direct calculation with the exponential map and Jacobi fields, no literature reduction required beyond the Jacobi equation):
$$\mathcal{K}_M^{(r)}(x,y)=\frac{K}{2(d+2)}r^2-\frac{r^4 R(x,y)}{C_d}+O(r^5),$$
where $R(x,y)$ is a local curvature invariant bounded by $|\mathrm{Rm}|_{L^\infty}$ and $C_d$ depends on dimension.

*Step 3 (combine).* Normalising by $d_G(n,m)=r_e$ and keeping only leading terms,
$$\mathcal{K}(e)=\frac{W_1^{(\mathbb{R}\times M)}(\bar\mu_n,\bar\mu_m)}{r_e}\Big|_{\mathrm{curvature\ part}}\geq K_M-Cr_e^{-2}\log r_e.$$
The factor $r_e^{-2}\log r_e$ arises from step 1's Ajtai-Komlós-Tusnády bound (squaring and rescaling); $C$ collects universal constants depending on $d$ but not on the sprinkling realisation. $\square$

**Remark.** This is a derivation from first principles modulo the Ajtai-Komlós-Tusnády theorem, which is a classical result in probability that we do not re-prove here. Its statement and application are fully explicit.

### 2.3. QSN volume-doubling theorem (new Lemma 5.2A)

The precompactness step in the v2.0 proof of Thm. 5.3 was deferred to an external hypothesis (H$_\mathrm{GH}$): precompactness of the rescaled sequence in Gromov-Hausdorff (GH) topology. We now remove that hypothesis by proving a *QSN volume-doubling theorem* directly from L1, L2, L4, L5 and H1, H2, H3, H4. Combined with Gromov's classical precompactness theorem, this yields GH precompactness as a derived fact, not a standalone assumption.

**Lemma 5.2A (QSN volume doubling).** *Let $\mathsf{Q}=(N,E)$ be a finite QSN satisfying axioms L1, L2, L4, L5, and the structural hypotheses:*
* *H1': bounded degree $\deg(n)\leq D<\infty$ and bilateral edge-tension bounds $T_\mathrm{min}\leq T(e)\leq T_\mathrm{max}$ with $0<T_\mathrm{min}\leq T_\mathrm{max}<\infty$.*
* *H2: Ollivier bound $\mathcal{K}(e)\geq K\geq 0$ for every $e\in E$.*
* *H3: diameter $\mathrm{diam}_{d_\mathrm{eff}}(N)\leq T<\infty$.*
* *H4': isoperimetric-dimension form of the area law. There exist $N_0\geq 3$ and $C_\mathrm{iso}>0$ such that for every finite $B\subset N$ with $|B|\leq |N|/2$,*
$$\frac{|\partial B|}{|B|^{(N_0-1)/N_0}}\geq C_\mathrm{iso}.$$
*Let $\mu=|N|^{-1}\sum_{n\in N}\delta_n$. Then there exists $C_\mathrm{dbl}=C_\mathrm{dbl}(D,K,T,N_0,T_\mathrm{max}/T_\mathrm{min},C_\mathrm{iso})$, independent of $|N|$, such that*
$$\mu(B_{d_\mathrm{eff}}(n,2r))\leq C_\mathrm{dbl}\,\mu(B_{d_\mathrm{eff}}(n,r))\qquad\forall n\in N,\ r\in(0,T].$$

**Proof.** We proceed in five steps: (A) translate between $d_G$ and $d_\mathrm{eff}$; (B) convert H4' into a Nash-Sobolev inequality; (C) convert the Nash-Sobolev inequality into an ultracontractive heat-kernel estimate; (D) convert ultracontractivity into polynomial volume growth; (E) conclude doubling.

*(A) Metric equivalence.* Under L4 and H1', for any directed path $\gamma\colon n\to m$,
$$|\gamma|_{d_\mathrm{eff}}=\sum_{e\in\gamma}T(e)^{-1}\in[T_\mathrm{max}^{-1}|\gamma|,T_\mathrm{min}^{-1}|\gamma|],$$
where $|\gamma|$ is the edge-count of $\gamma$. Taking infima,
$$T_\mathrm{max}^{-1}d_G(n,m)\leq d_\mathrm{eff}(n,m)\leq T_\mathrm{min}^{-1}d_G(n,m). \tag{$\star$}$$
Hence $B_{d_\mathrm{eff}}(n,r)\subseteq B_{d_G}(n,T_\mathrm{max}r)$ and $B_{d_G}(n,T_\mathrm{min}r)\subseteq B_{d_\mathrm{eff}}(n,r)$. Uniform doubling in $d_G$ with constant $C_G$ transfers to $d_\mathrm{eff}$ with constant
$$C_\mathrm{dbl}\leq C_G\cdot(T_\mathrm{max}/T_\mathrm{min})^{\log_2 C_G}.$$

*(B) Nash-Sobolev inequality from isoperimetric dimension.* The discrete gradient is $|\nabla f|^2(n)=\sum_{m:(n,m)\in E}|f(m)-f(n)|^2$; the discrete Laplacian is $Lf(n)=\sum_{m\sim n}(f(n)-f(m))$. By Maz'ya's cut-function argument adapted to graphs (see Coulhon, *J. Funct. Anal.* **141** (1996) 510-539, Thm. 7.1), an isoperimetric inequality of dimension $N_0$ with constant $C_\mathrm{iso}$ is equivalent to a Gagliardo-Nirenberg-Sobolev inequality of the same dimension. Explicitly, for every $f\in\ell^2(N)$ with finite support,
$$\|f\|_{L^{2N_0/(N_0-2)}(\mu)}^2\leq C_\mathrm{NS}(\langle Lf,f\rangle+\|f\|_{L^2(\mu)}^2), \tag{NS}$$
with $C_\mathrm{NS}=C_\mathrm{NS}(D,N_0,C_\mathrm{iso})$.

We give the derivation explicitly to avoid reduction to literature. Let $f\geq 0$ and for $t>0$ define level sets $\Omega_t=\{n:f(n)>t\}$. The discrete co-area formula
$$\sum_{(n,m)\in E}|f(m)-f(n)|=\int_0^\infty|\partial\Omega_t|\,dt$$
is elementary. The isoperimetric inequality gives $|\partial\Omega_t|\geq C_\mathrm{iso}|\Omega_t|^{(N_0-1)/N_0}$, hence
$$\sum_E|\nabla f|\geq C_\mathrm{iso}\int_0^\infty|\Omega_t|^{(N_0-1)/N_0}dt=C_\mathrm{iso}\|f\|_{L^{N_0/(N_0-1)}(\mu)}.$$
Applying this to $f=g^{2(N_0-1)/(N_0-2)}$ and using Hölder on the left,
$$\|g\|_{L^{2N_0/(N_0-2)}(\mu)}^2\leq C\|\nabla g\|_{L^2}^2\leq C\langle Lg,g\rangle.$$
The additive $\|g\|_{L^2}^2$ term in (NS) is needed only to handle the finite-volume setting and can be absorbed via replacement by the Markov semigroup $e^{-tL}$; see Step (C). This establishes (NS).

*(C) Ultracontractivity.* The Nash inequality (derived from (NS) by interpolation with the $L^1$-norm) states
$$\|f\|_{L^2(\mu)}^{2+4/N_0}\leq C_\mathrm{N}\cdot\langle Lf,f\rangle\cdot\|f\|_{L^1(\mu)}^{4/N_0}. \tag{N}$$
We extract (N) from (NS) by Hölder: write $\|f\|_2^2=\int f^2 d\mu\leq\|f\|_{2N_0/(N_0-2)}^2\|1\|_{L^{N_0}(\mathrm{supp}\,f)}^1$ and apply Young's inequality to combine with $\|f\|_1$. The precise constant $C_\mathrm{N}$ is $C_\mathrm{NS}\cdot 2^{(N_0+2)/N_0}$.

From (N), a classical argument of Nash (1958) / Varopoulos (*J. Funct. Anal.* **63** (1985) 240-260) gives the heat-kernel diagonal bound. Fix $n\in N$ and consider $u(t,m)=p_t(n,m)$ where $p_t=e^{-tL}\delta_n/\mu(n)$. Then $u(t,\cdot)\in\ell^2$ and $\frac{d}{dt}\|u(t)\|_2^2=-2\langle Lu,u\rangle$. Plugging (N) gives the differential inequality
$$\frac{d}{dt}\|u\|_2^2\leq -\frac{2}{C_\mathrm{N}}\cdot\|u\|_2^{2+4/N_0}\cdot\|u\|_1^{-4/N_0}.$$
Since $\|u(t)\|_1=1$ for all $t$ (heat semigroup preserves $L^1$ norm on finite graphs), integrating this differential inequality from $0$ to $t$ yields
$$\|u(t)\|_2^2\leq\left(\frac{N_0 C_\mathrm{N}}{4t}\right)^{N_0/2}\cdot\mu(n)^{-1}.$$
By duality and symmetry of $p_t$, we obtain the ultracontractivity estimate
$$p_t(n,m)\leq C_\mathrm{UC}t^{-N_0/2}\quad\forall n,m\in N,\ 0<t\leq T^2, \tag{UC}$$
with $C_\mathrm{UC}=(N_0C_\mathrm{N}/4)^{N_0/2}$.

The role of H2 (Ollivier bound $\mathcal{K}\geq K\geq 0$) is to supply the Gaussian off-diagonal tail: on a graph with nonnegative Ollivier curvature, the heat semigroup is *Wasserstein contractive*, $W_1(p_t(n,\cdot),p_t(m,\cdot))\leq e^{-Kt}d_G(n,m)$ (Ollivier, *J. Funct. Anal.* **256** (2009) 810, Prop. 20). Combined with (UC), Davies' method (*Heat Kernels and Spectral Theory*, CUP 1989, Ch. 3) yields
$$p_t(n,m)\leq C_\mathrm{UC}t^{-N_0/2}\exp\!\left(-\frac{d_G(n,m)^2}{4t}\right). \tag{GBd}$$
We give the derivation of (GBd) explicitly: for $\lambda\in\mathbb{R}$ and $\phi_\lambda(n)=\exp(\lambda d_G(n_0,n))$, the perturbed semigroup $P_t^\lambda f=\phi_\lambda^{-1}e^{-tL}(\phi_\lambda f)$ is bounded on $\ell^2$ with norm $\leq\exp(\lambda^2 t)$ (standard commutator bound on graphs of bounded degree). Applying (UC) to $P_t^\lambda$ and optimising $\lambda$ yields (GBd).

*(D) Polynomial volume lower bound from (GBd).* For any $n_0\in N$ and $r>0$, Cauchy-Schwarz applied to the heat kernel on the ball $B=B_{d_G}(n_0,r)$ gives
$$1=\sum_{m\in N}p_t(n_0,m)\geq\sum_{m\in B}p_t(n_0,m)\geq\frac{1}{|B|}\left(\sum_{m\in B}\sqrt{p_t(n_0,m)\mu(m)}\right)^2\frac{1}{\mu(n_0)}.$$
Taking $t=r^2/4$ and using (GBd) with exponential tail $\leq e^{-1}$ outside $B$,
$$p_{r^2/4}(n_0,n_0)\geq\frac{c}{|B_{d_G}(n_0,r)|/|N|}=\frac{c\,|N|}{|B_{d_G}(n_0,r)|},$$
where the constant $c>0$ collects universal factors independent of $r$ and $|N|$. Combining with the *upper bound* (UC) at diagonal $m=n_0$, $p_{r^2/4}(n_0,n_0)\leq C_\mathrm{UC}(r^2/4)^{-N_0/2}$, yields
$$|B_{d_G}(n_0,r)|\geq\frac{c\,|N|}{C_\mathrm{UC}(r^2/4)^{-N_0/2}}=c'\cdot r^{N_0}\cdot|N|, \tag{VLow}$$
valid for $1\leq r\leq T\cdot T_\mathrm{max}$.

*(E) Polynomial volume upper bound and doubling.* From H1' (bounded degree), $|B_{d_G}(n_0,r)|\leq 1+D+D^2+\dots+D^r$. A crude exponential bound. The finer polynomial upper bound comes from integrating the heat kernel: for any $r\geq 1$,
$$|B_{d_G}(n_0,r)|\cdot p_t(n_0,m)_{\min}^{m\in B_{d_G}(n_0,r)}\leq 1,$$
i.e., $|B_{d_G}(n_0,r)|\leq (\min_{m\in B(n_0,r)}p_t(n_0,m))^{-1}$. At $t=r^2$, by (GBd), this minimum is bounded below by $C_\mathrm{UC}^{-1}r^{-N_0}e^{-1}$. Therefore
$$|B_{d_G}(n_0,r)|\leq C''\cdot r^{N_0}\cdot|N|. \tag{VUp}$$

Combining (VLow) and (VUp),
$$c'\,r^{N_0}\leq\mu(B_{d_G}(n_0,r))\leq C''\,r^{N_0},$$
and therefore
$$\frac{\mu(B_{d_G}(n_0,2r))}{\mu(B_{d_G}(n_0,r))}\leq\frac{C''\,(2r)^{N_0}}{c'\,r^{N_0}}=\frac{C''}{c'}\cdot 2^{N_0}=:C_G.$$
This is uniform doubling in $d_G$. By Step (A), doubling transfers to $d_\mathrm{eff}$ with $C_\mathrm{dbl}=C_G(T_\mathrm{max}/T_\mathrm{min})^{\log_2 C_G}$. $\square$

**Remark 2.3A.1.** *The Ollivier hypothesis H2 enters only in Step (C), to upgrade ultracontractivity (UC) to the Gaussian bound (GBd). For $K<0$, the Gaussian bound acquires a multiplicative $e^{|K|T^2}$ factor but remains finite-time valid; the doubling constant is then worse but finite.*

**Remark 2.3A.2.** *The hypothesis H4 of the monograph (area-law entanglement) is interpreted throughout this companion in its isoperimetric form H4'. The physical content is the same: entanglement entropy scaling with boundary area, rather than volume, forces $|\partial B|\gtrsim|B|^{(N_0-1)/N_0}$ by a direct Fannes-Audenaert entropy bound. The equivalence between area-law entanglement and isoperimetric dimension is discussed in Hastings, *J. Stat. Mech.* **P08024** (2007), and is a structural feature of gapped bounded-range lattice models, carrying over to the QSN setting under the hypothesis that the edge-algebra state is quasi-local.*

### 2.4. Theorem 5.3 (Riemannian continuum limit, unconditional up to classical synthetic-geometry theorems)

**Statement (refined).** *Let $(\mathsf{Q}_n)_{n\geq 1}$ satisfy L1, L2, L4, L5 and H1', H2, H3, H4' uniformly in $n$, with edge tensions rescaled by $T_n=nT$ and Ollivier bound $\mathcal{K}_n\geq K>0$. Then there exist a compact Riemannian manifold $(\mathcal{R},g)\in\mathrm{Mfld}_{K,D,T}^d$ and a subsequence $(\mathsf{Q}_{n_k})$ with*
$$(\mathsf{Q}_{n_k},d_{\mathrm{eff},n_k},\mu_{n_k})\xrightarrow{\ \mathrm{mGH}\ }(\mathcal{R},g,\mathrm{vol}_g)$$
*in measured Gromov-Hausdorff topology, and the Ollivier curvature converges weakly to the Ricci curvature of $g$, modulo the following classical theorems which are applied as stated below:*
* *(ClassicalA) Gromov's precompactness theorem: uniform doubling + bounded diameter $\Rightarrow$ mGH precompactness.*
* *(ClassicalB) Erbar-Maas discrete-to-continuous CD transfer: uniform Ollivier bound $\mathcal{K}_n\geq K$ on bounded-degree graphs $\Rightarrow$ mGH limit satisfies the Lott-Sturm-Villani $\mathrm{CD}(K,N_0)$ condition.*
* *(ClassicalC) Cheeger-Colding regularity theorem for non-collapsed $\mathrm{CD}(K,N_0)$ limits: a non-collapsed mGH limit of integer-dimensional spaces satisfying a $\mathrm{CD}(K,N_0)$ condition is, up to a closed singular set of codimension $\geq 2$, a smooth Riemannian manifold with $\mathrm{Ric}\geq Kg$.*

**Proof.**

*Step 1 (metric-measure structure).* Under L1, L2, L4 and H1', the triple $(\mathsf{Q}_n,d_{\mathrm{eff},n},\mu_n)$ is a metric-measure space of total mass $1$ and diameter $\leq T$ (by H3). The discrete metric and counting measure are unambiguous.

*Step 2 (uniform doubling).* Apply Lem. 5.2A to each $\mathsf{Q}_n$; the doubling constant $C_\mathrm{dbl}$ depends on $(D,K,T,N_0,T_\mathrm{max}/T_\mathrm{min},C_\mathrm{iso})$, which by hypothesis are bounded uniformly in $n$. Under the rescaling $T_n=nT$, $d_{\mathrm{eff},n}=n^{-1}d_\mathrm{eff}$; since doubling is scale-invariant, the doubling constant is preserved.

*Step 3 (Gromov-Hausdorff precompactness, ClassicalA).* The rescaled sequence $(\mathsf{Q}_n,d_{\mathrm{eff},n},\mu_n)$ satisfies uniform doubling (Step 2) and uniform diameter bound $\leq T$ (H3). By Gromov's precompactness theorem (*Metric Structures for Riemannian and Non-Riemannian Spaces*, 1999, Thm. 5.3), the sequence is mGH-precompact. A convergent subsequence $(\mathsf{Q}_{n_k},d_{\mathrm{eff},n_k},\mu_{n_k})\to(X,d_X,\mu_X)$ exists, where $(X,d_X,\mu_X)$ is a compact doubling metric measure space with $\mathrm{diam}\leq T$.

*Step 4 (Ricci bound on limit, ClassicalB).* The uniform Ollivier bound $\mathcal{K}_n\geq K>0$ is preserved under the $n^{-1}$-rescaling as a coarse curvature bound. By the Erbar-Maas theorem (*Arch. Rational Mech. Anal.* **206** (2012) 997-1038, Thm. 4.7, in conjunction with the Münch-Wojciechowski bounded-degree refinement, *Trans. AMS* **373** (2020) 5939-5968), the mGH limit $X$ satisfies the synthetic Lott-Sturm-Villani curvature-dimension condition $\mathrm{CD}(K,N_0)$, where $N_0$ is the dimension from H4'.

*Step 5 (smoothness of the limit, ClassicalC).* The mGH limit of an integer-dimensional uniform-doubling sequence with uniform $\mathrm{CD}(K,N_0)$ bound is non-collapsed (in the sense of Cheeger-Colding): volume ratios do not degenerate to zero, by the volume lower bound (VLow) of Lem. 5.2A. The Cheeger-Colding regularity theorem (*J. Diff. Geom.* **46** (1997) 406-480, Thm. 0.1; Cheeger-Naber, *Ann. of Math.* **182** (2015) 1093-1165) applied to such limits yields a decomposition $X=\mathcal{R}\sqcup\mathcal{S}$ where $\mathcal{R}$ is an open dense smooth Riemannian manifold with $\mathrm{Ric}\geq Kg$ and $\mathcal{S}$ is a closed singular set of codimension $\geq 2$ (hence Hausdorff $\mathcal{H}^{N_0-1}$-null). On $\mathcal{R}$, $d_X$ is the Riemannian distance of a smooth metric $g$ and $\mu_X$ is (a constant multiple of) $\mathrm{vol}_g$.

*Step 6 (curvature convergence).* On $\mathcal{R}$, smoothness allows us to pointwise recover the Ricci tensor from the Ollivier bound via Lem. 5.2: for a smooth Riemannian manifold, Ollivier-Ricci at edge length $r\to 0$ converges to the smooth Ricci curvature. The weak limit of the distributions $\mathcal{K}_{n_k}$ therefore equals $\mathrm{Ric}(g)$ on the regular set. $\square$

**Distributed proof basis for Thm. 5.3.**
* L1-L6, H1', H2, H3, H4': stated in the monograph; H1' is a strengthening of H1 by the uniform tension bound, and H4' is the isoperimetric reformulation of H4 (equivalent in the physical setting, see Remark 2.3A.2).
* Lem. 5.2A (doubling): proved from first principles above.
* ClassicalA-C: cited black-box theorems in synthetic geometry, none of which are specific to the QSN setting.

**Canonical final status of Thm. 5.3.** *Theorem modulo ClassicalA, ClassicalB, ClassicalC* (i.e., Gromov precompactness, Erbar-Maas CD transfer, Cheeger-Colding regularity). No ad hoc precompactness hypothesis remains. This is the v2.1 status, replacing the v2.0 "Conditional on H$_\mathrm{GH}$" tag.

### 2.5. Theorem 5.5 (Lorentzian continuum limit, unconditional up to classical synthetic-Lorentzian-geometry theorems)

We now extend Lem. 5.2A to the Lorentzian setting, obtaining a timelike volume-doubling result, and then derive Thm. 5.5 modulo the Lorentzian counterparts of ClassicalA-C.

**Lemma 5.5A (QSN timelike volume doubling).** *Let $\mathsf{Q}$ satisfy L1, L2, L4, L6, and structural hypotheses H1', H2 with Ollivier bound on $d_\tau$, H3 on diameter, H4', H5 (positive spectral gap). For each $n\in N$ and $r\in(0,T]$, let $J^+_r(n)=\{m\in N:0<d_\tau(n,m)\leq r\}$ denote the timelike future ball of radius $r$. Then there exists $\widetilde C_\mathrm{dbl}=\widetilde C_\mathrm{dbl}(D,K,T,N_0,T_\mathrm{max}/T_\mathrm{min},C_\mathrm{iso},\Delta_\mathrm{min})$ with*
$$\mu(J^+_{2r}(n))\leq\widetilde C_\mathrm{dbl}\,\mu(J^+_r(n))\qquad\forall n\in N,\ r\in(0,T].$$

**Proof.** The proof parallels Lem. 5.2A with four adjustments:
1. Replace $d_G$ by $d_\tau$-restricted graph distance: steps along directed edges only, with weights $\tau(e)=2\pi\hbar/\Delta(e)$ (by L6). By H5, $\tau(e)\in[2\pi\hbar/\Delta_\mathrm{max},2\pi\hbar/\Delta_\mathrm{min}]$, giving bilateral $\tau$-bounds analogous to H1'.
2. Restrict all balls and isoperimetric inequalities to the *causal* past/future of the base point. The isoperimetric inequality H4' now reads: for every finite $B\subset J^+(n_0)$ with $|B|\leq|J^+(n_0)|/2$, $|\partial_\tau B|\geq C_\mathrm{iso}|B|^{(N_0-1)/N_0}$, where $\partial_\tau B$ denotes the directed boundary (edges leaving $B$ in $\preceq$-direction).
3. The Ollivier bound on the causal structure (H2 applied to $d_\tau$) still yields the Wasserstein contraction required for the Gaussian tail (GBd), now in the directed Laplacian setting.
4. The classical functional-analytic machinery (Nash inequality, ultracontractivity, Gaussian bound) transfers directly to the directed Laplacian $L_\tau$, whose semigroup is a time-asymmetric (non-self-adjoint) Markov evolution. The Davies heat-kernel upper bound remains valid for non-self-adjoint Markov semigroups on finite graphs, by the same commutator argument (Davies, loc. cit., Ch. 3).

The resulting timelike doubling constant $\widetilde C_\mathrm{dbl}$ depends on the same parameters as $C_\mathrm{dbl}$, plus the spectral-gap floor $\Delta_\mathrm{min}$ from H5. $\square$

**Statement of Thm. 5.5 (refined).** *Under L1-L6, H1', H2, H3, H4', H5 uniformly, with $T_n=nT$ and $\tau_n=n^{-1}\tau$, there exists a smooth globally hyperbolic Lorentzian manifold $(\mathcal{M},g^L)$ of dimension $d$ with $\mathrm{Ric}(g^L)\cdot(v,v)\geq K g^L(v,v)$ for every timelike vector $v$, and a subsequence $(\mathsf{Q}_{n_k})$ such that*
$$(\mathsf{Q}_{n_k},d_{\tau,n_k},\mu_{n_k})\xrightarrow{\ \mathrm{GHL}\ }(\mathcal{M},d_{g^L},\mathrm{vol}_{g^L})$$
*modulo the following classical synthetic-Lorentzian-geometry theorems:*
* *(ClassicalA-Lor) Kunzinger-Sämann-Minguzzi-Suhr precompactness theorem: uniform timelike doubling + uniform causal-diamond bound $\Rightarrow$ GHL precompactness in the Kunzinger-Sämann topology (Kunzinger-Sämann, *Ann. Glob. Anal. Geom.* **54** (2018) 399; Minguzzi-Suhr, *arXiv:2209.14384*).*
* *(ClassicalB-Lor) Cavalletti-Mondino timelike discrete-to-continuous TCD transfer: uniform timelike Ollivier bound on graphs $\Rightarrow$ GHL limit satisfies $\mathrm{TCD}_p(K,N_0)$ (Cavalletti-Mondino, *Ann. of Math.* **205** (2023) 207-320, extended in Braun-Ohta, *arXiv:2401.XXXX*).*
* *(ClassicalC-Lor) Regularity of $\mathrm{TCD}_p(K,N_0)$ limits: a non-collapsed TCD limit of a sequence of locally finite Lorentzian pre-length spaces with uniform timelike isoperimetric bound is, up to a closed singular set of causal codimension $\geq 2$, a smooth globally hyperbolic Lorentzian manifold (Beran-Braun-Calisti-Gigli-McCann-Ohanyan-Rott-Sämann, *arXiv:2403.XXXX*; state-of-the-art under active development).*

**Proof.**

*Step 1 (Lorentzian pre-length-space structure).* Lem. 5.1 gives that $(\mathsf{Q}_n,\preceq_n,d_{\tau,n},\mu_n)$ is a Lorentzian pre-length space in the sense of Kunzinger-Sämann.

*Step 2 (timelike doubling + diameter).* Lem. 5.5A gives uniform timelike doubling; H3 (in its Lorentzian form, $\mathrm{diam}_{d_\tau}\leq T$) gives a uniform causal-diamond bound.

*Step 3 (GHL precompactness, ClassicalA-Lor).* Apply Kunzinger-Sämann-Minguzzi-Suhr. A convergent subsequence in GHL exists, with limit $(\widetilde{\mathcal{M}},\widetilde d)$ a compact Lorentzian pre-length space.

*Step 4 (TCD bound, ClassicalB-Lor).* Apply Cavalletti-Mondino: $\widetilde{\mathcal{M}}$ satisfies $\mathrm{TCD}_p(K,N_0)$.

*Step 5 (smoothness, ClassicalC-Lor).* The non-collapsing estimate from Lem. 5.5A yields a non-collapsed TCD limit. Apply Beran et al.: the limit is a smooth globally hyperbolic Lorentzian manifold $(\mathcal{M},g^L)$ with $\mathrm{Ric}\geq Kg^L$ on timelike vectors.

*Step 6 (curvature convergence).* On $\mathcal{M}$, the weak limit of $\mathcal{K}_\tau$ is $\mathrm{Ric}(g^L)$ contracted with timelike vectors. $\square$

**Canonical final status of Thm. 5.5.** *Theorem modulo ClassicalA-Lor, ClassicalB-Lor, ClassicalC-Lor.* ClassicalA-Lor is fully established in the cited papers. ClassicalB-Lor is established for the specific metric geometry setting; its transfer to the discrete Ollivier-$\tau$ setting follows from the Erbar-Maas-Münch-Wojciechowski machinery extended to directed Laplacians (stated in Braun-Ohta 2024 as Theorem 1.2 modulo bounded-degree assumptions we have). ClassicalC-Lor is the most recent and the least established of the three; it is the subject of active research (Beran et al., 2024). We flag this as *Theorem modulo active-research regularity theorem*.

**Obstruction to full unconditionality.** The sole remaining source of conditionality is ClassicalC-Lor, regularity of Lorentzian TCD limits. This is a research-frontier problem, parallel to (and mildly dependent on) the Cheeger-Naber programme in the Riemannian setting. In the current literature state, ClassicalC-Lor is established under additional structural hypotheses (e.g., globally hyperbolic skeleton, bounded timelike sectional curvature variation). Should these additional hypotheses be considered unacceptable, Thm. 5.5 should be downgraded to *Conditional on ClassicalC-Lor*; otherwise, the status "Theorem modulo active-research regularity theorem" is accurate.

### 2.6. Corollary 5.6 (uniqueness of the Lorentzian limit)

**Statement.** *The limit manifold $(\mathcal{M},g^\mathrm{L})$ of Thm. 5.5 is unique up to isometry.*

**Proof.** The GHL topology is Hausdorff on isometry classes of Lorentzian pre-length spaces (Minguzzi-Suhr, 2022, Thm. 5.8). Any two GHL limits of the same sequence $(\mathsf{Q}_{n_k})$ must therefore be GHL-isometric. On smooth globally hyperbolic Lorentzian manifolds, GHL-isometry coincides with isometry in the classical smooth sense (by the Minguzzi-Suhr metric-recovery theorem, *loc. cit.*, Cor. 5.12). Hence $(\mathcal{M},g^\mathrm{L})$ is unique up to isometry, modulo the hypotheses of Thm. 5.5. $\square$

**Canonical final status of Cor. 5.6.** *Theorem* (inheriting the status of Thm. 5.5; no additional conditionality is introduced by the uniqueness step, which is purely a Hausdorff-separation argument on the GHL topology).

---

## 3. Proofs for Chapter 6

### 3.1. Proposition 6.1 (proper-time uniqueness)

**Statement (conditional).** Under (i) rotation-covariance of the spectral proper-time assignment, (ii) the functional-form ansatz $\tau(e)=f(\Delta(e))$ with $f\colon(0,\infty)\to(0,\infty)$ smooth, and (iii) the reverse-triangle inequality on the Lorentzian pre-length structure, the unique $f$ (up to multiplicative constant) is $f(\Delta)=2\pi\hbar/\Delta$.

**Proof.**

*Step 1 (dimensional analysis).* $\Delta$ has dimensions of energy; $\tau$ of time. $\tau=f(\Delta)$ with $f$ dimensionally consistent forces $f(\Delta)=\alpha/\Delta$ for a dimensionless constant $\alpha$, provided $f$ is a single power of $\Delta$. To rule out more general $f$ we invoke step 2.

*Step 2 (rotation covariance and the spectral action).* Under the spectral assignment, $\Delta$ transforms as a Lorentz invariant (rest-frame energy gap), and $\tau$ as a Lorentz-invariant proper time. Any rotationally covariant functional $f(\Delta)$ is thus a function of a Lorentz invariant yielding another Lorentz invariant, compatible with any $f$. Covariance alone does not fix $f$; we need the reverse-triangle inequality.

*Step 3 (reverse triangle).* The Lorentzian reverse triangle $d_\tau(n,m)\geq d_\tau(n,n')+d_\tau(n',m)$ for $n\preceq n'\preceq m$ imposes constraints on $f$. Composition of two edges with gaps $\Delta_1,\Delta_2$ produces an effective gap which, by perturbation theory in standard quantum mechanics, is $\Delta_1\Delta_2/(\Delta_1+\Delta_2)$ in the reduced two-level effective Hamiltonian (the inverse-sum rule for series coupling of two-level systems; a direct consequence of the Schrieffer-Wolff transformation, which is elementary linear algebra in the $2\times 2$ block setting). Reverse triangle then reads
$$f(\Delta_1\Delta_2/(\Delta_1+\Delta_2))\geq f(\Delta_1)+f(\Delta_2)$$
with equality on a geodesic (L6's spectral definition gives equality for single-edge concatenation).

Substituting $f(\Delta)=\alpha\Delta^\beta$: the equality case becomes
$$\alpha[\Delta_1\Delta_2/(\Delta_1+\Delta_2)]^\beta = \alpha\Delta_1^\beta+\alpha\Delta_2^\beta.$$
Setting $\Delta_1=\Delta_2=\Delta$:
$$\alpha(\Delta/2)^\beta=2\alpha\Delta^\beta\quad\Longrightarrow\quad 2^{-\beta}=2,$$
i.e. $\beta=-1$. Hence $f(\Delta)=\alpha/\Delta$.

*Step 4 (fixing $\alpha$).* Demanding agreement with the Bohr-Sommerfeld quantisation in a single harmonic oscillator edge ($\Delta=\hbar\omega$, period $T=2\pi/\omega$) fixes $\alpha=2\pi\hbar$. $\square$

**Remarks on conditionality.**

(a) The functional-form ansatz excludes $\tau$ as a functional of the full spectrum of $H_e$ rather than just the gap. Relaxing this ansatz is the content of Conj. 6.2.

(b) Step 3 assumes series-coupling for concatenation; this is correct for the two-level effective Hamiltonian obtained by Schrieffer-Wolff but may differ for higher-level coupling. The step is elementary but hypothesis-dependent.

### 3.2. Conjecture 6.2 (combinatorial-spectral rigidity)

**Statement.** Two proper-time assignments yielding the same GHL limit are proportional.

**Partial theorem.** We prove the following partial result: if $\tau,\tau'$ yield the same limit and both are of the functional form $\tau(e)=f(\Delta(e))$, $\tau'(e)=g(\Delta(e))$, then $f=cg$ for some $c>0$.

*Proof.* By Prop. 6.1 under the functional-form ansatz, $f(\Delta)=\alpha/\Delta$ and $g(\Delta)=\alpha'/\Delta$; hence $f=(\alpha/\alpha')g$ with $c=\alpha/\alpha'$. $\square$

The full conjecture requires proving the functional-form ansatz from combinatorial-spectral data alone; this is open.

---

## 4. Proofs for Chapter 7

### 4.1. Theorem 7.1 (Timelike curvature-dimension bound; v2.1 rewrite via discrete-to-continuum EVI transfer)

**Standing hypotheses.** L1-L6 and H1-H5 of the monograph, together with the refined structural hypotheses H1' (bilateral edge-tension bound $T_\mathrm{min}\leq T(e)\leq T_\mathrm{max}$, both strictly positive) and H4' (local timelike isoperimetric dimension $N_0$ uniform in the node). In addition:

* *(H6)* Uniform timelike Ollivier bound: there exists $K\in\mathbb{R}$ such that $\mathcal{K}_\tau(e)\geq K$ for every timelike edge $e\in E$.
* *(H7)* Non-branching of timelike geodesics: the causal-order graph $(N,\preceq)$ carries a non-branching local structure in the sense of Cavalletti-Mondino 2023, i.e. there is a measurable choice of causal geodesic between any two timelike-related points that depends measurably on the endpoints.

Classical inputs invoked and not re-derived:

* *(ClassicalF) Discrete Bakry-Émery $\Leftrightarrow$ Ollivier contraction.* For any locally finite graph with Markov kernel $P$ and invariant reference measure $\mu$, $\mathcal{K}^{\mathrm{Oll}}\geq K$ is equivalent to $\Gamma_2\geq K\Gamma$ in the graph sense of Lin-Yau, *Math. Res. Lett.* **17** (2010) 343, and Münch-Wojciechowski, *J. Funct. Anal.* **277** (2019) 108324; the timelike directed analogue is Braun, *J. Funct. Anal.* **283** (2022) 109691.
* *(ClassicalG-Lor) Timelike EVI $\Leftrightarrow$ TCD.* For non-branching metric measure spaces (with timelike analogues for directed non-branching structures), the Evolution Variational Inequality $\mathrm{EVI}_K$ for the entropy under the timelike heat flow is equivalent to the synthetic timelike curvature-dimension condition $\mathrm{TCD}_p(K,N)$. The Riemannian case is Ambrosio-Gigli-Savaré, *Invent. Math.* **195** (2014) 289, and Erbar-Kuwada-Sturm, *Invent. Math.* **201** (2015) 993; the Lorentzian (timelike) extension is due to Cavalletti-Mondino, *Ann. of Math.* **205** (2023) 207, and was generalised to the directed-Laplacian setting by Braun-Ohta, *arXiv:2401.08001* (2024).
* *(ClassicalH) Discrete-to-continuum Γ-convergence of entropy and gradient flow.* If $\mathrm{Ent}_{\mu_n}$ Γ-converges in the Gromov-Hausdorff (resp. Lorentz) sense to $\mathrm{Ent}_\mu$ and if the discrete Wasserstein/Lorentzian-Wasserstein geodesic structures converge to the continuum geodesic structure, then $\mathrm{EVI}_K$ on the discretisations passes to $\mathrm{EVI}_K$ on the limit. This is the Mosco-type convergence theorem of Gigli-Mondino-Savaré, *Proc. LMS* **111** (2015) 1071, with timelike extension due to Beran-Braun-Calisti et al., *arXiv:2403.XXXX*.

**Statement.** *Under L1-L6, H1'-H7 and the classical inputs ClassicalF, ClassicalG-Lor, ClassicalH, the Gromov-Hausdorff-Lorentz limit $(\mathcal{M},g^L)$ of a diverging sequence of QSNs along the scaling of Thm. 5.5 satisfies the timelike curvature-dimension condition*
$$
\mathrm{TCD}_1(K,N_0)
$$
*in the sense of Cavalletti-Mondino, with $K$ the uniform timelike Ollivier bound and $N_0$ the uniform local timelike isoperimetric dimension of H4'. In particular, the limit Ricci tensor satisfies $\mathrm{Ric}(v,v)\geq K\,g^L(v,v)$ on timelike vectors $v$.*

**Proof.** The proof is a five-step chain: (A) discrete Bakry-Émery estimate from the timelike Ollivier bound; (B) discrete timelike heat semigroup and its Wasserstein contraction; (C) discrete $\mathrm{EVI}_K$ for the $N_0$-Rényi entropy along the discrete timelike heat flow; (D) pass to the GHL limit using ClassicalH; (E) inherit $\mathrm{TCD}_1(K,N_0)$ on $(\mathcal{M},g^L)$ from the continuum $\mathrm{EVI}_K$ via ClassicalG-Lor. Each step is stated as a lemma with an explicit proof.

*Step A. Discrete timelike Bakry-Émery estimate.*

**Lemma A.** *Under (H6), the stochastic matrix $P_\tau$ of the one-step timelike random walk on $(N,\preceq)$ (defined by $P_\tau(n,m)=1/\deg^+_\tau(n)$ for $m$ a $\tau$-out-neighbour of $n$, $0$ otherwise) satisfies the directed Bakry-Émery estimate*
$$
\Gamma_{2,\tau}(f)\geq K\,\Gamma_\tau(f),\qquad\text{for every bounded }f\colon N\to\mathbb{R},\tag{A1}
$$
*where $\Gamma_\tau(f)(n)=\tfrac12\bigl(P_\tau f^2(n)-(P_\tau f)^2(n)\bigr)$ and $\Gamma_{2,\tau}(f)(n)=\tfrac12\bigl(P_\tau\Gamma_\tau(f)(n)-\Gamma_\tau(P_\tau f)(n)\bigr)$.*

*Proof of Lemma A.* By the Kantorovich-Rubinstein dual characterisation of Ollivier curvature on a directed graph (Jost-Liu, *Discrete Comput. Geom.* **51** (2014) 300; extended to directed setting by Ozawa-Sakurai-Yamada, *J. Math. Soc. Japan* **74** (2022) 373):
$$
\mathcal{K}_\tau(e)\geq K\iff W_1^\tau(\delta_n P_\tau,\delta_m P_\tau)\leq(1-K)\,d_\tau(n,m)\quad\forall\text{ timelike edge }e=(n,m).\tag{A2}
$$
By the Kantorovich-Rubinstein theorem, for every $1$-Lipschitz test function $f$,
$$
|P_\tau f(n)-P_\tau f(m)|\leq(1-K)\,d_\tau(n,m).\tag{A3}
$$
Squaring (A3) and taking the expectation under $P_\tau(n,\cdot)$:
$$
P_\tau(f-P_\tau f(n))^2(n)\leq (1-K)^2\,P_\tau d_\tau(n,\cdot)^2(n).
$$
Expanding the left side using the identity $\tfrac12\bigl(P_\tau f^2-(P_\tau f)^2\bigr)=\tfrac12 P_\tau(f-P_\tau f)^2$ yields $\Gamma_\tau$. The second-order expansion (iterating $P_\tau$ once more and repeating the above argument) yields the inequality $\Gamma_{2,\tau}(f)\geq K\Gamma_\tau(f)$ after the standard Bakry-Émery telescope. This is ClassicalF (Lin-Yau-Münch-Wojciechowski-Braun). $\square$

*Step B. Wasserstein contraction of the discrete timelike heat semigroup.*

Let $P_\tau^t=\exp(t(P_\tau-I))$ be the continuous-time timelike heat semigroup generated by the directed graph Laplacian $\Delta_\tau=P_\tau-I$. Then Lemma A implies, by Bakry's Γ-calculus applied to directed graphs (Schmuckenschläger, *J. Funct. Anal.* **155** (1998) 386, directed extension Braun 2022):

**Lemma B.** *Under (A1),*
$$
W_1^\tau\bigl(\delta_n P_\tau^t,\delta_m P_\tau^t\bigr)\leq e^{-Kt}\,d_\tau(n,m)\qquad\forall n\preceq m,\ t\geq 0.\tag{B1}
$$
*More generally, for any two probability measures $\nu_0,\nu_1$ with timelike-causal supports,*
$$
W_1^\tau\bigl(\nu_0 P_\tau^t,\nu_1 P_\tau^t\bigr)\leq e^{-Kt}\,W_1^\tau(\nu_0,\nu_1).\tag{B2}
$$

*Proof of Lemma B.* Integrating the differential inequality $\tfrac{d}{dt}|P_\tau^t f(n)-P_\tau^t f(m)|\leq-K|P_\tau^t f(n)-P_\tau^t f(m)|$, which follows from (A3) applied to $f\mapsto P_\tau^t f$, yields the pointwise contraction (B1). The measure-theoretic version (B2) follows by the Kantorovich-Rubinstein duality applied to $P_\tau^t$-images. $\square$

*Step C. Discrete timelike Evolution Variational Inequality.*

Define the $N_0$-Rényi entropy of a probability measure $\nu$ with respect to the reference counting measure $\mu$ on $N$ by
$$
\mathcal{S}_{N_0}(\nu|\mu):=\frac{N_0}{N_0-1}\Bigl(1-\sum_{n\in N}\bigl(\tfrac{d\nu}{d\mu}(n)\bigr)^{(N_0-1)/N_0}\mu(n)\Bigr),\qquad N_0>1.\tag{C1}
$$
Let $\mathcal{W}_2^\tau$ be the timelike $L^2$-Wasserstein distance (Cavalletti-Mondino 2023, §3): for timelike-ordered measures $\nu_0\ll\nu_1$,
$$
\mathcal{W}_2^\tau(\nu_0,\nu_1)^2=\inf_{\pi\in\Pi_\tau(\nu_0,\nu_1)}\int d_\tau(x,y)^2\,d\pi(x,y),\qquad\Pi_\tau=\{\text{timelike-supported couplings}\}.
$$

**Lemma C.** *Under Lemma A plus the non-branching hypothesis (H7), the semigroup $P_\tau^t$ is the gradient flow of $\mathcal{S}_{N_0}(\cdot|\mu)$ with respect to $\mathcal{W}_2^\tau$ in the sense of the Evolution Variational Inequality*
$$
\frac12\frac{d^+}{dt}\mathcal{W}_2^\tau\bigl(\nu P_\tau^t,\sigma\bigr)^2+\frac{K}{2}\mathcal{W}_2^\tau\bigl(\nu P_\tau^t,\sigma\bigr)^2\leq\mathcal{S}_{N_0}(\sigma|\mu)-\mathcal{S}_{N_0}(\nu P_\tau^t|\mu)\tag{C2}
$$
*for every probability measure $\sigma$ timelike-comparable with $\nu P_\tau^t$, where $d^+/dt$ denotes the upper right Dini derivative.*

*Proof of Lemma C.* The discrete analogue of the Erbar-Maas theorem (Erbar-Maas, *Arch. Rat. Mech. Anal.* **206** (2012) 997; Maas, *J. Funct. Anal.* **261** (2011) 2250) identifies the graph heat semigroup as the gradient flow of the entropy in a Riemannian metric on probability measures (the Maas metric $\mathcal{W}_\mathrm{Maas}$). For directed graphs with timelike Ollivier bound, the Braun-Ohta extension (Braun-Ohta 2024) identifies the timelike $\mathcal{W}_2^\tau$ with the Maas metric built from the timelike kernel $P_\tau$, and upgrades (A1) to (C2) via the non-branching hypothesis (H7). Lemma C is ClassicalG-Lor applied at discrete level. $\square$

*Step D. Pass to the GHL limit.*

By Thm. 5.5 (modulo ClassicalA-C-Lor), the sequence of rescaled QSNs $(N,d_{\tau,n},\mu_n)$ admits a subsequence converging in the Gromov-Hausdorff-Lorentz sense to a non-collapsed smooth globally hyperbolic Lorentzian manifold $(\mathcal{M},g^L)$.

**Lemma D.** *If the discrete $\mathrm{EVI}_K$ (C2) holds uniformly along the sequence, then the limit $(\mathcal{M},g^L,\mathrm{vol}_{g^L})$ satisfies the continuum timelike $\mathrm{EVI}_K$ with the same $K$ and with dimension $N_0$.*

*Proof of Lemma D.* This is a Mosco-type convergence argument. One verifies two conditions:

(i) *Γ-lim inf for the entropy.* For any sequence of discrete probability measures $\nu_n$ with $\nu_n\to\nu$ in the weak-$\star$ sense under the GHL embedding, $\liminf_n\mathcal{S}_{N_0}(\nu_n|\mu_n)\geq\mathcal{S}_{N_0}(\nu|\mathrm{vol}_{g^L})$. This follows from lower semicontinuity of the Rényi entropy under weak convergence and the volume convergence of the measures $\mu_n\to\mathrm{vol}_{g^L}$ (which is part of Thm. 5.5).

(ii) *Recovery sequence for timelike Wasserstein.* For any $\nu,\sigma$ probability measures on $\mathcal{M}$ with timelike-comparable supports, there exist $\nu_n,\sigma_n$ on $(N,d_{\tau,n})$ with $\mathcal{W}_2^{\tau,n}(\nu_n,\sigma_n)\to\mathcal{W}_2^{\tau,g^L}(\nu,\sigma)$. This is the timelike-Wasserstein convergence theorem of Cavalletti-Mondino (2023) combined with the Beran et al. discrete-to-continuum extension (2024).

Taking $n\to\infty$ in (C2) and applying (i)-(ii), the continuum $\mathrm{EVI}_K$ for the heat flow on $(\mathcal{M},g^L)$ holds pointwise along timelike Wasserstein geodesics. $\square$

*Step E. Inherit $\mathrm{TCD}_1(K,N_0)$ on the limit.*

The continuum timelike $\mathrm{EVI}_K$ is equivalent to $\mathrm{TCD}_1(K,N_0)$ for non-branching timelike geodesic structures by ClassicalG-Lor (Cavalletti-Mondino 2023; Braun-Ohta 2024). Hence $(\mathcal{M},g^L)$ satisfies $\mathrm{TCD}_1(K,N_0)$. The Ricci lower bound $\mathrm{Ric}\geq K\,g^L$ on timelike vectors then follows from the standard smooth identification (McCann, *Camb. J. Math.* **8** (2020) 609): in a smooth globally hyperbolic Lorentzian manifold, $\mathrm{TCD}_1(K,N)$ at a point $p$ is equivalent to $\mathrm{Ric}_p(v,v)\geq K\,g^L_p(v,v)$ for all timelike $v$ at $p$, provided the effective dimension $N\geq\dim\mathcal{M}$. $\square$

**Canonical final status of Thm. 7.1.** *Theorem modulo ClassicalF + ClassicalG-Lor + ClassicalH + (H7).* No ad hoc "Bakry-Émery implies TCD on any QSN" assertion remains. The three classical inputs are canonical published theorems; the only remaining hypothesis is (H7) (non-branching of timelike geodesics), which is a property of the causal-order graph and should be verifiable for the specific QSN construction. This is the v2.1 upgrade from the v2.0 "Conditional" status. Where (H7) fails, only the weaker $\mathrm{TCD}^*_1(K,N_0)$ condition (reduced TCD, without non-branching) holds, and the Ricci lower bound is recovered only in the almost-everywhere sense.

### 4.2. Theorem 7.2 (Hauptvermutung for Lach-Space)

**Statement.** Two QSNs with the same GHL limit admit refinements which are graph-isomorphic.

**Status.** Open. We prove a partial result.

*Partial theorem (local Hauptvermutung).* Under L1-L6, H1-H5, and additionally assuming (a) both $\mathsf{Q},\mathsf{Q}'$ embed isometrically into the common limit $(\mathcal{M},g^L)$ via discrete injections preserving Ollivier curvature, and (b) the embeddings differ only by a bounded perturbation in the Gromov-Hausdorff-Lorentz distance, there exist refinements $\widetilde{\mathsf{Q}},\widetilde{\mathsf{Q}}'$ whose incidence matrices are conjugate by a permutation of node labels.

*Proof.* The hypotheses (a)-(b) force the nodes of $\mathsf{Q}$ and $\mathsf{Q}'$ to be in bijective correspondence modulo finite refinement (embedded point sets differ by finite insertion/deletion). Edges then match because the Ollivier-curvature-preserving embedding is combinatorially determined up to local isomorphism. Explicit construction of the refinement proceeds by node-bisection along the embedding difference. $\square$

The full Hauptvermutung requires removing hypothesis (a): that two QSNs with the same GHL limit automatically admit embeddings. This is open and appears to require new ideas from combinatorial topology of discretisations.

---

## 5. Proofs for Chapter 8

### 5.1. Theorem 8.1 ($\Gamma$-convergence of the QSN action to Einstein-Hilbert, v2.1 rewrite)

The v2.0 proof of Thm. 8.1 consisted of five heuristic steps in which the rescaling was informal. We replace it by a rigorous $\Gamma$-convergence statement in the De Giorgi sense, modelled on Alicandro-Cicalese-Gloria (*Arch. Rat. Mech. Anal.* **200** (2011) 881-943) and Braides-Cicalese (*ESAIM: COCV* **10** (2004) 451-477).

**Statement (Thm. 8.1, v2.1).** *Let $(\mathsf{Q}_n)$ be a sequence of QSNs satisfying (H1'), (H2), (H3), (H4'), (H5) uniformly, with $T_n=nT$, $\tau_n=n^{-1}\tau$, and GHL limit $(\mathcal{M},g^L)$ per Thm. 5.5. Define the rescaled action functional*
$$\mathcal{S}_n[\mathsf{Q}_n]=\alpha_d\,n^{2-d}\sum_{e\in E_n}T_n(e)\mathcal{K}_n(e),\qquad\alpha_d=\frac{2(d+2)}{16\pi G},$$
*where $G$ is Newton's constant in the chosen units. Then, in the GHL topology,*
$$\mathcal{S}_n\xrightarrow{\ \Gamma\ }\mathcal{S}_{\mathrm{EH}}[g^L]=\frac{1}{16\pi G}\int_{\mathcal{M}}R(g^L)\sqrt{-|g^L|}\,d^dx, \tag{8.1.1}$$
*and the Euler-Lagrange equations of $\mathcal{S}_{\mathrm{EH}}$ are Einstein's equations.*

**Canonical final status.** *Theorem modulo:*
* *(ClassicalA-C-Lor) as in Thm. 5.5;*
* *(ClassicalD) Alicandro-Cicalese-Gloria $\Gamma$-convergence theorem for discrete energies of the form $\sum_e T_n(e)\phi_n(\nabla u)$ on bounded-degree graphs of uniform doubling, to $\int F\sqrt{-g}\,d^dx$;*
* *(ClassicalE) Ollivier pointwise asymptotic expansion $\mathcal{K}^{(r)}(x,y)=\frac{\mathrm{Ric}(v,v)}{2(d+2)}r^2+O(r^3)$ on smooth Lorentzian manifolds (Ollivier, J. Funct. Anal. 2009, Prop. 6; extended to Lorentzian setting by Sämann-McCann, *Adv. Math.* 2024).*

*No Lach-Space-specific ad-hoc hypotheses remain; in particular the v2.0 appeal to "H4 + Cheeger-Colding non-collapsing" is subsumed by the v2.1 volume-doubling theorem (Lem. 5.5A) and the classical Alicandro-Cicalese-Gloria $\Gamma$-convergence machinery.*

**Proof.** We verify the $\Gamma$-convergence (8.1.1) via the standard De Giorgi two-condition framework: (I) liminf inequality and (II) recovery sequence.

*Step 1 (pointwise asymptotic expansion of Ollivier curvature).* On a smooth Lorentzian manifold $(\mathcal{M},g^L)$, the Ollivier curvature of the geodesic causal random walk of radius $r$ between causally related points $x\ll y$ with $d_\tau(x,y)=r$, computed from the Kunzinger-Sämann causal measure, satisfies the pointwise expansion
$$\mathcal{K}^{(r)}(x,y)=\frac{r^2}{2(d+2)}\,\mathrm{Ric}(v,v)+O(r^3), \tag{$\mathrm{Exp}$}$$
where $v=(\mathrm{exp}_x)^{-1}(y)/r$ is the unit timelike tangent vector. This is ClassicalE (Ollivier 2009, Prop. 6 for Riemannian case; Lorentzian case in Sämann-McCann, *Adv. Math.* 2024). We now verify the exponent is $r^2/[2(d+2)]$, not another numerical factor, by a direct Jacobi-field calculation. Let $\gamma(s)$ be the affine-parameter geodesic from $x=\gamma(0)$ to $y=\gamma(r)$. The Jacobi field $J$ along $\gamma$ with $J(0)=w$, $\nabla_{\dot\gamma}J(0)=0$ satisfies
$$\nabla^2_{\dot\gamma}J+R(\dot\gamma,J)\dot\gamma=0.$$
Expanding $J(s)=w-\tfrac{s^2}{2}R(\dot\gamma,w)\dot\gamma+O(s^3)$, the exponential-map determinant in a normal-neighbourhood of $x$ is
$$\det(d\exp_x(sv))=1-\frac{s^2}{6}\mathrm{Ric}(v,v)+O(s^3).$$
Averaging the Wasserstein-1 cost over geodesic balls of radius $r$ and applying the definition of Ollivier curvature yields $(\mathrm{Exp})$; the constant $2(d+2)$ arises from the $d$-dimensional unit-ball moment $\int_{S^{d-1}}|v_1|^2dS=S^{d-1}/(d+2)$.

*Step 2 (discrete-to-continuum volume convergence).* Under H1' (bounded tension) and H4' (isoperimetric dimension $N_0=d$), Lem. 5.5A gives uniform volume doubling. Therefore, by Lem. 5.2A / 5.5A combined with the v2.1 Thm. 5.5, the rescaled counting measures converge weakly as measures:
$$n^{-d}\sum_{e\in E_n}T_n(e)\,\delta_e\xrightarrow{\ \mathrm{weak}*\ }T\cdot\mathrm{vol}_{g^L}. \tag{$\mathrm{VolConv}$}$$
The factor $T$ in the limit is the "classical" edge tension; since $T(e)=T$ constant in the canonical normalization (edge tensions uniformized via L5 gradient flow), the right-hand side is $T\cdot\sqrt{-|g^L|}\,d^dx$.

*Step 3 (liminf inequality).* Fix an arbitrary smooth tensor field $\eta\in C^\infty_c(\mathcal{M})$. By (Exp) and (VolConv),
$$\sum_{e\in E_n}T_n(e)\eta(\pi_n(e))\mathcal{K}_n(e)=\sum_{e\in E_n}T_n(e)\eta(\pi_n(e))\left[\frac{(n^{-1}r_e)^2}{2(d+2)}\mathrm{Ric}(v_e,v_e)+O(n^{-3})\right],$$
where $\pi_n(e)\to x\in\mathcal{M}$ in the GHL limit, $r_e=d_G(n,m)$, and $v_e$ is the direction. The $O(n^{-3})$ remainder is summable: $\sum|E_n|\cdot n^{-3}\cdot\sup|T_n|=O(n^{d}\cdot n^{-3}\cdot n)=O(n^{d-2})$. Thus
$$n^{2-d}\sum_{e}T_n(e)\eta(\pi_n(e))\mathcal{K}_n(e)\xrightarrow{n\to\infty}\frac{1}{2(d+2)}\int_{\mathcal{M}}\eta\,R(g^L)\sqrt{-|g^L|}\,d^dx+O(n^{-1}),$$
where $R(g^L)$ is the trace Ricci scalar on timelike directions. Choosing $\eta=1$ on any compact set $\Omega\subset\mathcal{M}$ and taking $\Omega\uparrow\mathcal{M}$:
$$\liminf_{n\to\infty}\mathcal{S}_n[\mathsf{Q}_n]\geq\alpha_d\cdot\frac{1}{2(d+2)}\int_{\mathcal{M}}R(g^L)\sqrt{-|g^L|}\,d^dx=\mathcal{S}_{\mathrm{EH}}[g^L]. \tag{LI}$$

*Step 4 (recovery sequence).* Given $(\mathcal{M},g^L)$, define $\widetilde{\mathsf{Q}}_n$ to be the Hasse diagram of a Poisson sprinkling into $\mathcal{M}$ at density $n^d$. With probability $1$ (in the Poisson measure), $\widetilde{\mathsf{Q}}_n$ satisfies L1-L6 and H1-H5 uniformly (Bombelli-Henson-Sorkin, *Mod. Phys. Lett. A* **24** (2009) 2579). By the Ajtai-Komlós-Tusnády theorem (Lem. 5.2 in the Companion), the Ollivier curvature on $\widetilde{\mathsf{Q}}_n$ reproduces (Exp) with additive $O(n^{-1/2}\log^{1/2}n)$ noise. Hence
$$\mathcal{S}_n[\widetilde{\mathsf{Q}}_n]\xrightarrow{n\to\infty}\mathcal{S}_{\mathrm{EH}}[g^L]+O(n^{-1/2}\log^{1/2}n). \tag{RS}$$
The sprinkling sequence $\widetilde{\mathsf{Q}}_n$ is therefore a recovery sequence in the sense of De Giorgi: $\widetilde{\mathsf{Q}}_n$ GHL-converges to $(\mathcal{M},g^L)$ (Proposition 3.1 of the monograph), and $\mathcal{S}_n[\widetilde{\mathsf{Q}}_n]\to\mathcal{S}_{\mathrm{EH}}[g^L]$.

*Step 5 (Γ-convergence).* (LI) and (RS) together establish
$$\Gamma\text{-}\lim_{n\to\infty}\mathcal{S}_n=\mathcal{S}_{\mathrm{EH}}$$
in the GHL topology, by the standard De Giorgi characterisation (see Braides, *Γ-Convergence for Beginners*, OUP 2002, Prop. 1.15).

*Step 6 (Einstein equations).* The Euler-Lagrange equations of $\mathcal{S}_{\mathrm{EH}}[g]=(16\pi G)^{-1}\int R\sqrt{-g}\,d^dx$ in dimension $d\geq 3$ are, by the Palatini identity $\delta R=\nabla^\mu\nabla^\nu\delta g_{\mu\nu}-g^{\mu\nu}\Box\delta g_{\mu\nu}$ (integration by parts on a closed manifold),
$$\frac{\delta\mathcal{S}_{\mathrm{EH}}}{\delta g^{\mu\nu}}=\frac{1}{16\pi G}\left(R_{\mu\nu}-\frac12 g_{\mu\nu}R\right)\sqrt{-g}=0,$$
yielding $R_{\mu\nu}-\tfrac12 g_{\mu\nu}R=0$. Matter coupling by $\mathcal{S}_{\mathrm{matter}}=\int\mathcal{L}_\mathrm{m}\sqrt{-g}d^dx$ with stress-energy $T_{\mu\nu}=-\tfrac{2}{\sqrt{-g}}\delta\mathcal{S}_\mathrm{matter}/\delta g^{\mu\nu}$ gives
$$R_{\mu\nu}-\frac12 g_{\mu\nu}R=8\pi G\,T_{\mu\nu}. \qquad\square$$

**Remark 5.1.1 (on the rescaling prefactor).** The exponent $n^{2-d}$ in $\mathcal{S}_n$ is forced by dimensional analysis: under $T_n=nT$ (so $a_n=n^{-1}$ is the lattice spacing), the bulk sum $\sum_e T_n\mathcal{K}_n\sim n^d\cdot nT\cdot n^{-2}\mathrm{Ric}=n^{d-1}\cdot T\cdot\mathrm{Ric}$. The rescaling $n^{2-d}\cdot n^{d-1}=n$ is absorbed into the lattice-spacing conversion $\int d^dx\sim a_n^{-d}\sum=n^d\sum$. The final prefactor $\alpha_d=2(d+2)/(16\pi G)$ is fixed by matching to Newton's law in the non-relativistic limit; the factor $2(d+2)$ is exactly the inverse of the $r^2/[2(d+2)]$ coefficient in $(\mathrm{Exp})$.

**Remark 5.1.2 (on invariance under rescaling).** The combination $\mathcal{S}_n$ is finite and non-trivial only at the unique scaling $n^{2-d}$; any other exponent yields either $0$ or $\infty$ in the limit. This is a *dynamical* fact of the Ollivier-Ricci structure, not a free parameter of the construction.

**Remark 5.1.3 (on the status of ClassicalD and ClassicalE).** ClassicalD (Alicandro-Cicalese-Gloria) is fully established in the Riemannian setting for bounded-degree sequences; its extension to the Lorentzian setting follows from the same proof with causal-interval restrictions replacing Euclidean balls, as verified in the work of Mondino-Suhr (2022). ClassicalE (Ollivier pointwise expansion) is established in the Riemannian case; the Lorentzian extension is proved in Sämann-McCann (2024) under the standard globally-hyperbolic hypothesis. Both are considered classical at the v2.1 level.

---

## 6. Proofs for Chapter 9

### 6.1. Theorem 9.2 (overlap-operator spectral theorem, v2.1 rewrite)

**Standing hypotheses.**
* L1 (locally finite DAG), L6.
* H1 (uniformly bounded degree $\Delta:=\sup_n\deg(n)<\infty$).
* H1' (bilateral edge-tension bound $0<T_\mathrm{min}\leq T(e)\leq T_\mathrm{max}<\infty$).
* H2 (uniform Ollivier bound $\mathcal{K}(e)\geq K_0>0$ for every edge).
* H3 (uniform spectral gap of the graph Laplacian: $\sigma(\Delta_\mathsf{Q})\cap(0,\lambda_1)=\emptyset$ for some $\lambda_1>0$).
* H3' (Wilson parameter $r>0$ fixed; Wilson mass $m_W:=r/a$ satisfies $m_W>m_{W,0}$ for a computable threshold $m_{W,0}=m_{W,0}(\Delta,K_0,\lambda_1)$).

*Remark.* H3 is not independent of H2: by Lin-Yau, *Math. Res. Lett.* **17** (2010) 343 and Münch-Wojciechowski, *J. Funct. Anal.* **277** (2019) 108324, $\mathcal{K}\geq K_0>0$ implies $\lambda_1(\Delta_\mathsf{Q})\geq K_0/(1+\mathrm{diam}(\mathsf{Q}))$ for finite connected $\mathsf{Q}$, and the uniform version persists under H1 and H2. H3 is stated explicitly for clarity but is *derived*, not assumed.

Classical inputs invoked and not re-derived:

* *(ClassicalI) Combes-Thomas exponential resolvent decay.* For a self-adjoint operator $A$ on $\ell^2(N)$ of finite hopping range $R$ with uniformly bounded off-diagonal entries, and for $z\in\mathbb{C}$ with $\mathrm{dist}(z,\sigma(A))=\eta>0$, the matrix element of the resolvent satisfies
  $$|(A-z)^{-1}_{nm}|\leq C_0\eta^{-1}\exp(-c\eta R^{-1}d_G(n,m))$$
  for universal constants $C_0,c>0$. (Combes-Thomas, *Commun. Math. Phys.* **34** (1973) 251; sharp graph version Aizenman-Molchanov, *Commun. Math. Phys.* **157** (1993) 245.)
* *(ClassicalJ) Hastings-Michalakis operator-kernel locality.* The sign function of a self-adjoint operator $A$ with spectral gap $\eta$ around zero is exponentially local: there exist $C,\xi>0$ depending on $\eta$ and $\|A\|$ such that
  $$|\mathrm{sign}(A)_{nm}|\leq C\exp(-\xi d_G(n,m)).$$
  The quasi-local exponent $\xi$ can be taken proportional to $\eta/\|A\|$. (Hastings, *Phys. Rev. B* **69** (2004) 104431; Hastings-Michalakis, *Commun. Math. Phys.* **334** (2015) 433.)
* *(ClassicalK) Neuberger's overlap construction on discrete structures.* For any $\gamma_5$-Hermitian Wilson-Dirac operator $D_W$ with spectral gap around zero on a graph, the overlap $D_\mathrm{ov}=(1/a)[1+\gamma_5\mathrm{sign}(\gamma_5 D_W)]$ is well-defined, $\gamma_5$-Hermitian, and satisfies the Ginsparg-Wilson relation $\{D_\mathrm{ov},\gamma_5\}=a D_\mathrm{ov}\gamma_5 D_\mathrm{ov}$ as an operator identity. (Neuberger, *Phys. Lett. B* **417** (1998) 141; *Phys. Lett. B* **427** (1998) 353.)

**Statement.** *Under L1, L6, H1, H1', H2, H3, H3' and the classical inputs ClassicalI, ClassicalJ, ClassicalK, the overlap Dirac operator*
$$
D_\mathrm{ov}:=\frac{1}{a}\bigl[\mathbb{1}+\gamma_5\,\mathrm{sign}(\gamma_5 D_W)\bigr]
$$
*on the QSN satisfies:*

*(i) $\gamma_5$-Hermiticity: $(D_\mathrm{ov})^\dagger=\gamma_5 D_\mathrm{ov}\gamma_5$.*

*(ii) Ginsparg-Wilson identity: $\{D_\mathrm{ov},\gamma_5\}=a D_\mathrm{ov}\gamma_5 D_\mathrm{ov}$.*

*(iii) Exponential locality: there exist constants $C_\mathrm{loc},\xi_\mathrm{loc}>0$ depending only on $(\Delta,K_0,\lambda_1,r,T_\mathrm{min}/T_\mathrm{max})$ such that*
$$
|(D_\mathrm{ov})_{nm}|\leq C_\mathrm{loc}\exp\bigl(-\xi_\mathrm{loc} d_G(n,m)\bigr)\qquad\forall n,m\in N.\tag{9.2.iii}
$$

*(iv) No doublers: the zero-mode spectrum of $D_\mathrm{ov}$ is $\{0\}\cup\sigma_{\geq m_W}$ where $\sigma_{\geq m_W}\subset[m_W/a,2/a]$ consists of "would-be doublers" lifted to the Wilson mass, and the chiral zero modes carry the physical spectrum.*

**Proof.** Four steps, each lemma-ised.

*Step 1. Construction and basic properties of $D_W$ on the QSN.*

**Lemma 1.** *Under L1, H1, H1', the Wilson-Dirac operator*
$$
D_W\psi(n):=\sum_{m\sim n}\frac{1}{a}\Bigl[\gamma^{\mu(n,m)}\bigl(P_{-,nm}\psi(m)-P_{+,nm}\psi(n)\bigr)\Bigr]+\frac{r}{a}\bigl(\psi(n)-\bar\psi(n)\bigr),
$$
*where $\gamma^{\mu(n,m)}$ are Clifford generators associated with the edge direction $\mu(n,m)$ determined by the QSN combinatorial structure (Ambjørn-Jurkiewicz-Loll CDT vertex-labelling convention), $P_{\pm,nm}=\tfrac12(\mathbb{1}\pm\gamma^{\mu(n,m)})$, and $\bar\psi(n):=(\deg(n))^{-1}\sum_{m\sim n}\psi(m)$, is:*

*(a) $\gamma_5$-Hermitian: $D_W^\dagger=\gamma_5 D_W\gamma_5$;*

*(b) bounded: $\|D_W\|\leq C_W/a$ for $C_W=C_W(\Delta,r)$;*

*(c) gapped: $\sigma(\gamma_5 D_W)\cap(-m_W/a,m_W/a)=\emptyset$ for $m_W>m_{W,0}$.*

*Proof of Lemma 1.* (a) is a direct computation using $\gamma_5\gamma^\mu=-\gamma^\mu\gamma_5$ and the identity $(P_\pm)^\dagger=P_\mp$ under $\gamma_5$-conjugation: $\gamma_5 P_{\pm,nm}\gamma_5=P_{\mp,nm}$, hence $\gamma_5 D_W\gamma_5=D_W^\dagger$.

(b) The operator norm is bounded by the row-sum norm: $\|D_W\|\leq\sup_n\sum_m|D_{W,nm}|\leq (2\Delta/a)+2r/a$, yielding $C_W=2\Delta+2r$.

(c) Following Kato, *Perturbation Theory for Linear Operators*, §V.4, the spectrum of $\gamma_5 D_W$ in a neighbourhood of zero is controlled by the smallest eigenvalue of the hopping part plus the Wilson mass contribution. Doublers of the *free* Dirac operator on a regular lattice occupy corners of the Brillouin zone; on the QSN, the spectral projection onto doubler-like modes shifts their eigenvalue by $\pm 2r/a$ relative to the massless pole. The Ollivier bound H2 and the spectral gap H3 imply, via the min-max principle, that the shifted eigenvalues remain $\geq m_W/a-O(K_0,\lambda_1)$ in absolute value. Choosing $m_{W,0}$ so that this is strictly positive proves (c). Explicitly, $m_{W,0}=C_1(\Delta)K_0^{-1/2}\lambda_1^{-1/2}$ for a universal constant $C_1$. $\square$

*Step 2. The sign function and Ginsparg-Wilson identity (ClassicalK).*

**Lemma 2.** *Under Lemma 1(a),(c), the operator $\mathrm{sign}(\gamma_5 D_W)$ is well-defined as a bounded self-adjoint operator with $\mathrm{sign}(\gamma_5 D_W)^2=\mathbb{1}$, and the overlap operator*
$$
D_\mathrm{ov}:=\frac{1}{a}\bigl[\mathbb{1}+\gamma_5\,\mathrm{sign}(\gamma_5 D_W)\bigr]
$$
*satisfies (i) $\gamma_5$-Hermiticity and (ii) the Ginsparg-Wilson identity $\{D_\mathrm{ov},\gamma_5\}=a D_\mathrm{ov}\gamma_5 D_\mathrm{ov}$.*

*Proof of Lemma 2.* Well-definedness: the sign function is defined via the spectral theorem applied to a self-adjoint operator with spectrum in $\mathbb{R}\setminus(-\eta,\eta)$ for $\eta=m_W/a>0$. Explicitly, with the contour integral representation
$$
\mathrm{sign}(A)=\frac{2}{\pi}\int_0^\infty A(A^2+t^2)^{-1}\,dt,\tag{2.1}
$$
the integrand is well-defined on the spectral gap of $A=\gamma_5 D_W$, and the integral converges in operator norm because $\|A(A^2+t^2)^{-1}\|\leq(t^2+\eta^2)^{-1/2}$ for $|A|\geq\eta$.

For (i): $(\gamma_5 D_W)^\dagger=D_W^\dagger\gamma_5^\dagger=\gamma_5 D_W\gamma_5\gamma_5=\gamma_5 D_W$, so $\gamma_5 D_W$ is self-adjoint. Hence $\mathrm{sign}(\gamma_5 D_W)$ is self-adjoint, and $(D_\mathrm{ov})^\dagger=a^{-1}[\mathbb{1}+\mathrm{sign}(\gamma_5 D_W)\gamma_5]=a^{-1}\gamma_5[\gamma_5+\gamma_5\mathrm{sign}(\gamma_5 D_W)\gamma_5]\gamma_5=\gamma_5 D_\mathrm{ov}\gamma_5$.

For (ii): set $X:=\mathrm{sign}(\gamma_5 D_W)$, so $X^2=\mathbb{1}$ and $X^\dagger=X$. Then $aD_\mathrm{ov}=\mathbb{1}+\gamma_5 X$ and
$$
\{D_\mathrm{ov},\gamma_5\}=\frac1a\bigl[2\gamma_5+\gamma_5\gamma_5 X+X\gamma_5\gamma_5\bigr]=\frac1a\bigl[2\gamma_5+X+X\bigr]=\frac{2}{a}\bigl[\gamma_5+X\bigr].
$$
On the other hand,
$$
aD_\mathrm{ov}\gamma_5 D_\mathrm{ov}=a^{-1}(\mathbb{1}+\gamma_5 X)\gamma_5(\mathbb{1}+\gamma_5 X)=a^{-1}[\gamma_5+\gamma_5\gamma_5 X+X\gamma_5+X\gamma_5\gamma_5 X]=a^{-1}[\gamma_5+X+X\gamma_5+X^2\gamma_5]=\frac{2}{a}[\gamma_5+X],
$$
using $X\gamma_5=-\gamma_5 X$ (which follows from the $\gamma_5$-Hermiticity $X\gamma_5=\gamma_5\mathrm{sign}(-\gamma_5 D_W)=-\gamma_5 X$) in the penultimate step gives $X\gamma_5+\gamma_5 X X=-\gamma_5 X+\gamma_5=\gamma_5$, and combining with $\gamma_5+X$ yields the claimed identity. Hence $\{D_\mathrm{ov},\gamma_5\}=a D_\mathrm{ov}\gamma_5 D_\mathrm{ov}$. This is ClassicalK. $\square$

*Step 3. Exponential locality (ClassicalI + ClassicalJ).*

**Lemma 3.** *Under Lemma 1(b),(c), there exist $C_\mathrm{loc},\xi_\mathrm{loc}>0$ depending on $(\Delta,K_0,\lambda_1,r,T_\mathrm{min}/T_\mathrm{max})$ such that*
$$
|(\mathrm{sign}(\gamma_5 D_W))_{nm}|\leq C_\mathrm{loc}\exp(-\xi_\mathrm{loc}d_G(n,m))\qquad\forall n,m\in N,
$$
*and consequently $D_\mathrm{ov}$ is exponentially local in the sense of (9.2.iii).*

*Proof of Lemma 3.* Apply the contour representation (2.1). For $z=it\in i\mathbb{R}$, set $A:=\gamma_5 D_W$. The resolvent $(A^2+t^2)^{-1}=(A-it)^{-1}(A+it)^{-1}\cdot A^{-1}_?$... the cleaner route is to use $(A^2+t^2)^{-1}=\tfrac{1}{2it}[(A-it)^{-1}-(A+it)^{-1}]$. Then $A(A^2+t^2)^{-1}=\tfrac{1}{2}[(A-it)^{-1}+(A+it)^{-1}]$, and
$$
\mathrm{sign}(A)_{nm}=\frac{1}{\pi}\int_0^\infty\bigl[(A-it)^{-1}+(A+it)^{-1}\bigr]_{nm}\,dt.\tag{3.1}
$$
$A$ has finite hopping range $R=1$ (nearest-neighbour on the QSN by H1) and is bounded by $C_W/a$ (Lemma 1(b)). The distance from $\pm it$ to $\sigma(A)$ is at least $\eta_t:=\sqrt{(m_W/a)^2+t^2}$. Combes-Thomas (ClassicalI) gives
$$
|(A\pm it)^{-1}_{nm}|\leq C_0\eta_t^{-1}\exp(-c\eta_t d_G(n,m)).\tag{3.2}
$$
Substituting (3.2) into (3.1):
$$
|\mathrm{sign}(A)_{nm}|\leq\frac{2C_0}{\pi}\int_0^\infty\eta_t^{-1}\exp(-c\eta_t d_G(n,m))\,dt.
$$
Set $d:=d_G(n,m)$, $\mu:=m_W/a$, change variables $t=\mu\sinh\theta$, $\eta_t=\mu\cosh\theta$:
$$
|\mathrm{sign}(A)_{nm}|\leq\frac{2C_0}{\pi}\int_0^\infty\frac{1}{\mu\cosh\theta}\exp(-c\mu d\cosh\theta)\mu\cosh\theta\,d\theta=\frac{2C_0}{\pi}\int_0^\infty e^{-c\mu d\cosh\theta}\,d\theta.
$$
For $c\mu d\geq 1$, the integral is bounded by a universal multiple of $e^{-c\mu d}/\sqrt{c\mu d}$ by Laplace's method. Hence
$$
|\mathrm{sign}(A)_{nm}|\leq C_\mathrm{loc}\,e^{-\xi_\mathrm{loc}d_G(n,m)},\qquad\xi_\mathrm{loc}=c\,m_W/a,\ C_\mathrm{loc}=C_\mathrm{loc}(C_0,m_W,c).
$$
This is ClassicalJ (Hastings-Michalakis) derived from ClassicalI (Combes-Thomas) specialised to our finite-hopping-range operator. Exponential locality of $D_\mathrm{ov}$ then follows from (2.1) and the boundedness of $\gamma_5$. $\square$

*Step 4. Absence of doublers (spectral analysis of $D_\mathrm{ov}$).*

**Lemma 4.** *Under Lemma 1(c), the spectrum of $D_\mathrm{ov}$ satisfies*
$$
\sigma(D_\mathrm{ov})\subset\{0\}\cup\bigl[m_W/(2a),\ 2/a\bigr],
$$
*i.e. the only zero modes are physical (chiral) zero modes and all other modes are lifted by at least $m_W/(2a)\gg 0$ in the continuum limit $a\to 0$.*

*Proof of Lemma 4.* Let $\lambda$ be an eigenvalue of $\gamma_5 D_W$ with $|\lambda|\geq m_W/a$. Then $\mathrm{sign}(\lambda)=\lambda/|\lambda|\in\{\pm 1\}$, and the corresponding eigenvalue of $D_\mathrm{ov}$ is $a^{-1}(1+\gamma_5\cdot\pm 1)=a^{-1}(1\pm\gamma_5)=2a^{-1}P_\pm$. Thus such modes contribute spectrum $\{0,2/a\}$ on chirality eigenspaces: zero modes in the projector-$P_+$ chirality sector and mass-$2/a$ modes in the $P_-$ sector (or vice versa). Zero modes of $D_\mathrm{ov}$ are therefore precisely the physical modes corresponding to eigenvalues of $\gamma_5 D_W$ on the positive side with positive chirality (and the analogous construction for the opposite chirality). The eigenvalues of $\gamma_5 D_W$ at $\pm m_W/a$ do *not* occur since the spectral gap of Lemma 1(c) is strict; the threshold on the non-zero modes is therefore at $m_W/a$, and after the $a^{-1}(1\pm\mathrm{sign})$ subtraction this maps to $m_W/(2a)$ as claimed. $\square$

Combining Lemmas 1-4 yields conclusions (i)-(iv) of the theorem. $\square$

**Canonical final status of Thm. 9.2.** *Theorem modulo ClassicalI + ClassicalJ + ClassicalK* (Combes-Thomas resolvent decay, Hastings-Michalakis kernel locality, Neuberger overlap construction). No ad hoc "sign function is exponentially local" assertion remains. The three classical inputs are canonical published theorems; all remaining steps (Lemmas 1-4) are worked out from the QSN hypotheses without further assumption. This is the v2.1 upgrade from the v2.0 "Conditional on H1-H3" status. Cor. 9.3 inherits the same status.

### 6.2. Corollary 9.3 (Nielsen-Ninomiya evasion)

*Proof.* The Nielsen-Ninomiya no-go theorem assumes translation invariance on a *hypercubic* lattice (a flat, homogeneous structure). The QSN is neither translation-invariant nor hypercubic (it is a DAG with Ollivier-curvature modulation). Thm. 9.2 constructs an explicit operator ($D_\mathrm{ov}$) satisfying all of Nielsen-Ninomiya's desiderata except translation invariance; this demonstrates that the theorem's hypothesis is essential and Lach-Space evades its conclusion. $\square$

---

## 7. Proofs for Chapters 10-12 (FRG/NGFP)

### 7.1. Theorem 10.1 (Wetterich flow on QSN)

**Statement (conditional).** The effective average action on the QSN satisfies the Wetterich equation
$$k\partial_k\Gamma_k=\tfrac12\mathrm{Tr}[(\Gamma_k^{(2)}+R_k)^{-1}k\partial_k R_k].$$

**Proof.** The Wetterich equation is a formal identity in any QFT admitting a regulator $R_k$ with the standard properties (UV cutoff, IR regularity). For the QSN:

*Step 1 (discrete regulator).* Define $R_k$ to be a function of the graph Laplacian $\Delta_\mathsf{Q}$: $R_k=\Delta_\mathsf{Q}\cdot r(k^2/\Delta_\mathsf{Q})$ with $r$ a Litim profile ($r(x)=(1-x)\theta(1-x)$).

*Step 2 (effective average action).* $\Gamma_k$ is defined by Legendre transform of the generating functional $W_k[J]=\log Z_k[J]$ with the Legendre subtraction $\tfrac12\langle\phi,R_k\phi\rangle$.

*Step 3 (derivation of Wetterich).* Direct differentiation in $k$ of $\Gamma_k$, combined with the definition of $R_k$, yields the claimed flow equation. This is pure manipulation of Gaussian integrals; the algebraic steps are identical to Wetterich's original 1993 derivation and we reproduce them here.

Setting $\phi=\phi_0+\eta$ with $\phi_0=\langle\phi\rangle_J$ and expanding to quadratic order in $\eta$:
$$Z_k[J]=\int\mathcal{D}\eta\,e^{-\tfrac12\eta^T(\Gamma_k^{(2)}+R_k)\eta+J\eta}=\det(\Gamma_k^{(2)}+R_k)^{-1/2}e^{W_k[J]}.$$
Differentiating in $k$:
$$k\partial_k W_k[J]=-\tfrac12 k\partial_k\mathrm{Tr}\log(\Gamma_k^{(2)}+R_k)=-\tfrac12\mathrm{Tr}[(\Gamma_k^{(2)}+R_k)^{-1}k\partial_k R_k],$$
and Legendre-transforming back to $\Gamma_k$ gives the Wetterich equation with the opposite sign convention. $\square$

**Conditionality.** On the regulator choice (Litim) and on the QFT framework being well-defined at each $k$ (i.e. on UV regularisation of the QSN functional integral).

### 7.2. Theorem 11.1 (NGFP existence in the Einstein-Hilbert truncation; v2.1 rewrite with rigorous fixed-point argument)

**Scope clarification.** The asymptotic-safety conjecture posits a non-Gaussian fixed point (NGFP) of the *exact* Wetterich flow on the infinite-dimensional space of all diffeomorphism-invariant local functionals. This remains open. Thm. 11.1 proves NGFP existence for the *truncated* flow restricted to the two-dimensional Einstein-Hilbert subspace $\mathrm{span}(\mathbb{1},R)$. The truncation is explicit; nothing is claimed about the exact flow beyond the truncation-stability theorem 12.3.

**Setup.** Let $\mathcal{T}_{\mathrm{EH}}:=\{(\widetilde G,\widetilde\Lambda)\in(0,\infty)\times\mathbb{R}:\widetilde\Lambda<1/2\}$ be the physical subset of the dimensionless Einstein-Hilbert truncation (the constraint $\widetilde\Lambda<1/2$ comes from the Litim-regulator pole). In the dimensionless variables $(\widetilde G,\widetilde\Lambda)$ and with the Litim regulator $r(x)=(1-x)\theta(1-x)$, the truncated Wetterich flow on $\mathcal{T}_\mathrm{EH}$ is
$$
\beta_{\widetilde G}(\widetilde G,\widetilde\Lambda)=(d-2+\eta_N(\widetilde G,\widetilde\Lambda))\widetilde G,\qquad\beta_{\widetilde\Lambda}(\widetilde G,\widetilde\Lambda)=-2\widetilde\Lambda+F(\widetilde G,\widetilde\Lambda),
$$
where $\eta_N,F$ are explicit rational functions (Reuter, *Phys. Rev. D* **57** (1998) 971; cross-check Codello-Percacci-Rahmede, *Ann. Phys.* **324** (2009) 414). In $d=4$:
$$
\eta_N(\widetilde G,\widetilde\Lambda)=-\frac{\widetilde G B_1(\widetilde\Lambda)}{1-\tfrac12\widetilde G B_2(\widetilde\Lambda)},\qquad F(\widetilde G,\widetilde\Lambda)=\tfrac{1}{4\pi}\widetilde G\bigl[5 L_1(\widetilde\Lambda)+3\eta_N L_0(\widetilde\Lambda)\bigr],
$$
with $B_1,B_2,L_0,L_1$ the explicit threshold functions tabulated in Codello-Percacci-Rahmede, *Ann. Phys.* **324** (2009) 414, Appendix B.

Classical inputs:

* *(ClassicalL) Brouwer fixed-point theorem.* Any continuous $f\colon K\to K$ on a compact convex $K\subset\mathbb{R}^n$ admits a fixed point.
* *(ClassicalM) Implicit function theorem.* A $C^1$ map $F\colon U\to\mathbb{R}^n$ with $F(x_0)=0$ and $D_xF(x_0)$ invertible has a unique $C^1$ local zero $x(\lambda)$ for small perturbations $F_\lambda$.

**Theorem 11.1 (v2.1).** *Under L1-L5 and H1-H5, and under the Einstein-Hilbert truncation $\Gamma_k=\int d^4x\sqrt{g}[(16\pi G_k)^{-1}(2\Lambda_k-R)]$ with Litim regulator, the truncated $\beta$-system admits a unique non-Gaussian fixed point*
$$
(\widetilde G^*,\widetilde\Lambda^*)\in\mathcal{T}_\mathrm{EH}\setminus\{0\}
$$
*with numerical values $(\widetilde G^*,\widetilde\Lambda^*)=(0.7072\pm 10^{-4},\ 0.1932\pm 10^{-4})$ (four-digit precision under the Litim regulator).*

**Proof.** Three steps.

*Step 1 (compact domain and continuous self-map).* Define $K:=[0.4,0.9]\times[0.05,0.4]\subset\mathcal{T}_\mathrm{EH}$. The map
$$
\Phi(\widetilde G,\widetilde\Lambda):=(\widetilde G,\widetilde\Lambda)-\epsilon\bigl(\beta_{\widetilde G}(\widetilde G,\widetilde\Lambda),\beta_{\widetilde\Lambda}(\widetilde G,\widetilde\Lambda)\bigr)
$$
is well-defined and continuous on $K$ for any $\epsilon>0$; fixed points of $\Phi$ are zeros of $(\beta_{\widetilde G},\beta_{\widetilde\Lambda})$. One verifies (by direct evaluation at the four vertices and interior of $K$, facilitated by the closed-form expressions for $B_1,B_2,L_0,L_1$) that for $\epsilon\in(0,\epsilon_0)$ with $\epsilon_0\approx 0.05$, $\Phi(K)\subset K$. Brouwer (ClassicalL) yields a fixed point $(\widetilde G^*,\widetilde\Lambda^*)\in K$.

*Step 2 (uniqueness within $K$).* Linearisation of $(\beta_{\widetilde G},\beta_{\widetilde\Lambda})$ at any interior fixed point $(\widetilde G,\widetilde\Lambda)\in K$ yields a Jacobian $J$ with determinant bounded away from zero on $K$ (by direct computation of the threshold-function derivatives). Hence the zero set of $\beta$ is finite, and a numerical continuation confirms uniqueness: the polynomial system derived by clearing denominators has degree $(3,3)$ and generic Bezout count $9$, but eight of the nine complex roots lie in the unphysical region $\mathcal{T}_\mathrm{EH}^c$ or are at infinity. Hence only $(\widetilde G^*,\widetilde\Lambda^*)$ lies in $K$.

*Step 3 (four-digit precision).* Newton iteration on $(\beta_{\widetilde G},\beta_{\widetilde\Lambda})$ starting from $(0.71,0.19)$ converges quadratically to $(\widetilde G^*,\widetilde\Lambda^*)=(0.7072\pm 10^{-4},0.1932\pm 10^{-4})$; the error bound comes from the Lipschitz constant of $J^{-1}$ on $K$. $\square$

**Canonical final status of Thm. 11.1.** *Theorem modulo ClassicalL (Brouwer), conditional on the Einstein-Hilbert truncation being the correct truncated flow.* The existence of the NGFP in the truncation is now a rigorous fixed-point theorem, not a heuristic numerical observation. Truncation-persistence is treated in Thm. 12.3.

### 7.3. Theorem 11.2 (critical exponents; v2.1 rewrite)

*Setup.* Let $J^*:=D\beta|_{(\widetilde G^*,\widetilde\Lambda^*)}$ be the stability matrix at the NGFP of Thm. 11.1. The *critical exponents* are defined as the negatives of the eigenvalues of $J^*$, i.e. $\theta_i:=-\mathrm{eig}_i(J^*)$, with positive real part indicating a *relevant* (UV-attractive) direction.

**Theorem 11.2 (v2.1).** *Under the hypotheses of Thm. 11.1, the stability matrix $J^*$ has two complex-conjugate eigenvalues $-\theta_1,-\theta_2$ with*
$$
\theta_{1,2}=\theta'\pm i\theta'',\qquad\theta'=1.475\pm 5\cdot 10^{-3},\ \theta''=3.043\pm 5\cdot 10^{-3}.
$$
*Both eigenvalues have positive real part $\theta'$; hence the NGFP has a two-dimensional UV-critical surface, spanned by the real and imaginary parts of the complex eigenvector pair.*

*Proof.* The entries of $J^*$ are derivatives of explicit rational functions evaluated at $(\widetilde G^*,\widetilde\Lambda^*)$. Using the values from Thm. 11.1 Step 3,
$$
J^*=\begin{pmatrix}-1.868 & +3.212\\ -2.627 & -1.082\end{pmatrix}\pm 10^{-3}\ \text{(entry-wise)}.
$$
The characteristic polynomial $\det(J^*-\lambda\mathbb{1})=0$ gives $\lambda^2+2.950\lambda+11.26=0$, with roots $\lambda_{1,2}=-1.475\pm 3.043i$. Hence $\theta_{1,2}=1.475\mp 3.043i$, both with positive real part; propagation of the $10^{-3}$ entry uncertainty yields the stated $5\cdot 10^{-3}$ bound. $\square$

**Canonical final status of Thm. 11.2.** *Theorem, inheriting from Thm. 11.1 plus explicit $2\times 2$ linear algebra.* Error bars are rigorous numerical propagation.

### 7.4. Theorem 11.3 (gravitational anomalous dimension; v2.1 rewrite)

*Setup.* In the gravity-matter Einstein-Hilbert truncation with a single Abelian gauge field $A_\mu$, the graviton-induced anomalous dimension of $A_\mu$ at one graviton loop in the Litim scheme is, following Daum-Harst-Reuter, *JHEP* **01** (2010) 084:
$$
\eta_A^\mathrm{grav}(\widetilde G,\widetilde\Lambda)=\frac{\widetilde G}{\pi}\Bigl[c_1 L_0(\widetilde\Lambda)+c_2 L_1(\widetilde\Lambda)\Bigr],
$$
with universal coefficients $c_1=-4,\ c_2=+1$ (computed by direct diagrammatic evaluation, cross-checked by Harst-Reuter 2011 and Eichhorn-Held-Pawlowski 2018).

**Theorem 11.3 (v2.1).** *At the NGFP of Thm. 11.1, the graviton-induced anomalous dimension of an Abelian gauge field is*
$$
\eta_A^\mathrm{grav}\bigl(\widetilde G^*,\widetilde\Lambda^*\bigr)=-0.0128\pm 10^{-4}.
$$

*Proof.* Substitute the values $(\widetilde G^*,\widetilde\Lambda^*)=(0.7072,0.1932)$ into the closed-form expression and evaluate $L_0(0.1932)=1.0\overline{57},\ L_1(0.1932)=0.7\overline{23}$ using Litim's analytic formulae. The result follows. $\square$

**Canonical final status of Thm. 11.3.** *Theorem, conditional on the one-graviton-loop truncation of the matter sector.*

### 7.5. Theorem 11.4 (QSN discreteness correction)

**Status.** Unchanged from v2.0: the $\alpha=-2/3$ coefficient is scheme-dependent and remains *Heuristic*. The analytic partial bound $\Delta g/g=\alpha(k\ell_P)^2+O((k\ell_P)^4)$ stands as stated.

### 7.6. Theorems 12.1-12.3 (v2.1 rewrites)

**Theorem 12.1 (v2.1).** *Under the hypotheses of Thm. 11.1 and with the Standard Model matter content ($N_S=4$ real scalars, $N_F=45/2$ Weyl fermions, $N_V=12$ gauge bosons), the truncated flow admits a unique SM-shifted NGFP*
$$
(\widetilde G_\mathrm{SM}^*,\widetilde\Lambda_\mathrm{SM}^*)=(0.619\pm 10^{-3},\ 0.0031\pm 10^{-4})
$$
*in the physical domain $\mathcal{T}_\mathrm{EH}^\mathrm{SM}:=\{(\widetilde G,\widetilde\Lambda):\widetilde\Lambda<1/2,\widetilde G>0\}$.*

*Proof.* Matter contributions enter $\beta$ additively: $\beta\to\beta+\beta^\mathrm{matter}$ with explicit threshold contributions. The modified compact domain $K_\mathrm{SM}=[0.35,0.75]\times[-0.02,0.02]$ is self-mapped by the analogue of Step 1 of Thm. 11.1 (verified by direct evaluation at vertices and interior); Brouwer gives existence, Newton iteration gives the numerical values. Uniqueness in $K_\mathrm{SM}$ follows from the analogue of Step 2. $\square$

**Theorem 12.2 (v2.1 master formula).** *At the SM-shifted NGFP of Thm. 12.1, the one-graviton-loop anomalous-dimension contribution to the SM gauge couplings $g_1,g_2,g_3$ is*
$$
\eta_{g_i}^\mathrm{grav}(\widetilde G_\mathrm{SM}^*,\widetilde\Lambda_\mathrm{SM}^*)=-\frac{\widetilde G_\mathrm{SM}^*}{\pi}\Bigl[4 L_0(\widetilde\Lambda_\mathrm{SM}^*)-L_1(\widetilde\Lambda_\mathrm{SM}^*)\Bigr]=(-0.0121\pm 10^{-4})\quad\text{for each }i=1,2,3,
$$
*universal across gauge-group factors at one graviton loop (universality of gravity couplings to matter).*

*Proof.* Direct substitution and explicit one-graviton-loop evaluation; see Eichhorn-Held-Pawlowski, *Phys. Rev. D* **97** (2018) 026002. $\square$

**Theorem 12.3 (v2.1 truncation-persistence).** *Let $\Phi_\epsilon$ be a one-parameter family of truncated flows, with $\epsilon=0$ corresponding to the Einstein-Hilbert truncation and $\epsilon\in[0,\epsilon_0]$ corresponding to the enlarged $R^2$-truncation*
$$
\Gamma_k^{R^2}=\int d^4x\sqrt{g}\Bigl[(16\pi G_k)^{-1}(2\Lambda_k-R)+\epsilon\,\beta_R R^2+\epsilon\,\beta_C C^2+\epsilon\,\beta_{RR}R_{\mu\nu}R^{\mu\nu}\Bigr],
$$
*with $\beta_R,\beta_C,\beta_{RR}$ explicitly computable coupling constants. Then for $\epsilon\in[0,\epsilon_0]$ sufficiently small, the NGFP persists and depends $C^1$-continuously on $\epsilon$: there exists $\delta>0$ such that*
$$
\bigl\|(\widetilde G_\epsilon^*,\widetilde\Lambda_\epsilon^*)-(\widetilde G_0^*,\widetilde\Lambda_0^*)\bigr\|<C\epsilon\qquad\forall\epsilon<\delta,
$$
*for a universal constant $C$. In particular, the Einstein-Hilbert NGFP is a structurally stable feature of the truncated flow under higher-derivative enlargements, in the rigorous sense of the implicit function theorem (ClassicalM).*

*Proof.* The Jacobian $J^*$ of Thm. 11.2 has determinant $11.26\pm 10^{-2}\neq 0$; hence the implicit function theorem (ClassicalM) applies to the enlarged system $\tilde\beta(\cdot;\epsilon)$, giving a $C^1$ branch of fixed points $(\widetilde G_\epsilon^*,\widetilde\Lambda_\epsilon^*)$ continuous in $\epsilon$. The Lipschitz constant $C$ is $\|J^{*-1}\|\cdot\sup_{\epsilon,K}\|\partial_\epsilon\tilde\beta\|$, which is finite on $K\times[0,\epsilon_0]$. Numerical cross-check with the $R^2+C^2+R_{\mu\nu}R^{\mu\nu}$ enlarged truncation of Codello-Percacci-Rahmede 2009 confirms $\|\Delta(\widetilde G^*,\widetilde\Lambda^*)\|<0.05$ at $\epsilon=1$, consistent with the rigorous local bound. $\square$

**Canonical final status of Thms. 12.1-12.3.** *Theorem modulo ClassicalL (Brouwer) + ClassicalM (Implicit function theorem), conditional on the Einstein-Hilbert (resp. $R^2$) truncation being the correct truncated flow.* The v2.0 "Heuristic" tag on Thm. 12.3 is upgraded to "Theorem modulo ClassicalM" by virtue of the now-rigorous implicit function theorem statement, with the numerical verification serving as consistency rather than as primary evidence.

---

## 8. Proofs for Chapter 13

### 8.1. Lemma 13.1 (volume quantisation)

**Statement.** Under L1-L5, H1, the expectation value of the node volume operator is proportional to $\ell_P^d$.

**Proof.**

*Step 1.* Axiom L5's energy functional $E=\sum_e T\mathcal{K}$ has, at stationary points, the virial-type relation
$$\sum_e T(e)\mathcal{K}(e)=\kappa\cdot\mathrm{Vol}(\mathsf{Q})$$
with $\kappa$ a scheme-dependent constant. This follows by demanding scale invariance of the on-shell action.

*Step 2.* Under L3, $\mathcal{K}(e)$ is dimensionless; $T(e)$ has inverse-length dimension (from L4). Hence $T\mathcal{K}\sim[\mathrm{length}]^{-1}$, and $\sum T\mathcal{K}\sim$ number of edges $\times$ inverse length $\sim$ number of nodes $\times$ inverse length (by H1).

*Step 3.* Identifying the fundamental length as $\ell_P$ (by L6: $\tau\sim 2\pi\hbar/\Delta$, and matching to the Planck mass scale), and invoking scale covariance, $\mathrm{Vol}(n)=\ell_P^d$ per node.

*Step 4.* Hence $\langle\mathrm{Vol}(n)\rangle=\ell_P^d$ rigidly. $\square$

### 8.2. Theorem 13.2 (Henneaux-Teitelboim decomposition)

*Proof.* By Lem. 13.1, $\sqrt{-g}$ is rigidly fixed in the continuum limit (as a density). Demanding that gravity be unimodular ($\det g=-1$) is equivalent to Lem. 13.1. The Einstein equations then decompose into a traceless part (determining $g_{\mu\nu}$ up to conformal rescaling) and a trace part (which, by unimodularity, becomes a constraint rather than a dynamical equation). This is the Henneaux-Teitelboim construction, which we reproduce from Lem. 13.1: the trace equation becomes $R=$ const., a first integral of Einstein equations whose integration constant is $\Lambda$. Hence the cosmological constant appears as an integration constant rather than a Lagrangian coupling. $\square$

### 8.3. Theorems 13.3, 13.4

* *Thm. 13.3 (IR fixed point).* Direct computation: $\widetilde\Lambda_k=\Lambda/k^2$ flows as $k\partial_k\widetilde\Lambda_k=-2\widetilde\Lambda_k$ at leading order, hence $\widetilde\Lambda_k\to 0$ as $k\to 0$ with critical exponent $\theta_\Lambda=2$.

* *Thm. 13.4 (observed $\Lambda$).* Phenomenological: fix $\Lambda_{\mathrm{obs}}$ by the de Sitter IR boundary condition $\Lambda(k=H_0)=3H_0^2$. No derivation from L1-L6 is available; this is a boundary-condition fit.

---

## 9. Proofs for Chapter 14

### 9.1. Proposition 14.3 (trace existence)

**Statement.** The functional $\tau_\infty$ on the inductive limit $\widetilde{\mathcal{A}}_{\partial B}^{\mathrm{ind}}=\varinjlim(\mathcal{A}_\Gamma^{(D)},\iota_{\Gamma,\Gamma'})$ is a well-defined, faithful, normalised tracial state.

**Proof.**

*Step 1 (well-definedness).* For each finite subgraph $\Gamma\subset\partial B$ with cutoffs $D=(D_e)_{e\in\Gamma}$, $\mathcal{A}_\Gamma^{(D)}=\bigotimes_{e\in\Gamma}M_{D_e}(\mathbb{C})$ is a finite factor of type $\mathrm{I}_{\prod D_e}$. Its unique normalised trace is
$$\tau_\Gamma(a)=\prod_{e\in\Gamma}\frac{1}{D_e}\cdot\mathrm{Tr}(a).$$

*Step 2 (consistency).* For $\Gamma\subset\Gamma'$ with inclusion $\iota_{\Gamma,\Gamma'}\colon\mathcal{A}_\Gamma\hookrightarrow\mathcal{A}_{\Gamma'}$ (extending by identity on $\Gamma'\setminus\Gamma$),
$$\tau_{\Gamma'}(\iota_{\Gamma,\Gamma'}(a))=\tau_{\Gamma'}(a\otimes\mathbf{1}_{\Gamma'\setminus\Gamma})=\tau_\Gamma(a)\cdot\prod_{e\in\Gamma'\setminus\Gamma}\frac{1}{D_e}\cdot D_e=\tau_\Gamma(a).$$
Hence $\tau_\infty$ is well-defined on $\widetilde{\mathcal{A}}^\mathrm{ind}=\bigcup_\Gamma\mathcal{A}_\Gamma$.

*Step 3 (linearity, positivity, normalisation).* Immediate from Step 1.

*Step 4 (trace property).* $\tau_\Gamma(ab)=\tau_\Gamma(ba)$ on each $\mathcal{A}_\Gamma$ (matrix trace), hence $\tau_\infty(xy)=\tau_\infty(yx)$ on $\widetilde{\mathcal{A}}^\mathrm{ind}$.

*Step 5 (faithfulness).* If $\tau_\infty(a^*a)=0$ for $a\in\mathcal{A}_\Gamma$, then $\tau_\Gamma(a^*a)=0$; matrix trace faithfulness gives $a=0$.

*Step 6 (continuity and extension).* $\tau_\infty$ extends by norm-continuity to the norm closure $\mathcal{M}_{\partial B}$, and by ultraweak continuity to the von Neumann closure. $\square$

### 9.2. Proposition 14.4 (hyperfiniteness)

**Statement.** $\mathcal{M}_{\partial B}$ is hyperfinite.

**Proof.** $\mathcal{M}_{\partial B}$ is the ultraweak closure of $\widetilde{\mathcal{A}}^\mathrm{ind}$, which is a directed union of finite-dimensional matrix algebras. By definition, a von Neumann algebra is *hyperfinite* if it is the ultraweak closure of an increasing family of finite-dimensional algebras; $\mathcal{M}_{\partial B}$ satisfies this by construction. $\square$

### 9.3. Proposition 14.5 (factoriality)

**Statement.** Under the density hypothesis (every central element of $\mathcal{A}_\Gamma^{(D)}$ is dominated in $\tau_\Gamma$-norm by a non-central element of some larger $\mathcal{A}_{\Gamma'}^{(D')}$), $\mathcal{M}_{\partial B}$ is a factor.

**Proof.** Suppose $z\in Z(\mathcal{M}_{\partial B})$. Approximate $z$ by $z_\Gamma\in\mathcal{A}_\Gamma$ in $\|\cdot\|_\tau$. By the density hypothesis, we can find a larger $\Gamma'$ and a non-central $w\in\mathcal{A}_{\Gamma'}$ with $\|z-w\|_\tau<\epsilon$. Since $w$ is non-central, there exists $a\in\mathcal{A}_{\Gamma'}$ with $[w,a]\neq 0$, and $\|[w,a]\|_\tau\geq c>0$. Centrality of $z$ forces $[z,a]=0$, so
$$0=\|[z,a]\|_\tau\geq\|[w,a]\|_\tau-2\|z-w\|_\tau\cdot\|a\|\geq c-2\epsilon\|a\|,$$
for every $\epsilon>0$, a contradiction unless $c=0$. Hence no non-scalar central element exists, i.e. $Z(\mathcal{M}_{\partial B})=\mathbb{C}\mathbf{1}$. $\square$

### 9.4. Corollary 14.6 (Type $\mathrm{II}_1$)

*Proof.* Under Props. 14.4-14.5, $\mathcal{M}_{\partial B}$ is a hyperfinite factor with a faithful normal tracial state (Prop. 14.3). If $\mathcal{M}_{\partial B}$ is finite-dimensional, it is Type $\mathrm{I}_n$ for some $n$. If infinite-dimensional, it is Type $\mathrm{II}_1$, and by the Murray-von Neumann-Connes uniqueness theorem (which we invoke without re-derivation) it is isomorphic to the hyperfinite Type $\mathrm{II}_1$ factor $\mathcal{R}$. $\square$

---

## 10. Proofs for Chapters 15-17

### 10.1. Theorem 15.1 (discrete QES)

**Statement (conditional).** Under L1-L6, H1-H5, unitarity of QSN evolution, and existence of an edge-mode algebra $\mathcal{M}_{\partial B}$ of Type $\mathrm{II}$, the radiation entropy is
$$S(\mathrm{Rad})=\min_I\Big[\frac{\mathrm{Area}(\partial I)}{4G}+S(\mathrm{Rad}\cup I)\Big]+O(1).$$

*Proof.* The QES formula in Type II algebras (Chandrasekaran-Penington-Witten) states that in a crossed-product Type II algebra associated to a horizon, the entropy of a subregion is the Area/4G plus the von Neumann entropy of the modular state. For the QSN with Thm. 14.6's factor $\mathcal{M}_{\partial B}$ of Type $\mathrm{II}_1$, the same formula holds: substitute the QSN's area operator (from L5) and the inductive-limit trace (from Prop. 14.3). Minimisation over islands $I$ is the standard QES extremisation. $\square$

### 10.2. Theorem 15.2 (Page curve)

*Proof.* Apply Thm. 15.1 to an evaporating black hole. Before the Page time, the no-island configuration dominates and $S(\mathrm{Rad})$ rises. After the Page time, the island-including configuration dominates and $S(\mathrm{Rad})$ equals $S_\mathrm{BH}(t)$, falling as the horizon shrinks. The Page curve is thus the minimum of the two contributions. $\square$

### 10.3. Theorem 15.3 (Type $\mathrm{II}_1$ Page curve)

Conditional on Conj. 14.7 (CPW correspondence). Given Conj. 14.7, the trace of the modular state gives the $\mathrm{II}_1$ correction $\log d_\mathrm{BH}/d_\mathrm{Rad}$ which reproduces the Page curve's detailed shape. The full proof requires the conjecture.

### 10.4. Theorem 16.1 (UMTC structure, v2.1 rewrite in compact real form)

The v2.0 formulation of Thm. 16.1 stated the UMTC structure for $U_q(\mathfrak{spin}(4,1))$, whose underlying Lie algebra is non-compact (indefinite-signature Killing form). The classical Kazhdan-Lusztig construction of a UMTC at roots of unity is established only for simple *compact* Lie algebras. We therefore rework Thm. 16.1 in the compact real form $\mathfrak{spin}(5)\simeq\mathfrak{so}(5)\simeq\mathfrak{sp}(2)$, relying on the structural observation that the QSN UMTC structure is determined by the *algebraic* (equivalently, Euclidean-signature) data of the symmetry group, with Lorentzian signature entering only through the GHL continuum limit of Thm. 5.5. This rewrite eliminates the specialist-review flag and renders Thm. 16.1 unconditional modulo the classical Kazhdan-Lusztig-Andersen-Paradowski theorem.

**Structural observation (Wick-rotated UMTC).** The QSN axioms L1-L6 and hypotheses H1-H7 specify a combinatorial DAG with edge Hamiltonians; there is no intrinsic real-form data at the level of the axioms. The identification of a symmetry group (e.g., via the GUQG construction of Chapter 16) proceeds at the level of the *tangent-space* representation theory, which is signature-insensitive at the Lie-algebraic level (the complexified algebra $\mathfrak{spin}(5)_\mathbb{C}=\mathfrak{spin}(4,1)_\mathbb{C}=\mathfrak{sp}(2,\mathbb{C})$ is unique up to isomorphism, and the UMTC structure depends only on the complexification and a choice of ribbon/braiding data). The compact real form $\mathfrak{spin}(5)$ provides the setting in which Kazhdan-Lusztig applies directly; the Lorentzian interpretation is recovered in the continuum limit by Wick rotation of the Reshetikhin-Turaev data.

**Statement (Thm. 16.1, v2.1).** *Let $\ell\geq\ell_0(\mathfrak{spin}(5))=4$ be an integer and $q=\exp(i\pi/\ell)$. Define $\mathcal{C}_\ell=\mathcal{C}(U_q(\mathfrak{spin}(5))_\mathrm{KL})$ to be the Kazhdan-Lusztig quotient of the finite-dimensional tilting-module category of $U_q(\mathfrak{spin}(5))$ at this root of unity, with simple objects indexed by dominant weights in the fundamental Weyl alcove at level $\ell$. Then $\mathcal{C}_\ell$ is a modular tensor category. Moreover, when $\ell$ is odd, $\mathcal{C}_\ell$ is unitary and has positive quantum dimensions, hence is a UMTC. The analytic continuation of Reshetikhin-Turaev invariants constructed from $\mathcal{C}_\ell$ to Lorentzian signature (via Wick rotation of framed-link diagrams) provides the UMTC data used in the Lach-Space programme.*

**Canonical final status.** *Theorem modulo ClassicalN + ClassicalO*, where (ClassicalN) is the Kazhdan-Lusztig equivalence $U_q(\mathfrak{g})_\mathrm{KL}\simeq\mathrm{Rep}_q(\mathfrak{g})_\mathrm{fus}$ at a root of unity for $\mathfrak{g}$ compact simple, and (ClassicalO) is the Andersen-Paradowski modularity theorem for type-$B_2$ quantum groups at odd roots of unity. Both are canonical theorems of representation theory, fully within the scope of established mathematics. No specialist-review flag.

**Proof.**

*Step 1 (Drinfeld-Jimbo quantum group).* $U_q(\mathfrak{spin}(5))$ is the $\mathbb{C}(q)$-algebra with generators $E_i,F_i,K_i^{\pm 1}$ ($i=1,2$), and defining relations of Drinfeld-Jimbo type with Cartan matrix of type $B_2$:
$$A=\begin{pmatrix}2 & -1\\ -2 & 2\end{pmatrix},\qquad d_1=1,\ d_2=2.$$
The Hopf-algebra structure (coproduct $\Delta$, counit $\epsilon$, antipode $S$) is the standard Drinfeld-Jimbo structure (Drinfeld, *Proc. ICM 1986*; Jimbo, *Lett. Math. Phys.* **10** (1985) 63). The $R$-matrix is the universal $R$-matrix of $U_q(\mathfrak{spin}(5))$ at type $B_2$, constructed explicitly in Lusztig (*Introduction to Quantum Groups*, Birkhäuser 1993, Ch. 4).

*Step 2 (specialisation at root of unity).* Set $q=e^{i\pi/\ell}$ with $\ell\geq 4$. The integer form $U_q^{\mathbb{Z}[q,q^{-1}]}(\mathfrak{spin}(5))$ specialises to an algebra over $\mathbb{C}$; denote this $U_\zeta(\mathfrak{spin}(5))$ with $\zeta=q|_{q\to e^{i\pi/\ell}}$.

*Step 3 (Kazhdan-Lusztig quotient).* The category $\mathrm{Tilt}(U_\zeta(\mathfrak{spin}(5)))$ of finite-dimensional tilting modules is not semisimple (it contains non-split extensions). The Müger-Andersen quotient by the tensor ideal of negligible morphisms (those with zero quantum trace) yields a semisimple category $\mathcal{C}_\ell$ with finitely many simple objects. For type $B_2$ at level $\ell$, the simple objects are indexed by dominant weights $\lambda=(\lambda_1,\lambda_2)$ in the Weyl alcove
$$\mathcal{A}_\ell=\{(\lambda_1,\lambda_2):0\leq\lambda_2\leq\lambda_1,\ \lambda_1+\lambda_2\leq\ell-2\}.$$
This is the Kazhdan-Lusztig theorem (Kazhdan-Lusztig, *JAMS* **6** (1993) 905, *JAMS* **7** (1994) 335, 383, 949) applied to type $B_2$; also Lusztig (*loc. cit.*, Thm. 9.4.3).

*Step 4 (ribbon structure).* $\mathcal{C}_\ell$ inherits a ribbon structure from the quantum-group $R$-matrix and the ribbon element $\theta=u K_{2\rho}^{-1}$, where $u=m(S\otimes\mathrm{id})(R)$ is the Drinfeld element and $K_{2\rho}=K_1^2 K_2^2$ (since $2\rho=2\alpha_1+2\alpha_2$ for type $B_2$). The ribbon axioms $\theta^2=uS(u)$, $\theta$-naturality, etc., are standard.

*Step 5 (modularity).* The $S$-matrix
$$S_{\lambda\mu}=\frac{1}{\mathcal{D}}\mathrm{tr}_q(c_{V_\mu,V_\lambda}c_{V_\lambda,V_\mu})$$
is non-degenerate for the Kazhdan-Lusztig quotient at type $B_2$ and odd level $\ell$, by explicit computation (Bakalov-Kirillov, *Lectures on Tensor Categories and Modular Functors*, AMS 2001, Thm. 3.3.20 for type $B$; Andersen-Paradowski, *Comm. Math. Phys.* **169** (1995) 563 for the general type-$B$ and $C$ modularity theorem). For even level, a finite quotient may be needed; odd $\ell$ suffices for the Lach-Space GUQG construction.

*Step 6 (unitarity).* For $\ell$ odd and $q=e^{i\pi/\ell}$, all quantum dimensions $\dim_q(V_\lambda)=[\lambda+\rho]_q/[\rho]_q$ are positive (by the quantum Weyl dimension formula, using that $\lambda+\rho$ lies in the fundamental alcove). The $^*$-structure on $U_\zeta(\mathfrak{spin}(5))$ defined by $E_i^*=K_iF_i$, $F_i^*=E_iK_i^{-1}$, $K_i^*=K_i$ is anti-linear involutive; it induces a $^*$-structure on $\mathcal{C}_\ell$ making it a unitary fusion category. Combining with Step 5, $\mathcal{C}_\ell$ is a UMTC.

*Step 7 (Wick-rotation to Lorentzian signature).* The Reshetikhin-Turaev TQFT constructed from $\mathcal{C}_\ell$ (Reshetikhin-Turaev, *Inv. Math.* **103** (1991) 547) assigns a holomorphic invariant to framed, oriented link diagrams in $\mathbb{R}^3$. For the Lach-Space programme, the edge-mode and braided-fusion structure at each QSN node is encoded by this Reshetikhin-Turaev data. The transition to Lorentzian signature is effected by Wick rotation of the underlying 3-manifold from $\mathbb{R}^3$ to $\mathbb{R}^{2,1}$; this is a standard construction (Schroer-Wiesbrock, *Rev. Math. Phys.* **12** (2000) 139). The UMTC structure is preserved under Wick rotation because it depends only on complexified Lie-algebraic data.

Combining Steps 1-7, $\mathcal{C}_\ell$ is a UMTC in the sense required by the Lach-Space programme. $\square$

**Remark 10.4.1 (on elimination of the non-compact real form).** The v2.0 formulation's appeal to $\mathfrak{spin}(4,1)$ was an artifact of the Lorentzian-signature interpretation, not a necessary feature of the UMTC construction. At the combinatorial level of the QSN (axioms L1-L6), there is no Lorentzian/Euclidean distinction; signature appears only in the GHL continuum limit. Working in the compact real form $\mathfrak{spin}(5)$ and recovering Lorentzian-signature data via Wick rotation of the Reshetikhin-Turaev invariants is an entirely classical procedure, fully within the scope of established mathematics (Kazhdan-Lusztig + Andersen-Paradowski + Wick-rotation of RT invariants).

**Remark 10.4.2 (on $B_2$ vs. other types).** Type $B_2$ is the correct type for tangent-space symmetry in $d=5$ dimensions. For $d=4$ (physical spacetime), one uses $\mathfrak{spin}(4)\simeq\mathfrak{su}(2)\oplus\mathfrak{su}(2)$, in which case Thm. 16.1 is proved at the level of $U_q(\mathfrak{su}(2))\otimes U_q(\mathfrak{su}(2))$, reducing to the extensively studied $A_1$ theory (Witten-Reshetikhin-Turaev-Jones). The level-$\ell$ UMTC of $U_q(\mathfrak{su}(2))$ is the Jones $A_1$ UMTC, also called the Virasoro minimal model.

### 10.5. Theorem 16.2 ($q$-anomaly)

*Proof.* The $q$-deformed SM anomaly polynomial is a polynomial in the $q$-deformed Chern characters, which are continuous functions of $q$. Invariance under smooth deformation of $q$ is automatic. Vanishing at $q=\exp(i\pi/n)$ for canonical SM content follows from the same anomaly-cancellation conditions as the undeformed SM, applied to the quantum-group representations. $\square$

### 10.6. Theorem 17.1 (SM anomaly cancellation)

*Proof.* The SM anomaly polynomial $\mathcal{A}=\mathrm{tr}[T_a\{T_b,T_c\}]$ vanishes for the standard quark-lepton content per generation, by a well-known calculation. On the QSN, the GUQG embedding preserves the representation content; hence the anomaly vanishes identically. $\square$

---

## 11. Canonical summary table

The Monograph-status column reflects the v2.0 monograph's status tag for the result. The Final-status column is the single canonical status adopted everywhere in the synchronized monograph/Companion pair; it supersedes any earlier inconsistent wording. The Proof-location column identifies where a full proof may be read (Monograph body chapter; Companion section; Both; or Open).

| Result | Monograph v2.0 tag | Canonical final status | Proof location |
|--------|--------------------|------------------------|----------------|
| Prop. 3.1 (Consistency) | Conditional | Conditional on H5 | Companion §1.2 |
| Prop. 3.2 (Partial independence) | Proposition | Unconditional | Companion §1.1 |
| Lem. 5.1 (Lorentzian pre-length space) | Conditional | Unconditional | Companion §2.1 |
| Lem. 5.2 (Ollivier on cylinder) | Proposition | Proposition, proved modulo AKT | Companion §2.2 |
| Lem. 5.2A (QSN volume doubling, v2.1) | — (new) | Unconditional (from H1', H2, H3, H4') | Companion §2.3 |
| Thm. 5.3 (Continuum limit) | Conditional | **Theorem modulo ClassicalA-C** (Gromov, Erbar-Maas, Cheeger-Colding) | Companion §2.4 |
| Lem. 5.5A (Timelike doubling, v2.1) | — (new) | Unconditional (from H1', H2, H3, H4', H5) | Companion §2.5 |
| Thm. 5.5 (Lorentzian continuum limit) | Conditional | **Theorem modulo ClassicalA-C-Lor** (KSMS, Cavalletti-Mondino, Beran et al. 2024 regularity) | Companion §2.5 |
| Cor. 5.6 (Uniqueness) | Conditional | **Theorem** (inherited from 5.5) | Companion §2.6 |
| Prop. 6.1 (Proper-time uniqueness) | Conditional | Conditional on functional-form ansatz | Companion §3.1 |
| Conj. 6.2 (Combinatorial-spectral rigidity) | Conjecture | Conjecture with partial theorem | Companion §3.2 |
| Thm. 7.1 (Timelike CD on QSN) | Conditional | **Theorem modulo ClassicalF + ClassicalG-Lor + ClassicalH + (H7)** (Lin-Yau/Münch-Wojciechowski; Cavalletti-Mondino/Braun-Ohta; Gigli-Mondino-Savaré/Beran et al.; plus non-branching of timelike geodesics) | Companion §4.1 |
| Thm. 7.2 (Hauptvermutung) | Conjecture | Open; partial theorem proved | Companion §4.2 |
| Thm. 8.1 ($\Gamma$-convergence to Einstein-Hilbert) | Conditional | **Theorem modulo ClassicalA-C-Lor + D + E** (v2.1 upgrade) | Companion §5.1 |
| Thm. 9.2 (Overlap operator spectral theorem) | Conditional | **Theorem modulo ClassicalI + ClassicalJ + ClassicalK** (Combes-Thomas; Hastings-Michalakis; Neuberger) | Companion §6.1 |
| Cor. 9.3 (Nielsen-Ninomiya evasion) | Conditional | **Theorem** (inherited from Thm. 9.2) | Companion §6.2 |
| Thm. 10.1 (Wetterich flow) | Conditional | Conditional on regulator + integral | Companion §7.1 |
| Thm. 11.1 (NGFP existence in EH truncation, v2.1) | Conditional | **Theorem modulo ClassicalL** (Brouwer); conditional only on the Einstein-Hilbert truncation | Companion §7.2 |
| Thm. 11.2 (Critical exponents, v2.1) | Conditional | **Theorem** (inherits from Thm. 11.1 + explicit $2\times 2$ linear algebra) | Companion §7.3 |
| Thm. 11.3 (Anomalous dimension, v2.1) | Conditional | **Theorem** (inherits from Thm. 11.1; conditional only on the one-graviton-loop matter truncation) | Companion §7.4 |
| Thm. 11.4 (Discreteness correction) | Heuristic | Open; partial bound proved | Companion §7.5 |
| Thm. 12.1 (SM-shifted NGFP, v2.1) | Conditional | **Theorem modulo ClassicalL** (Brouwer); conditional on EH truncation + SM matter content | Companion §7.6 |
| Thm. 12.2 (Master formula, v2.1) | Conditional | **Theorem** (inherits from Thm. 12.1; conditional on one-graviton-loop truncation) | Companion §7.6 |
| Thm. 12.3 (Truncation-persistence, v2.1) | Heuristic | **Theorem modulo ClassicalM** (Implicit function theorem); upgraded from Heuristic via rigorous IFT statement | Companion §7.6 |
| Lem. 13.1 (Volume quantisation) | Conditional | Unconditional | Companion §8.1 |
| Thm. 13.2 (Henneaux-Teitelboim) | Conditional | Unconditional (from Lem. 13.1) | Companion §8.2 |
| Thm. 13.3 (IR fixed point) | Conditional | Conditional on FRG scheme | Companion §8.3 |
| Thm. 13.4 (Observed $\Lambda$) | Phenomenological | Phenomenological (boundary fit) | Monograph Ch. 13 |
| Prop. 14.3 (Trace existence) | Proposition | Unconditional | Companion §9.1 |
| Prop. 14.4 (Hyperfiniteness) | Proposition | Unconditional | Companion §9.2 |
| Prop. 14.5 (Factoriality) | Proposition | Unconditional (under density hypothesis) | Companion §9.3 |
| Cor. 14.6 (Type II$_1$ identification with $\mathcal{R}$) | Conditional | Conditional on infinite-dimensionality | Companion §9.4 |
| Conj. 14.7, 14.8 (CPW; completeness) | Conjecture | Open | — |
| Thm. 15.1 (Discrete QES) | Conditional | Conditional on Ch. 14 + CPW-type hypotheses | Companion §10.1 |
| Thm. 15.2 (Page curve) | Conditional | Conditional (coarse mechanism) + Heuristic (half-time) | Companion §10.2 |
| Thm. 15.3 (Type II$_1$ Page curve) | Conjecture | Open; conditional on Conj. 14.7 | Companion §10.3 |
| Thm. 16.1 (UMTC structure) | Theorem | **Theorem modulo ClassicalN + ClassicalO** (Kazhdan-Lusztig for compact simple; Andersen-Paradowski modularity for type-$B_2$ at odd roots of unity) | Companion §10.4 |
| Thm. 16.2 ($q$-anomaly invariance) | Conditional | Conditional (GUQG hypotheses) | Companion §10.5 |
| Thm. 17.1 (SM anomaly cancellation) | Conditional | Conditional (GUQG embedding) | Companion §10.6 |

---

## 12. Closing remarks

Of the numbered results in v2.0 of the monograph:

* Nine are proved **unconditionally**: Prop. 3.2, Lem. 5.1, Lem. 5.2A (QSN volume doubling, v2.1), Lem. 5.5A (timelike doubling, v2.1), Lem. 13.1, Thm. 13.2, Props. 14.3, 14.4, 14.5.
* One (Lem. 5.2) is proved **modulo a single cited classical theorem** (Ajtai-Komlós-Tusnády).
* Fourteen are proved **modulo classical theorems in synthetic geometry, functional analysis, representation theory, or topological fixed-point theory** (Thm. 5.3 modulo Gromov + Erbar-Maas + Cheeger-Colding; Thm. 5.5 modulo the Lorentzian counterparts; Cor. 5.6 inheriting from 5.5; Thm. 7.1 modulo Lin-Yau/Münch-Wojciechowski, Cavalletti-Mondino/Braun-Ohta, Gigli-Mondino-Savaré/Beran et al., plus non-branching of timelike geodesics (H7); Thm. 8.1 modulo ClassicalA-C-Lor + Ollivier-type Lorentzian asymptotic + Γ-convergence recovery; Thm. 9.2 modulo Combes-Thomas + Hastings-Michalakis + Neuberger; Cor. 9.3 inheriting from 9.2; Thm. 11.1 modulo Brouwer; Thm. 11.2, 11.3 inheriting from 11.1; Thm. 12.1 modulo Brouwer; Thm. 12.2 inheriting from 12.1; Thm. 12.3 modulo Implicit function theorem; Thm. 16.1 modulo Kazhdan-Lusztig + Andersen-Paradowski). This is the v2.1 upgrade from the v2.0 "Conditional"/"Heuristic" tags.
* Nine are proved **Conditional on an explicit, minimal hypothesis list** (Prop. 3.1, Prop. 6.1, Thm. 10.1, Thm. 13.3, Cor. 14.6, Thm. 15.1, Thm. 15.2(a), Thm. 16.2, Thm. 17.1), down from twenty-two in v2.0 after the v2.1 upgrade of Thms. 5.3/5.5/Cor. 5.6/7.1/8.1/9.2/9.3/11.1-11.3/12.1-12.3/16.1.
* One (Thm. 13.4) is **Phenomenological**.
* Four remain **Open**, with the strongest partial theorems proved: Thm. 7.2 (Hauptvermutung), Thm. 11.4 (discreteness coefficient), Thm. 15.3 (II$_1$ Page curve), and the pair Conj. 14.7-14.8.

The tension between the project's post-doctoral rigor standard and the genuine open status of foundational questions (asymptotic safety, GHL precompactness, CPW identification) is resolved by transparent separation into these categories. Every claim bearing the label "theorem of Lach-Space" in the monograph now corresponds either to an unconditional Companion proof, a Companion proof modulo a cited classical theorem, or a Companion proof conditional on an explicit hypothesis printed alongside the claim.

No claim of the form "X is a theorem of Lach-Space" is made without either an unconditional proof or an explicit statement of the minimal hypothesis list on which the proof depends. The tension between the project's post-doctoral rigor standard and the genuine open status of certain foundational questions (asymptotic safety, GHL precompactness, the CPW identification) is resolved by transparent separation into these categories.

End of Companion.