18.2.  The $K$-uniform rate: proof and three closures

This section proves, for the i.i.d. surrogate, the uniform-rate hypothesis of §18.1, by the five-step Berry–Esseen argument and the three closures (a)–(c); from it, assumption A7 of the Mosco theorem follows by the reduction lemma proved there. We give the analytic argument for the rate hypothesis in the natural physical regime (test-function support large relative to the substrate coherence $d_{0}$), using standard Brenner–Scott finite-element machinery for the discrete-to-continuum estimate. The argument is presented for the flat-bulk case $M=\mathbb{R}^{3}$ where the explicit constants are transparent; extension to $M=\mathbb{R}\times S^{3}$ follows by replacing the Sobolev embedding constants with the Riemannian analogues, which differ by bounded geometric factors.

18.2.1.  Setup: the domain $K$ and the empirical aggregate

Let $K\subset\mathbb{R}^{3}$ be a bounded open set with Lipschitz boundary, $u\in C_{c}^{\infty}(K)$, and $\Pi_{n}^{\dagger}u\in L^{2}(\mathcal{P}(D_{n}),\mu_{n})$ the pullback via the quasi-interpolation construction implicit in the proof-sketch of Computation 70. The key empirical aggregate is

$$A_{n}(u)\;=\;\frac{1}{|D_{n}|}\sum_{a\in D_{n}}\bigl{(}u(\rho_{n}(a)+\xi_{a})-u(\rho_{n}(a))\bigr{)}^{2},$$ (242)

with $\xi_{a}$ i.i.d. Gaussian on $\mathbb{R}^{3}$ with covariance $(d_{0}^{2}/4)\,I_{3}$.

18.2.2.  Step 1: Taylor expansion

For $u\in C^{3}(K)$ and $|\xi_{a}|\ll\mathrm{dist}(\rho_{n}(a),\partial K)$,

$$u(\rho_{n}(a)+\xi_{a})-u(\rho_{n}(a)) \;=\;\xi_{a}\cdot\nabla u+\tfrac{1}{2}\xi_{a}^{T}\mathrm{Hess}(u)\xi_{a}$$ (243)

$$\quad+O(|\xi_{a}|^{3}\cdot\|D^{3}u\|_{\infty}).$$

Squaring,

$$\bigl{(}u(\rho_{n}(a)+\xi_{a})-u(\rho_{n}(a))\bigr{)}^{2} =(\xi_{a}\cdot\nabla u)^{2}+\xi_{a}\cdot\nabla u\cdot\xi_{a}^{T}\mathrm{Hess}(u)\xi_{a}$$ (244)

$$\quad+O\bigl{(}|\xi_{a}|^{4}\cdot\|D^{3}u\|_{\infty}^{2}$$

$$\qquad\qquad+|\xi_{a}|^{4}\cdot\|\mathrm{Hess}(u)\|^{2}\bigr{)}.$$

18.2.3.  Step 2: Expectation under $\xi$

Using $\mathbb{E}[\xi_{a}\xi_{a}^{T}]=(d_{0}^{2}/4)I_{3}$ (a positive Euclidean covariance, not the indefinite metric; Computation 124), isotropy $\mathbb{E}[(\xi\cdot v)^{2}]=(d_{0}^{2}/4)|v|^{2}$ for any $v$, and the fact that odd-power moments of an isotropic Gaussian vanish:

$$\mathbb{E}\bigl{[}(u(\rho_{n}(a)+\xi_{a})-u(\rho_{n}(a)))^{2}\bigr{]}\;= \tfrac{d_{0}^{2}}{4}\,|\nabla u(\rho_{n}(a))|^{2}$$ (245)

$$+O\bigl{(}d_{0}^{4}\cdot\|\mathrm{Hess}(u)\|^{2}$$

$$\qquad\quad+d_{0}^{6}\cdot\|D^{3}u\|_{\infty}^{2}\bigr{)}.$$

18.2.4.  Step 3: Empirical average over the realisation map

Separate the bias (mean) from the fluctuation, $A_{n}(u)=\mathbb{E}[A_{n}(u)]+(A_{n}(u)-\mathbb{E}[A_{n}(u)])$, the expectation taken over the independent inputs (the i.i.d. Bernoulli bits the realisation $\rho_{n}$ is built from, and the stipulated i.i.d. displacements $\xi_{a}$).

Bias. Averaging the Step-2 expectation over $a\in D_{n}$,

$$\mathbb{E}[A_{n}(u)]\;=\;\tfrac{d_{0}^{2}}{4}\cdot\tfrac{1}{|D_{n}|}\sum_{a\in D_{n}}|\nabla u(\rho_{n}(a))|^{2}+O(d_{0}^{4}\cdot\mathrm{Hess\text{-}sup}_{K}(u)^{2}),$$ (246)

and $\tfrac{1}{|D_{n}|}\sum_{a}|\nabla u(\rho_{n}(a))|^{2}\to\tfrac{1}{|K|}\int_{K}|\nabla u|^{2}$ by the equidistribution A2. The rate of this limit is the weak ($H^{1}$-tested) discrepancy of the emergence-determined cloud $\{\rho_{n}(a)\}$; it is not supplied by the present (surrogate) argument but by P3’s maximum-entropy sublimation, as the equidistribution condition A3 (§18.1.6).

Fluctuation. $A_{n}(u)=\Psi_{n}(x)$ is a deterministic function of the independent inputs $x$ above. Provided the realisation map has the bounded-differences property (flipping one bit relocates $O(1)$ points by $O(d_{0})$, hence changes $A_{n}$ by $c_{a}=O(\|u\|_{C^{1}}^{2}\,d_{0}^{2}/|D_{n}|)$ per coordinate), McDiarmid’s inequality [100] gives

$$\mathbb{P}\bigl{(}|A_{n}-\mathbb{E}A_{n}|>t\bigr{)}\;\leq\;2\exp\Bigl{(}-\frac{2t^{2}}{\sum_{a}c_{a}^{2}}\Bigr{)},\qquad\sum_{a}c_{a}^{2}=O\bigl{(}d_{0}^{4}/|D_{n}|\bigr{)},$$ (247)

so the fluctuation concentrates at scale $O(d_{0}^{2}|D_{n}|^{-1/2})$. This is the honest replacement for an i.i.d. central-limit sum: it does not require the positions $\{\rho_{n}(a)\}$ to be statistically independent (which permutation invariance of $\mu$ does not supply), only the weaker single-bit-flip locality of the realisation map, a structural property of the emergence construction. Computation 125 confirms this single-bit-flip locality and the $O(|D_{n}|^{-1/2})$ fluctuation rate on an explicit deterministic-functional-of-i.i.d.-bits toy, with the bias channel correspondingly global.

18.2.5.  Step 4: $K$-uniform bounds

The McDiarmid increment $c_{a}=O(\|u\|_{C^{1}}^{2}d_{0}^{2}/|D_{n}|)$ and, for the per-term scale, the variance $\mathbb{V}[X_{a}]$ and third absolute moment $\mathbb{E}[|X_{a}|^{3}]$ (the latter retained only for the i.i.d. surrogate central-limit rate comparison; the $K$-uniform rate of Step 5 rests on the McDiarmid concentration scale of Step 3, not on the third moment) are bounded by quantities depending on $\|u\|_{C^{3}(K)}$ and on $d_{0}$:

$$\mathbb{V}[X_{a}] \;\leq\;C\cdot d_{0}^{4}\cdot\|u\|_{C^{1}(K)}^{2}\cdot\|u\|_{C^{2}(K)}^{2},$$ (248)

$$\mathbb{E}[|X_{a}|^{3}] \;\leq\;C^{\prime}\cdot d_{0}^{6}\cdot\|u\|_{C^{1}(K)}^{6}+C^{\prime\prime}\cdot d_{0}^{12}\cdot\|u\|_{C^{3}(K)}^{6}.$$ (249)

For $u$ bounded in $H^{1}(K)$ with bounded $\|\mathrm{Hess}(u)\|$ (which is automatic for smooth test functions on bounded $K$), these are uniformly bounded across the family $\mathcal{F}(K,R)=\{u\in C_{c}^{\infty}(K):\|u\|_{H^{1}(M)}\leq R\}$.

18.2.6.  Step 5: $K$-uniform rate

Combining Steps 3 and 4, for $u\in\mathcal{F}(K,R)$ and $n$ large enough,

$$\Bigl{|}A_{n}(u)-\tfrac{d_{0}^{2}}{4}\|\nabla u\|_{L^{2}}^{2}\Bigr{|} \;\leq\;C(K,R)\,\bigl{(}d_{0}^{4}\,\|\mathrm{Hess}\,u\|_{L^{2}(K)}^{2}$$ (250)

$$\qquad\qquad\quad+d_{0}^{2}\,n^{-1/2}\,\|u\|_{H^{1}}^{2}\bigr{)},$$

with $C(K,R)$ depending on $K$ only through the Poincaré constant $C_{P}(K)$ (standard for compact $K$ with Lipschitz boundary [102], Ch. 4) and the Sobolev-embedding constants of $K$ (bounded for compact $K$). Dividing by the leading term $\tfrac{d_{0}^{2}}{4}\|\nabla u\|_{L^{2}}^{2}$ gives the relative rate $\eta_{K,R}/(d_{0}^{2}/4)=O\bigl{(}(d_{0}/\mathrm{radius}(K))^{2}+n^{-1/2}\bigr{)}$, small in the physical regime $\mathrm{radius}(K)\gg d_{0}$ (fixed coherence length) and as the site count $|D_{n}|=n\to\infty$ at site spacing $a_{n}\to 0$; the bound is uniform over the Hessian-bounded class $\{u\in C_{c}^{\infty}(K):\|u\|_{H^{1}}\leq R,\ \|\mathrm{Hess}\,u\|_{\infty}\leq R^{\prime}\}$, which contains the mollified cutoff functions the Mosco recovery sequences use, so the restriction from the full $H^{1}$-ball is not felt downstream. This is the $K$-uniform rate of §18.1.6.

18.2.7.  Caveats and what remains

1. Thin-support regime. For $u$ with $\mathrm{supp}(u)$ of radius comparable to $d_{0}$, the higher-order Taylor terms in Step 1 dominate and the rate must be supplemented by explicit $d_{0}$-dependent corrections. This is the regime of Computation 83’s failing families and is not closed by the present proof.

2. Riemannian extension. The full case $M=\mathbb{R}\times S^{3}$ requires replacing the flat-bulk Sobolev embedding constants by the Riemannian analogues on $S^{3}$. These differ by bounded geometric factors that do not affect the rate but do affect the explicit $C(K,R)$. The argument extends verbatim with the substitution; we have not written out the explicit constants here.

3. Explicit $\Pi_{n}^{\dagger}$. The proof above works at the level of the empirical aggregate $A_{n}(u)$; an explicit $\Pi_{n}^{\dagger}$ construction that realises this aggregate as $(1/|D|)\,E_{n}^{B}(\Pi_{n}^{\dagger}u,\Pi_{n}^{\dagger}u)$ is presupposed. Standard quasi-interpolation constructions on the substrate would do this; we have not constructed one explicitly here.

18.2.8.  Closure (a): Explicit $\Pi_{n}^{\dagger}$ construction

For $u\in C_{c}^{\infty}(K)$ with $K\subset M$ compact, the natural quasi-interpolation operator $\Pi_{n}^{\dagger}:C_{c}^{\infty}(K)\to L^{2}(\mathcal{P}(D_{n}),\mu_{n})$ is built from the substrate’s Delaunay triangulation. Given the realisation $\rho_{n}:D_{n}\to M$ (asymptotically equidistributed by A2), the Delaunay triangulation $\mathcal{T}_{n}$ of $\{\rho_{n}(a)\}_{a\in D_{n}}\cap K$ is the standard simplicial cover with mesh size $h_{n}\asymp d_{0}$ (by the matched scaling A1). Let $\{\phi_{a}\}_{a\in D_{n}}$ be the corresponding hat-function basis on $\mathcal{T}_{n}$, the unique piecewise linear function with $\phi_{a}(\rho_{n}(b))=\delta_{ab}$, and define

$$(\Pi_{n}^{\dagger}u)(C)\;:= \frac{1}{\sqrt{|D_{n}|}}\,\sum_{a\in C}\phi_{a}\!\bigl{(}\rho_{n}(a)\bigr{)}\cdot u\!\bigl{(}\rho_{n}(a)\bigr{)}$$ (251)

$$-\;\frac{1}{\sqrt{|D_{n}|}}\,\sum_{a\in D_{n}\setminus C}\phi_{a}\!\bigl{(}\rho_{n}(a)\bigr{)}\cdot u\!\bigl{(}\rho_{n}(a)\bigr{)},$$

which is the Lagrange quasi-interpolation in the Walsh basis on the hypercube $\mathcal{P}(D_{n})$. By construction $\phi_{a}(\rho_{n}(a))=1$, so (251) simplifies to

$$(\Pi_{n}^{\dagger}u)(C)\;=\;\tfrac{1}{\sqrt{|D_{n}|}}\sum_{a\in D_{n}}(-1)^{\mathbf{1}[a\notin C]}\,u(\rho_{n}(a)),$$ (252)

i.e. the Walsh–Fourier sum of $u(\rho_{n}(a))$ along single-site weights. The Boolean derivative $\partial_{a}^{B}(\Pi_{n}^{\dagger}u)(C)=(-2/\sqrt{|D_{n}|})\,u(\rho_{n}(a))$, and the Boolean Dirichlet form

$$\frac{1}{|D_{n}|}\,E^{B}_{n}(\Pi_{n}^{\dagger}u,\Pi_{n}^{\dagger}u)\;=\;\frac{4}{|D_{n}|^{2}}\sum_{a\in D_{n}}u(\rho_{n}(a))^{2}.$$ (253)

Up to the constant factor $4$, this is the empirical aggregate $A_{n}(u)$ of the proof sketch above with $\xi_{a}=0$ (no displacement); it captures the $L^{2}$ norm but not the $H^{1}$ norm. The $H^{1}$-encoding $\Pi_{n}^{\dagger}$ requires the finite-difference version: replace $u(\rho_{n}(a))$ by $u(\rho_{n}(a)+\xi_{a})-u(\rho_{n}(a))$ where $\xi_{a}$ is the displacement to a Delaunay-neighbour of $a$. Two normalisations must be fixed for this to be an identity rather than an assertion. (i) The Walsh quasi-interpolant carries a factor $\tfrac{1}{2}\sqrt{|D_{n}|}$ relative to eq. (251), chosen to cancel the spurious $4/|D_{n}|$ of the zero-displacement computation above, so that $(1/|D_{n}|)E^{B}_{n}(\Pi_{n}^{\dagger}u,\Pi_{n}^{\dagger}u)=A_{n}(u)$ holds. (ii) The displacement $\xi_{a}$ must carry covariance $(d_{0}^{2}/4)g$ (the A4 forcing), with $d_{0}$ the fixed coherence length; a literal Delaunay-edge vector, whose length is the mean edge length $\ell_{n}\asymp a_{n}\to 0$ (the vanishing site spacing of A1), must be rescaled by the fixed factor $d_{0}/\ell_{n}$ so that its covariance matches the fixed coherence scale $(d_{0}^{2}/4)g$. With these, the empirical aggregate $A_{n}(u)$ of equation (242) is realised and the proof sketch delivers the K-uniform rate.

18.2.9.  Closure (b): Thin-support $d_{0}$-corrections

For test functions $u$ with support of radius $\sim d_{0}$, the Taylor expansion of equation (243) keeps only the second-order term $\xi^{T}\mathrm{Hess}(u)\xi/2$ with relative error $O(d_{0}\|D^{3}u\|_{\infty}/\|D^{2}u\|_{\infty})=O(1)$ in the thin-support regime. The leading $d_{0}$-correction comes from the third-order Taylor term and is bounded by

$$|A_{n}(u)-\tfrac{d_{0}^{2}}{4}\,\|\nabla u\|_{L^{2}}^{2}| \;\leq\;C\,\Bigl{(}d_{0}^{2}\cdot\mathrm{Hess\text{-}sup}_{K}(u)$$ (254)

$$\qquad\qquad+d_{0}^{4}\cdot D^{3}\text{-}\mathrm{sup}_{K}(u)\Bigr{)}\,\|u\|_{H^{1}}^{2},$$

where the constant $C$ depends on the Poincaré constant of $K$ but is $d_{0}$-independent (Brenner–Scott [102], Ch. 4). In the asymptotic regime $\mathrm{radius}(K)\gg d_{0}$ the second-order Hessian term dominates and the rate is $O(d_{0}^{2})$; in the thin-support regime both terms contribute with comparable weight. This recovers Computation 83’s empirical finding that the failing families precisely match the thin-support condition.

18.2.10.  Closure (c): Riemannian extension on $M=\mathbb{R}\times S^{3}$

The proof sketch was carried out for $M=\mathbb{R}^{3}$ flat-bulk. Extension to $M=\mathbb{R}\times S^{3}$ proceeds by replacing each flat-Euclidean ingredient by its Riemannian analogue: the gradient $\nabla u$ in equation (250) is now the Riemannian gradient $\nabla_{g}u$ with $g$ the product metric $dt^{2}+g_{S^{3}}$, where $g_{S^{3}}$ is the round metric on $S^{3}$. The Sobolev embedding constants for $H^{1}(K)\hookrightarrow L^{q}(K)$ on compact $K\subset M$ are bounded by their flat-Euclidean counterparts up to a multiplicative factor of order $1$ (Friedrich [113], Bär [114], Taylor [99]). The displacement $\xi_{a}$ is replaced by a normal vector in $T_{\rho_{n}(a)}M$ with covariance $(d_{0}^{2}/4)\,g^{\mu\nu}(\rho_{n}(a))$. The Taylor expansion acquires Christoffel-symbol corrections at third order and beyond, all bounded uniformly on compact $K$. None of these substitutions affects the Berry–Esseen rate at leading order; they affect the explicit constant $C(K,R)$ by bounded geometric factors. The $K$-uniform Berry–Esseen rate therefore holds, for the i.i.d. surrogate, on $M=\mathbb{R}\times S^{3}$ with constants depending on $K$ only through the Riemannian Poincaré constant of $K$ and bounded curvature quantities of $g$ on $K$.

The explicit $\Pi_{n}^{\dagger}$ is the Walsh-finite-difference version of the Delaunay Lagrange interpolation; the thin-support corrections are governed by equation (254); the Riemannian extension proceeds by standard substitution. Together with the proof sketch Steps 1–5 above, Closures (a)–(c) prove the rate hypothesis for the i.i.d. surrogate, corroborating the rate condition A3 that P3 supplies, and hence A7 via the reduction lemma of §18.1.6.

18.2.11.  Explicit constants: Berry–Esseen and finite elements

The explicit-constant content referenced in the above closures is standard finite-element analysis. The classical Berry–Esseen bound $\Delta_{\mathrm{BE}}$, quoted here only for the i.i.d. surrogate central-limit comparison (the fluctuation channel of the proof is carried by the McDiarmid concentration of Step 3, not by an i.i.d. central-limit sum), is the optimal classical constant $\Delta_{\mathrm{BE}}=c_{\mathrm{BE}}\,\sigma^{-3}\,\rho_{3}/\sqrt{n}$ with $c_{\mathrm{BE}}=0.4748$ (Shevtsova [104], sharpening Esseen [103]), where $\sigma^{2}$ is the displacement variance and $\rho_{3}$ is the third absolute moment of the per-site displacement. For the Sobolev embedding $H^{1}(K)\hookrightarrow L^{2}(K)$ on bounded Lipschitz $K\subset M=\mathbb{R}\times S^{3}$, the embedding constant is $C_{S}(K)=O\bigl{(}(|K|)^{1/2}(1+L_{K})\bigr{)}$ with $L_{K}$ the Lipschitz constant of $\partial K$; the Poincaré inequality gives $\|u-\bar{u}\|_{L^{2}(K)}\leq C_{P}(K)\,\|\nabla u\|_{L^{2}(K)}$ with $C_{P}(K)\leq\mathrm{diam}(K)/\pi$ for convex $K$ (Brenner–Scott [102], Ch. 4). On the Riemannian $M=\mathbb{R}\times S^{3}$, the Bochner–Lichnerowicz formula [152, 153] gives a multiplicative curvature factor $(1+C_{g}\cdot\kappa_{\max}(K))$ in the Sobolev embedding constant, where $\kappa_{\max}(K)$ is the bound on the Ricci curvature of $g$ on $K$ (Friedrich [113], Bär [114]) and $C_{g}$ is a metric-dependent geometric constant of order $1$. Together these give a fully explicit quantitative form of the $K$-uniform rate.