18.1.  Mosco convergence: theorem and assumptions

The Mosco-convergence statement used above (§7.8) is made precise here: the seven assumptions A1–A7 below are stated explicitly, the structural ones are verified, and the equidistribution condition A3 is supplied by P3’s maximum-entropy sublimation. Computation 4 supplies finite-$|D|$ numerical verification of the spin-structure component. A fully rigorous proof of the full Mosco theorem with explicit functional-analytic estimates would expand this section to a companion paper of its own (cf. [98, 113] for the analogous arguments in Riemannian settings). §18.2 supplies the proof-detail core of that programme: it proves the load-bearing $K$-uniform Berry–Esseen rate, the single analytic estimate to which A7 reduces, for the i.i.d. surrogate, via the five-step Berry–Esseen argument and the three closures (a)–(c). Thus the present subsection states the theorem and the reduction of A7 to that rate, and §18.2 proves it for the surrogate at proof-detail level, corroborating the rate that P3’s maximum-entropy sublimation supplies for the emergence-determined case. The present form makes precise what PST claims, what is structural, and what P3 supplies.

18.1.1.  Setup: the substrate sequence and matched scaling

Let $D_{n}$ be a sequence of substrate sites with $|D_{n}|\to\infty$ and matched scaling parameter $a_{n}$ (the substrate site spacing, distinct from the fixed coherence length $d_{0}$ of Chapter 15 and of the A4 displacement covariance) such that $|D_{n}|\cdot a_{n}^{4}\to V_{M}$ (the Riemannian volume of $M$, read locally as in A1 below) and $a_{n}\to 0$ (the standard quantum Gromov–Hausdorff (QGH) convergence scaling [98]; cf. Computation 62). Define the configuration-space probability space $(\mathcal{P}(D_{n}),\mu_{n})$ with $\mu_{n}=\bigotimes_{a\in D_{n}}\mathrm{Bern}(\tfrac{1}{2})$; the Boolean Dirichlet form $E^{B}_{n}:L^{2}(\mathcal{P}(D_{n}),\mu_{n})\to[0,\infty]$ as in eq. (55); the realisation map $\rho_{n}:D_{n}\to M=\mathbb{R}\times S^{3}$ with image $\rho_{n}(D_{n})$ asymptotically equidistributed with respect to the Riemannian volume of $M$; the pullback $\Pi_{n}^{\dagger}:C_{c}^{\infty}(M)\to L^{2}(\mathcal{P}(D_{n}),\mu_{n})$ via the realisation map; and the target Dirichlet form $E^{\mathrm{Riem}}:H^{1}(M)\to[0,\infty]$ generating the Laplace–Beltrami operator $\Delta_{g}$.

18.1.2.  The seven assumptions A1–A7

The structural and analytic inputs needed for Mosco convergence collect into seven explicit assumptions.

A1 (Matched scaling).

The sequence $(D_{n},a_{n})$, with site spacing $a_{n}\to 0$, satisfies $|D_{n}|\cdot a_{n}^{4}\to V_{M}$, the Riemannian volume of $M$. This is the QGH-matching condition (cf. Computation 62) and is an input from the PST architecture, not a substrate consequence. (Since $M=\mathbb{R}\times S^{3}$ is non-compact, $V_{M}$ is infinite, so A1 and A2 are read locally. The local node density is $|D_{n}\cap K|/\mathrm{vol}(K)=\lambda_{n}=1/a_{n}^{4}\to\infty$ on every compact $K$, the sites densifying as $a_{n}\to 0$. That is all the rate estimates require, being carried out on compact $K$ with $u\in C_{c}^{\infty}(K)$.)

A2 (Equidistribution of the realisation map).

For every $f\in C_{c}^{\infty}(M)$,

$$\frac{1}{|D_{n}|}\sum_{a\in D_{n}}f(\rho_{n}(a))\;\longrightarrow\;\frac{1}{V_{M}}\int_{M}f\,\sqrt{-g}\,d^{4}x.$$ (232)

Verified for the i.i.d. surrogate by the strong law of large numbers. For the emergence-determined $\rho_{n}$ the substrate’s permutation invariance of $\mu$ delivers equidistribution in measure; the quantitative weak ($H^{1}$-tested) discrepancy rate the convergence argument uses is then exactly the rate condition A3, supplied by P3’s maximum-entropy sublimation (§18.1.6).

A3 (Equidistribution condition, supplied by P3).

For the emergence-determined cloud $\rho_{n}$, the metric-covariant unevenness of the realised node measure against $\sqrt{-g}\,\mathrm{d}^{4}x$ stays within the closure margin, i.e. the rate hypothesis of §18.1.6 holds with some $\eta_{K,R}<d_{0}^{2}/4$. This is supplied by P3’s maximum-entropy sublimation, not a separately adopted premise. A2 establishes equidistribution in measure (forced by the permutation symmetry of $\mu$); maximum-entropy sublimation strengthens this to the realised configuration itself, selecting the $\mu$-typical instantiated configuration, the analogue of smoothness (or generic initial data) in general relativity and of typicality in statistical mechanics. It is a typicality principle internal to P3, not a fourth postulate and not an analytic derivation of the rate from more primitive structure. The i.i.d. surrogate computations (Computations 70, 83) corroborate the rate P3 supplies. The same clause supplies one thing more: that the realised positions be independent, Poisson-type, which equidistribution in measure does not fix (a lattice satisfies A2 and selects a frame) and which is the hypothesis under which the sprinkling theorem of causal set theory makes PST’s discreteness Lorentz invariant (Computation 208). Read as a law on the realised cloud, maximum entropy given the density is independence, by the subadditivity of entropy, and the rate A3 states conditions only the modes above the Mosco scale: the conditioned cloud is Poisson at the grain in every frame and hyperuniform only at the constraint’s scale (Computation 211). The one explicit placement the book has, classical scaling of the signed-tension metric, cannot test it, its Gram form being a negative multiple of the direction Gram matrix at leading order (Computation 209); the condition is supplied by the clause, not by a placement.

A4 (Isotropy / covariance forcing).

The covariance matrix of the displacement $\rho_{n}(a)^{\mu}-x^{\mu}$ satisfies eq. (56):

$$\frac{1}{|D_{n}|}\sum_{a\in D_{n}}\bigl{(}\rho_{n}(a)^{\mu}-x^{\mu}\bigr{)}\bigl{(}\rho_{n}(a)^{\nu}-x^{\nu}\bigr{)}$$ (233)

$$\qquad\;\longrightarrow\;\frac{d_{0}^{2}}{4}\,g^{\mu\nu}(x).$$ (234)

Follows from A2 plus the absence of any preferred direction in $V_{7}$ surviving the configuration-space integral under $\mu$. Structural.

A5 (Regularity of the Landau–Ginzburg quartic potential).

The Landau–Ginzburg free energy $F[\psi,\varepsilon]$ of (1) is quadratic at leading order around $\mathcal{V}$ and has bounded Hessian on bounded domains. Structural from P3.

A6 (Density of pulled-back test functions).

The pullback

$$\Pi_{n}^{\dagger}C_{c}^{\infty}(M)\;\subset\;L^{2}(\mathcal{P}(D_{n}),\mu_{n})$$ (235)

is dense in the form domain of $E^{B}_{n}$ as $|D_{n}|\to\infty$. Standard but technical: follows from A2 plus the Stone–Weierstrass density of $C_{c}^{\infty}(M)$ in $L^{2}(M,dvol_{g})$, combined with the explicit construction of $\Pi_{n}^{\dagger}$ via spherical-harmonic partial sums on $S^{3}$. A full proof requires explicit convergence-rate control of the truncation error.

A7 (Equicoercivity / Poincaré-type inequality).

There exists a constant $C>0$, independent of $n$, such that for every $u\in C_{c}^{\infty}(M)$,

$$\frac{1}{|D_{n}|}\,E^{B}_{n}(\Pi_{n}^{\dagger}u,\Pi_{n}^{\dagger}u)\;\geq\;C\,\|u\|_{H^{1}(M)}^{2}.$$ (236)

A theorem given A1, A2 and A4 together with A3, the last supplied by P3: the reduction lemma of §18.1.6 derives A7 with $C_{K}>0$ from A1, A2, A4 together with A3. A3 is the rate hypothesis on the realised cloud (see the A3 entry above). Equicoercivity for the Boolean Dirichlet form is the discrete analogue of a Poincaré inequality on the substrate configuration space.

18.1.3.  The Mosco-convergence theorem

18.1.3.1.  Theorem (Mosco convergence, under A1–A7)

Under assumptions A1–A7 above, the rescaled Boolean Dirichlet forms $\bigl{(}\tfrac{1}{|D_{n}|}E^{B}_{n}\bigr{)}$ converge in the Mosco sense to $\tfrac{d_{0}^{2}}{4}\,E^{\mathrm{Riem}}$ on $L^{2}(M,dvol_{g})$, where $d_{0}$ is the fixed coherence length (the A4 displacement covariance width, Chapter 15), unrelated to the vanishing site spacing $a_{n}$ of A1:

1. (Liminf inequality). For every sequence $u_{n}\in L^{2}(\mathcal{P}(D_{n}),\mu_{n})$ with $u_{n}\to u$ in the limit identification implied by A2,

   $$\liminf_{n\to\infty}\tfrac{1}{|D_{n}|}\,E^{B}_{n}(u_{n},u_{n})\;\geq\;\tfrac{d_{0}^{2}}{4}\,E^{\mathrm{Riem}}(u,u).$$ (237)

2. (Recovery sequence). For every $u\in H^{1}(M)$ there exists $u_{n}\in L^{2}(\mathcal{P}(D_{n}),\mu_{n})$ with $u_{n}\to u$ and

   $$\lim_{n\to\infty}\tfrac{1}{|D_{n}|}\,E^{B}_{n}(u_{n},u_{n})\;=\;\tfrac{d_{0}^{2}}{4}\,E^{\mathrm{Riem}}(u,u).$$ (238)

By Mosco’s theorem [98], this implies strong resolvent convergence of the generators, $\bigl{(}\tfrac{1}{|D_{n}|}\,L^{B}_{n}+\lambda\bigr{)}^{-1}\,\Pi_{n}^{\dagger}\to\Pi\bigl{(}\tfrac{d_{0}^{2}}{4}\,\Delta_{g}+\lambda\bigr{)}^{-1}$ for all $\lambda>0$, and the generator of the limiting form is the Laplace–Beltrami operator $\Delta_{g}$, a Laplace-type operator with principal symbol $g^{\mu\nu}$.

18.1.3.2.  Proof sketch of the Mosco theorem

Liminf inequality. Given $u_{n}\to u$, write $E^{B}_{n}(u_{n},u_{n})=\sum_{a\in D_{n}}\int(\partial_{a}^{B}u_{n})^{2}\,d\mu_{n}$. By A2 and the matched scaling A1, the sum over $a\in D_{n}$ approximates the integral over $M$ with respect to $dvol_{g}$. By A4, the covariance matrix forces the leading order of $(\partial_{a}^{B}u_{n})^{2}$ pulled back to $x=\rho_{n}(a)$ to equal $(d_{0}^{2}/4)g^{\mu\nu}\partial_{\mu}u\partial_{\nu}u$, with $d_{0}$ the fixed displacement covariance width. Together, $\tfrac{1}{|D_{n}|}E^{B}_{n}(u_{n},u_{n})\to\tfrac{d_{0}^{2}}{4}E^{\mathrm{Riem}}(u,u)$. The liminf (rather than equality) version requires lower semicontinuity of $u_{n}\mapsto E^{B}_{n}(u_{n},u_{n})$ under A2-convergence, which follows from A7.

Recovery sequence. Given $u\in H^{1}(M)$, set $u_{n}=\Pi_{n}^{\dagger}u$. By A6, $u_{n}\to u$. By A2, A4 and the construction of $\Pi_{n}^{\dagger}$, $\tfrac{1}{|D_{n}|}E^{B}_{n}(u_{n},u_{n})\to\tfrac{d_{0}^{2}}{4}E^{\mathrm{Riem}}(u,u)$ with equality at the limit. A standard density argument extends from $C_{c}^{\infty}(M)$ to $H^{1}(M)$.

Strong resolvent convergence. Apply Mosco’s theorem [98]; the generator of the limiting form is $\tfrac{d_{0}^{2}}{4}\Delta_{g}$ by direct identification.

The standing of A1 to A7 is as their entries above record: A1 is architectural input; A2, A4 and A5 are structural; A3 is supplied by P3’s maximum-entropy sublimation; A6 is technical but standard, a full proof wanting the spherical-harmonic-partial-sum projector with convergence-rate control; and A7 (equicoercivity) is a theorem given A1, A2, A4 and A3, via the reduction lemma of §18.1.6.

18.1.4.  Numerical verification on finite models

Computation 4 verifies the spin-structure component of the Mosco limit explicitly on $\mathbb{R}\times S^{3}$: the Friedrich–Bär Dirac spectrum $\pm(n+3/2)$ on $S^{3}$ is recovered from finite-$|D|$ Boolean Laplacian eigenvalues along the matched scaling. Computation 62 verifies the spectral-action invariance at finite $|D|$ as a consistency check on the matched-scaling convergence. No discrete finite-$|D|$ test currently corroborates A3 across $|D|$ for the emergence-determined cloud; such a test would be a natural concrete corroboration on the analytic side.

18.1.5.  What the theorem establishes

The theorem above converts the heuristic statement “Mosco convergence yields $M=\mathbb{R}\times S^{3}$ as the Laplace-type limit” (used in §7.8 and downstream) into a precise theorem whose equidistribution condition A3 is supplied by P3’s maximum-entropy sublimation. Under A1–A7, the strong-resolvent convergence is rigorous and the Laplace–Beltrami operator is identified as the generator of the limiting form, so the Mosco limit closes under A1–A7. It is not a theorem of P1–P3 alone: A1 is architectural input, and A6 still wants explicit convergence-rate control of the truncation error.

Equicoercivity A7 follows from A3 by the reduction lemma; the i.i.d. surrogate corroborates the rate (Computations 70, 83), and the corresponding discrete-geometry analytic quantification, whose continuum analogue is standard [113, 99], is a concrete deliverable rather than a gap. Consequently the downstream statements that depend on the Mosco limit (Einstein–Hilbert action from Seeley–DeWitt, the Connes spectral-triple structure of $M$, the Casimir prediction insofar as it relies on the kernel structure of $\Pi(L^{B})$) inherit that standing, conditional on A1–A7.

The structural content of PST does not depend on which level of analytic rigour one prefers, and the explicit assumption block makes the inputs, and their supply by P1–P3, transparent.

18.1.6.  Sharpening A7: a uniform-rate inequality

The form of A7 stated above (“$\exists\,C>0$ s.t. $(1/|D_{n}|)\,E^{B}_{n}(\Pi_{n}^{\dagger}u,\Pi_{n}^{\dagger}u)\geq C\,\|u\|_{H^{1}}^{2}$ for every $u\in C_{c}^{\infty}(M)$, independent of $n$”) is the canonical Poincaré-type bound a Mosco-type proof would require. Under A1–A4 the proof-sketch argument above already gives the pointwise convergence

$$\frac{1}{|D_{n}|}\,E^{B}_{n}(\Pi_{n}^{\dagger}u,\Pi_{n}^{\dagger}u)\;\xrightarrow[n\to\infty]{}\;\frac{d_{0}^{2}}{4}\,\|\nabla u\|_{L^{2}(M,g)}^{2},\qquad u\in C_{c}^{\infty}(M),$$ (239)

which together with the standard Poincaré inequality for compactly-supported smooth functions reduces A7 to a uniformity question. We state the reduction explicitly.

18.1.6.1.  Lemma (A7 reduction)

Assume A1, A2 and A4 above, which give the pointwise convergence (239). The equidistribution condition A3, supplied by P3’s maximum-entropy sublimation, is precisely the statement that for each compact $K\subset M$ and each $R>0$ there exist $N_{0}(K,R)$ and $\eta_{K,R}>0$ with $\eta_{K,R}<d_{0}^{2}/4$ such that for all $n\geq N_{0}(K,R)$ and all $u\in C_{c}^{\infty}(K)$ with $\|u\|_{H^{1}(M)}\leq R$,

$$\left|\frac{1}{|D_{n}|}\,E^{B}_{n}(\Pi_{n}^{\dagger}u,\Pi_{n}^{\dagger}u)-\frac{d_{0}^{2}}{4}\,\|\nabla u\|_{L^{2}(M,g)}^{2}\right|\;\leq\;\eta_{K,R}\,\|u\|_{H^{1}(M)}^{2}.$$ (240)

Then A7 holds with $K$-dependent constant $C_{K}=\bigl{(}d_{0}^{2}/4-\eta_{K,R}\bigr{)}\,/\,\bigl{(}C_{P}(K)^{2}+1\bigr{)}$, where $C_{P}(K)$ is the Poincaré constant of $K\subset(M,g)$ for test functions supported in $K$.

Proof.

Standard Poincaré on $C_{c}^{\infty}(K)$, $\|u\|_{L^{2}(K)}\leq C_{P}(K)\,\|\nabla u\|_{L^{2}(K)}$ [113, 99], gives $\|u\|_{H^{1}}^{2}=\|u\|_{L^{2}}^{2}+\|\nabla u\|_{L^{2}}^{2}\leq(C_{P}(K)^{2}+1)\,\|\nabla u\|_{L^{2}}^{2}$. Combining with (240):

$$\frac{1}{|D_{n}|}\,E^{B}_{n}(\Pi_{n}^{\dagger}u,\Pi_{n}^{\dagger}u)\;\geq\;\frac{d_{0}^{2}}{4}\,\|\nabla u\|_{L^{2}}^{2}-\eta_{K,R}\,\|u\|_{H^{1}}^{2}\;\geq\;\frac{d_{0}^{2}/4-\eta_{K,R}}{C_{P}(K)^{2}+1}\,\|u\|_{H^{1}}^{2},$$ (241)

which is A7 with constant $C_{K}=(d_{0}^{2}/4-\eta_{K,R})/(C_{P}(K)^{2}+1)>0$. ∎

18.1.7.  The physical content of the bound

The equicoercivity constant $C_{K}=(d_{0}^{2}/4-\eta_{K,R})/(C_{P}(K)^{2}+1)$ is not mere bookkeeping; it is the margin by which the emergent geometry stays non-degenerate. $C_{K}>0$, equivalently the leading-coefficient inequality $\eta_{K,R}<d_{0}^{2}/4$, is exactly the condition that the rescaled Boolean Dirichlet form remains coercive in the Mosco limit, hence that the limit is a genuine continuum Dirichlet form rather than a collapsed one, the analytic statement that spacetime closes as a manifold. The rate condition is then not bookkeeping but a quantitative threshold: it measures how much equidistribution defect of the emergence map $\rho_{n}$, how much unevenness in the realised node measure against the Riemannian volume $\sqrt{-g}\,\mathrm{d}^{4}x$, the continuum geometry can absorb while still closing. As $\eta_{K,R}\uparrow d_{0}^{2}/4$ the constant $C_{K}\downarrow 0$ and the coercive lower bound is lost; past the threshold the construction no longer secures a non-degenerate limit. Read this way the rate condition is physical content: the equidistribution bound is the closure margin of emergent spacetime, the condition asks precisely where the $T$-determined cloud sits relative to it, and P3’s maximum-entropy sublimation answers the question: the realised configuration is a typical one under $\mu$, its cloud equidistributed at the rate A3 states, so the realised cloud sits inside the margin. Concretely, this residual unevenness is texture at the node (site) scale $a_{n}$: the permutation symmetry of the substrate measure evens it in measure, and the discreteness of the realised cloud smooths it away in the continuum limit, so the condition is precisely the rate at which that node-scale texture is erased.

The lemma converts the A7 problem from “find a uniform Poincaré-type spectral gap for the rescaled Boolean Laplacian” into “establish a uniform-rate quantitative version of the pointwise-in-$u$ convergence already known under A1–A4”. Three points on the resulting reduced problem follow: its $K$-uniformity, its analytic content, and its numerical verification.

18.1.8.  $K$-uniformity of the constant

The lemma delivers A7 with a constant $C_{K}$ that depends on the support $K\subset M$ of the test function through the Poincaré constant $C_{P}(K)$ of $K$. Because $M=\mathbb{R}\times S^{3}$ is non-compact (a finite-volume compact factor $S^{3}$ times a non-compact $\mathbb{R}$), the Poincaré constant is bounded on any compact $K$ but can grow without bound as $K$ exhausts $M$ in the temporal direction. For the downstream Mosco theorem only test functions of compact support are needed (recovery sequences are constructed by mollified cutoff arguments in $C_{c}^{\infty}(M)$ [98]), so the $K$-dependence is not an obstruction. A genuinely $K$-uniform constant would require a stronger statement than is needed for the downstream applications.

18.1.9.  The analytic content of the rate hypothesis

The pointwise convergence (239) relies on the law-of-large-numbers limit in A2 and the covariance identity A4. Both have explicit finite-$n$ correction rates for the i.i.d. surrogate (Berry–Esseen rate $|D_{n}|^{-1/2}$ [101]). For the emergence-determined $\rho_{n}$ the fluctuation of the empirical aggregate $A_{n}$ (equation (242)) carries the same $|D_{n}|^{-1/2}$ rate by McDiarmid concentration on the i.i.d. bits under a single-bit-flip locality hypothesis (§18.2, no statistical independence of positions assumed; Computation 125); the bias (mean) rate, with the A4 covariance-scale convergence, is the content that P3’s maximum-entropy sublimation supplies (corroborated on the i.i.d. surrogate).

The reduced problem (240) is therefore quantitative: it asks for a Berry–Esseen-type rate of order $|D_{n}|^{-\gamma}$ for some $\gamma>0$, uniform on the $H^{1}$-ball of $C_{c}^{\infty}(K)$. For the i.i.d. surrogate this is a discrete-analytic statement of the same form as the finite-element rate estimates for elliptic PDEs [102], closed at proof-detail level by the five-step Berry–Esseen argument below. For the emergence-determined $\rho_{n}$ it is sharper than an exponent: the lemma needs the leading-coefficient inequality $\eta_{K,R}<d_{0}^{2}/4$ (not merely $\eta_{K,R}\to 0$), so a universality result giving only the rate $\gamma$ does not suffice.

A covariance-decay (spectral-gap) mixing estimate is, however, the wrong instrument for this residual. A spectral gap is a relaxation rate, and bounds only the statistical fluctuation of $A_{n}$ about its mean; the stationary mean is the kernel eigenvector, decoupled from the gap. No mixing exponent therefore reaches the leading coefficient. Computation 129 shows this directly: the gap is invariant under a full sweep of the stationary mean, while the systematic coefficient is flat in $|D_{n}|$ as the fluctuation decays at $|D_{n}|^{-1/2}$. The binding content is therefore the systematic mean deviation: a deterministic equidistribution defect of the emergence-determined $\rho_{n}$ in the invariant sector $\ker L$. What is needed is the leading-coefficient (equidistribution) inequality $\eta_{K,R}<d_{0}^{2}/4$ itself, not a quantitative ergodic theorem. The spectral-gap and mixing route is therefore foreclosed for the coefficient, as pinning $T$ and local dependence are, and it is exactly this invariant-sector coefficient that P3’s maximum-entropy sublimation supplies.

This equidistribution defect is concretely the uniform-weight nodal quadrature error of the emergence cloud against the continuum measure. Writing $c_{u}=|D_{n}|^{-1}\sum_{a}|\nabla u(\rho_{n}(a))|^{2}\big{/}\bigl{(}|K|^{-1}\int_{K}|\nabla u|^{2}\bigr{)}$, one has $c_{u}\to 1$ iff the realised points cell-volume-equidistribute: the deterministic sibling of the A2 weak ($H^{1}$-tested) discrepancy. It is not the Delaunay finite-element interpolation error, which mesh shape-regularity would control; shape-regularity is here neither necessary nor sufficient. The matched-scaling assumption A1 does not supply it: A1 fixes only the average density $|D_{n}|/V_{M}\sim a_{n}^{-4}\to\infty$, not the minimum separation or the local equidistribution, and a matched-count cloud carrying a void or density excess retains a nonzero $c_{u}$ (Computation 130). The strong separation channel, by contrast, does not bind the weak discrepancy. The directed metric (30) manufactures vanishing-separation pairs on the null cone, but that locus is codimension $\geq 1$ on pair-space, and leaves the single-node empirical measure, hence $c_{u}$, unchanged (Computation 131). It therefore does not obstruct A7.

The bias half therefore rests on a single named deterministic gauge-invariant lemma, the codimension-zero cell-volume equidistribution (uniform bulk fibre-occupation of the node measure over the $4$-volume) of the $T$-determined cloud, which matched scaling does not force and which is precisely what P3’s maximum-entropy sublimation supplies. Five constructions are each provably blind to it: the globally-rigid quotient embedding, the displacement-covariance operator, the Lorentzian (30), the Mosco recovery sequence, and the tension functional itself. A positive-volume void leaves every pairwise tension value, the magnitude ordering of $|T(C)|$, and the global bias $\hat{\tau}$ invariant (Computation 132), so it is invisible to all three channels through which $T$ reaches physics. The theory’s one genuinely variational term, the Landau–Ginzburg gradient energy, is minimised over the modal order parameter on the configuration lattice, never over spatial node density.

Permutation symmetry of the substrate measure forces equidistribution in measure, which forbids a systematic asymmetric bias but supplies neither the positive-rate weak ($H^{1}$-tested) discrepancy nor the leading-coefficient inequality $\eta_{K,R}<d_{0}^{2}/4$, the precise, fully-located content beyond symmetry. This is a gauge-invariant analytic rate, not a new postulate: the positive-rate weak ($H^{1}$-tested) discrepancy of the $T$-determined $\rho_{n}$ against the Riemannian volume $\sqrt{-g}\,\mathrm{d}^{4}x$ is supplied by P3’s maximum-entropy sublimation and corroborated on the i.i.d. surrogate.

A diagnosis in terms of an irreducible coordinate bulk-density freedom would be a category error: the node-quadrature $c_{u}$ above carries flat weights and is therefore chart-dependent, so a metric-preserving (diffeomorphic) void biases it only by dropping the chart’s own $g^{-1}$ and $\sqrt{g}$ factors, whereas the metric-covariant defect is identically void-blind (Computations 133 and 134); a void and its co-transforming metric form a single gauge orbit and carry no independent freedom. A7 therefore needs no bulk-density axiom and no new postulate beyond P1–P3. It is the intrinsic, gauge-invariant equicoercivity of the Boolean Dirichlet form $E^{B}_{n}$, of which $A_{n}$ is the exact pullback value (Closure (a)), resting on one diffeomorphism-invariant equidistribution rate. That rate is the standard quantitative-realisation input of any discrete-to-continuum limit, supplied for the emergence-determined $\rho_{n}$ by P3’s maximum-entropy sublimation and corroborated on the i.i.d. surrogate. A genuine fixed-metric density modulation, as opposed to a coordinate void, would bias the metric-covariant defect (Computation 134); the $\mu$-typical configuration carries none.

Two honest provisos attach to the rates just quoted. The displacement is modelled as i.i.d. Gaussian of covariance $(d_{0}^{2}/4)g$, the A4 covariance forcing, and the $|D_{n}|^{-1/2}$ rates use the i.i.d. structure of the realisation. Permutation invariance of $\mu$ delivers equidistribution in measure but not statistical independence, so the quantitative forms of A2 (a positive-rate weak ($H^{1}$-tested) discrepancy of $\rho_{n}$) and A4 (a positive-rate covariance isotropy of $\rho_{n}$) are not consequences of symmetry alone. They are realisation properties of the $\mu$-typical configuration, exactly what P3’s maximum-entropy sublimation asserts of the instantiated substrate, and the i.i.d. surrogate is the vehicle on which they are corroborated (Computations 70, 83).

18.1.10.  Numerical verification of the rate

Computation 70 verifies (240) numerically on an explicit flat-bulk toy model. Three compactly-supported smooth test functions of bounded $H^{1}$-norm (Gaussian, $C_{c}^{\infty}$ bump, $(1{-}r^{2})^{2}\cos(\pi r^{2})$) are tested at $|D_{n}|\in\{10^{4},\ldots,3\times 10^{6}\}$. Two findings:

1. The empirical std-error of $A_{n}(u)$ across independent realisations scales as $|D_{n}|^{-1/2}$ (empirical exponent $-0.467$ for the Gaussian, within the Berry–Esseen prediction $-0.5$). The statistical part of $|A_{n}-A_{\infty}|$ is exactly the central limit theorem (CLT) rate predicted above.

2. The systematic part of $|A_{n}-A_{\infty}|$ scales as $O(d_{0}^{4})$ in the fixed coherence length $d_{0}$, the next-order Taylor correction beyond the leading covariance forcing A4. It is a fixed finite-$d_{0}$ bias, small in the physical regime $\mathrm{radius}(K)\gg d_{0}$ (relative correction $O((d_{0}/\mathrm{radius}(K))^{2})$), and is bounded by $\|u\|_{H^{1}}^{2}$ with an explicit constant involving $\|\mathrm{Hess}(u)\|$.

Together these supply, on the Gaussian-bump-trigonometric class, the systematic error $O(d_{0}^{4}\,\|\mathrm{Hess}\,u\|^{2})$ and statistical error $O(d_{0}^{2}\,|D_{n}|^{-1/2})$, i.e. a relative rate $\eta_{K,R}/(d_{0}^{2}/4)=O\bigl{(}(d_{0}/\mathrm{radius}(K))^{2}+|D_{n}|^{-1/2}\bigr{)}$, small in the physical regime $\mathrm{radius}(K)\gg d_{0}$ and as $|D_{n}|\to\infty$, confirming the analysis above quantitatively. Under the rate hypothesis with a relative rate below unity the Lemma above closes A7, with a $K$-dependent constant, which is all the recovery-sequence step requires.