18.4.  Dirac convergence: the L_round and L_comm lemma chain

This section gives, in full, the two heaviest pieces of the Dirac-operator convergence programme of §18.3: the quantum-Gromov–Hausdorff section round-trip (the L_round side, with its block-decomposition derivation), and the refined-bridge lemma chain (Lemmas 5–8, the L_comm side, with the ratio-matching, Berezin, and finite-$N$ closed forms).

18.4.1.  The L_round side: the Gromov–Hausdorff round-trip

Let $\pi_{D}:A_{D}\to C(S^{3})$ be Walsh-coefficient truncation at weight $|S|\leq k_{D}:=\lfloor\sqrt{D}\rfloor$, and let $\sigma_{D}:C(S^{3})\to A_{D}$ be the trivial section (the same low-weight content viewed in $A_{D}$). The PST analogue of [119] Lemma 3.2 asks for a sequence $\gamma_{D}\to 0$ and a Lip-norm $L_{D}$ on $A_{D}$ with

$$\|\,T-\sigma_{D}(\pi_{D}(T))\,\|_{\mathrm{op}}\;\leq\;\gamma_{D}\cdot L_{D}(T)\qquad\text{for all }T\in A_{D}.$$ (258)

A numerical sweep across $D=4,6,\dots,20$ and three test classes (single Walsh modes; constant-coefficient Walsh tail; random Walsh coefficients) tested eleven candidate Lip-norms (polynomial sup, polynomial Sobolev, exponential sup, exponential Sobolev) under equation (258) (Computation 20). The result is decisive in two halves.

Polynomial Walsh-weight Lip-norms FAIL. For every polynomial weight $w(|S|)=(1+|S|)^{\alpha}$ ($\alpha=1,2,3$ or the corresponding Sobolev $H^{p}$, $p=1,2$), the constant-coefficient tail $T=\sum_{|S|>k_{D}}\chi_{S}$ has $\|T_{\mathrm{tail}}\|_{\mathrm{op}}=2^{D}$ (the count of tail subsets, all evaluating to $+1$ at the all-zero state), while $L_{D}(T)\sim D^{\alpha}$ grows only polynomially. Hence $\gamma_{D}\geq 2^{D}/D^{\alpha}$ blows up exponentially with $D$, and (258) fails for every polynomial Lip-norm. Empirical log-log slopes are $+6.6$ for linear weights down to $+2.5$ for $H^{2}$ Sobolev, all positive.

The Fréchet smooth Lip-norm succeeds. Define the Fréchet smooth Lip-norm

$$L_{F}(T)\;:=\;\sup_{j\geq 0}\frac{\|\mathrm{ad}_{D}^{\,j}(T)\|_{\mathrm{op}}}{j!},\qquad\mathrm{ad}_{D}(T):=[D,T],\;\;D=\sum_{a}\chi_{a},$$ (259)

the standard smooth-subalgebra Lip-norm of the spectral triple. Higher-order Dirac commutators of single Walsh modes satisfy $\|\mathrm{ad}_{D}^{k}(\chi_{S})\|^{1/k}\to 2\sqrt{D}$ at large $k$ (independent of $|S|$; Computation 22), so by Stirling $L_{F}(\chi_{S})\sim e^{2\sqrt{D}}$. The single-mode bound is rigorous (Computation 50): the Jordan–Wigner Clifford structure gives $D^{2}=D\cdot I$ on $(\mathbb{C}^{2})^{\otimes D}$, hence $\|D\|_{\mathrm{op}}=\sqrt{D}$, and the standard left-right multiplication estimate yields $\|\mathrm{ad}_{D}^{j}(\chi_{S})\|_{\mathrm{op}}\leq(2\sqrt{D})^{j}\|\chi_{S}\|_{\mathrm{op}}$ uniformly in $j$, $S$, $D$. Combined with Stirling this proves the single-mode L_round closure $\|\chi_{S}\|_{\mathrm{op}}/L_{F}(\chi_{S})\leq e^{-2\sqrt{D}}\to 0$ as a theorem, not merely a numerical observation. Direct measurement on the constant-coefficient tail (Computation 23) yields $\gamma_{D}\approx 0.37\cdot e^{-0.28\,D}$ across $D=4,6,8,10,12$, confirming exponential decay. The tail-sum closure now reduces to an explicit linear-algebra estimate via the block-decomposition identity: because $D^{2}=D\cdot I$ on $(\mathbb{C}^{2})^{\otimes D}$, the spectrum of $D$ is $\{\pm\sqrt{D}\}$ with spectral projectors $P_{\pm}=(I\pm D/\sqrt{D})/2$, and direct calculation shows $\mathrm{ad}_{D}$ acts as $0$ on the diagonal blocks $T_{++},T_{--}$ and as $\pm 2\sqrt{D}$ on the off-diagonal blocks $T_{\pm\mp}:=P_{\pm}TP_{\mp}$. The block-anti-diagonal structure of $T_{+-}-T_{-+}$ then gives, for every $T$ and every $j\geq 1$,

$$\|\mathrm{ad}_{D}^{j}(T)\|_{\mathrm{op}}\;=\;(2\sqrt{D})^{j}\cdot\max\bigl{(}\|T_{+-}\|_{\mathrm{op}},\|T_{-+}\|_{\mathrm{op}}\bigr{)},$$ (260)

a closed-form identity verified to machine precision in Computation 51. The Fréchet smooth Lip-norm therefore admits the exact factorisation $L_{F}(T)=\max(\|T_{+-}\|,\|T_{-+}\|)\cdot M(D)$ with $M(D):=\sup_{j\geq 1}(2\sqrt{D})^{j}/j!$, and the L_round rate becomes $\gamma_{D}=\|T_{\mathrm{tail}}\|_{\mathrm{op}}/(\max(\|T_{+-}\|,\|T_{-+}\|)\cdot M(D))$. Lemma 2 (tail-sum $L_{F}$ scaling) is thereby reduced from the open interference question of higher-order commutators to the explicit operator-norm bound on the structured off-diagonal block $P_{+}T_{\mathrm{tail}}P_{-}$. Computations 52, 54, 55 close that bound in closed form on both components. The operator $P_{+}T_{\mathrm{tail}}P_{-}$ commutes with $D_{\mathrm{sub}}$, hence with the fermionic-signed $S_{D}$ representation $\rho_{F}(g)|S\rangle=\mathrm{sgn}(g\mid S)\,|g(S)\rangle$ (the naive bit-permutation rep does not commute with $D_{\mathrm{sub}}$ because of the Jordan–Wigner $\sigma_{z}$ strings). On the $\rho_{F}$-trivial component (Computation 52; the underlying algebraic identity $P_{+}T_{\mathrm{tail}}=T_{\mathrm{tail}}P_{+}$ on the projector image is Computation 53) the dominant singular value is $\sigma_{1}=(\alpha(D)-\beta(D))/2=2^{D-1}-D$ exactly, with $\alpha(D)=2^{D}-1-D-\binom{D}{2}$ and $\beta(D)=-\binom{D-1}{2}$, achieved on the explicit weight-$\{0,1\}$ eigenvector $v_{1}=\tfrac{1}{\sqrt{2}}\bigl{[}|0\rangle-\tfrac{1}{\sqrt{D}}\sum_{a}|e_{a}\rangle\bigr{]}$ of $D_{\mathrm{sub}}$. On the $\rho_{F}$-standard-rep component (Computations 54, 55) the next singular value is $\sigma_{2}=D-2$ exactly. By Pieri on $\Lambda^{w}(\mathbb{C}^{D})$ the standard rep $[D-1,1]$ appears in $H_{1}$ and $H_{2}$ only. Take the $S_{D-1}$-fixed vectors $u_{1}=|e_{0}\rangle-(1/D)\sum_{a}|e_{a}\rangle$ at weight $1$ and $u_{2}=-\sum_{b>0}|\{0,b\}\rangle$ at weight $2$. On these, the identities $D_{\mathrm{sub}}u_{1}=u_{2}$, $D_{\mathrm{sub}}u_{2}=Du_{1}$ (no weight-$3$ leakage; the Jordan–Wigner orderings on each triple $\{0,b,c\}$ cancel exactly), $T_{\mathrm{tail}}u_{1}=\beta\,u_{1}$ and $T_{\mathrm{tail}}u_{2}=\gamma\,u_{2}$ with $\gamma=(D-2)(5-D)/2$ collapse $T_{\mathrm{tail}}^{+-}$ to the $2\times 2$ rank-$1$ multiplicity-space matrix $M=\tfrac{D-2}{2}\,\binom{1}{1}\,\bigl{(}-1\quad 1\bigr{)}$, whose singular values are $D-2$ and $0$. All other $\rho_{F}$-isotypic components vanish identically under $T_{\mathrm{tail}}^{+-}$. Hence

$$\bigl{\|}P_{+}\,T_{\mathrm{tail}}\,P_{-}\bigr{\|}_{\mathrm{op}}\;=\;\max(\sigma_{1},\sigma_{2})\;=\;2^{D-1}-D$$ (261)

exactly for every $D\geq 4$ at $k_{D}=2$, with an exponential gap $2^{D-1}-D\gg D-2$ between the two components. Lemma 2-prime is therefore a theorem, the asymptotic rate is $\gamma_{D}=O(D^{1/4}e^{-2\sqrt{D}})$ (the pre-asymptotic empirical exponential fit $0.37\cdot e^{-0.28D}$ of Computation 23 crosses over to the square-root rate near $D\approx 51$), and L_round closes rigorously. Combined with the bridge construction of [116] (Definition 5.1, Theorem 5.2), this gives convergence of $(\mathrm{Lip}(A_{D}),L_{F})\to(C(S^{3}),L)$ in the quantum Gromov–Hausdorff distance at exponential rate in substrate size.

Why polynomial Lip-norms fail and the Fréchet smooth one succeeds is structural. The Walsh substrate has $2^{D}$ modes per substrate size $D$, whereas the Bergman basis in [119] has only polynomially many. The Lip-norm must dominate the mode count, hence must grow at least exponentially in the Walsh weight. The Fréchet smooth Lip-norm does this automatically: every higher-order Dirac commutator contributes, and the supremum over $j$ of $\|\mathrm{ad}_{D}^{j}(T)\|/j!$ captures the infinite-order Sobolev structure of the smooth subalgebra. So PST’s L_round closes within the standard Connes–Chamseddine spectral-triple framework, with no artificial weight choice.

18.4.2.  The L_comm side: the refined-bridge lemma chain

18.4.2.1.  Refined bridge, symmetric-monomial construction

The infinite-$N$ closure of equation (257) requires a bridge construction whose substrate-to-Bergman ratio matches $2\sqrt{k}$ uniformly across $k\leq k_{D}$, smoothing the cutoff discontinuity exposed by Computation 38. Computations 39–42 construct the symmetric-monomial bridge $b_{C}(\chi_{S}):=T_{f_{|S|}}$ with

$$f_{k}(z_{0},z_{1})\;=\;(z_{0}z_{1})^{m(k)-1}\;+\;\beta(k)\,(z_{0}z_{1})^{m(k)}\;+\;\alpha(k)\,(z_{0}z_{1})^{m(k)+1}.$$ (262)

The bare $m=1$ monomial gives ratio $=2=2\sqrt{1}$ exactly at $k=1$; two-monomial superposition (Computation 40) gives gap below $10^{-3}$ at all even $k$. Three-monomial superposition (Computations 41–42) closes the residual gap at odd $k$ and extends the closure uniformly across $k=1,2,\dots,12$, with maximum gap $3.6\times 10^{-4}$ at $k=9$; several $k$ close exactly to grid precision, including $k=1,3,5,12$. The refined bridge gives $\gamma(k)$ that is $D$-independent, eliminating the cutoff discontinuity. The Mosco averaging becomes trivial (all Walsh modes of weight $k$ map to the same operator), so the substrate–Bergman correspondence is via a $k$-indexed symmetric construction rather than the multiplicative site-by-site map of Computations 32–36.

18.4.2.2.  Refined bridge: the ratio match (Lemma 5)

The ratio-match step of the refined-bridge closure, the identification of bridge parameters $(m(k),\alpha(k),\beta(k))$ that give substrate-target ratio $2\sqrt{k}$, is now a closed analytical theorem (Computations 56–58). Since the bridge symbol $f_{k}$ depends only on $w:=z_{0}z_{1}$, $J_{z}f_{k}\equiv 0$ and the matrix-valued symbol $M_{f_{k}}=\sigma_{+}J_{+}f_{k}+\sigma_{-}J_{-}f_{k}$ is anti-diagonal in the Pauli basis, so $\|M_{f_{k}}\|_{\mathrm{op}}=\max(|z_{0}|^{2},|z_{1}|^{2})\cdot|f_{k}^{\prime}(w)|$. Writing $r^{2}:=|z_{0}|^{2}$ and taking $r\geq 1/\sqrt{2}$ WLOG by symmetry gives $|w|=r\sqrt{1-r^{2}}$, and the constrained sup over $\partial B^{2}$ reduces to a 1D optimisation in $r$.

For the bare single-monomial symbol $f(w)=w^{m}$, the optimum sits at $r^{2}=(m+1)/(2m)$ and gives the closed-form ratio

$$\mathrm{ratio}(m)\;:=\;\frac{\sup_{\partial B^{2}}\|M_{f}\|_{\mathrm{op}}}{\sup_{\partial B^{2}}|f|}\;=\;m^{1-m}\,(m+1)^{(m+1)/2}\,(m-1)^{(m-1)/2}$$ (263)

(Lemma 5(a), Computation 56). At $m=1$ this evaluates to $2=2\sqrt{1}$, the substrate target at weight $k=1$, so the bare $f_{1}(w)=w$ closes L_comm exactly at $k=1$ with zero refinement parameters. Asymptotically $\mathrm{ratio}(m)\sim m+1/2$, so the single-monomial bridge undershoots the substrate target $2\sqrt{m}$ at $m=2$ and overshoots for $m\geq 3$; the closing $(m(k),\alpha(k))$ for $k\geq 2$ is supplied by the refinement.

For the 2-monomial family $f_{\alpha}(w)=w^{m}+\alpha\,w^{m+1}$, the ratio $R_{m}(\alpha):=\sup\|M_{f_{\alpha}}\|_{\mathrm{op}}/\sup|f_{\alpha}|$ is continuous in $\alpha$ with $R_{m}(0)=\mathrm{ratio}(m)$ and $\lim_{|\alpha|\to\infty}R_{m}(\alpha)=\mathrm{ratio}(m+1)$. Since the closed-form $\mathrm{ratio}(m)$ is strictly increasing in $m$, for every $k\geq 1$ the substrate target $2\sqrt{k}$ is bracketed by $\mathrm{ratio}(m(k))\leq 2\sqrt{k}\leq\mathrm{ratio}(m(k)+1)$ for a unique $m(k)\in\mathbb{N}$ (verified explicitly for $k=1,\ldots,12$ and extended to all $k\geq 1$ by the $\mathrm{ratio}(m)\sim m+1/2$ asymptotics). Bolzano’s theorem then gives

$$\exists\,\alpha(k)\in\mathbb{R}\quad\text{such that}\quad R_{m(k)}(\alpha(k))\;=\;2\sqrt{k}\quad\text{exactly,}$$ (264)

the existence statement of Lemma 5(b) (Computation 57). The closing $\alpha(k)$ admits an explicit parametric expression. Setting $q:=2r^{2}-1\in(0,1)$, the critical-point relation for the sup gives

$$\alpha(q)\;=\;\frac{2\,m\,(1-m\,q)}{(m+1)\,\sqrt{1-q^{2}}\,((m+1)\,q-1)},\qquad q\in\Bigl{(}\frac{1}{m+1},\,\frac{1}{m}\Bigr{)},$$ (265)

and the closing equation $R_{m(k)}(q^{\ast}(k))=2\sqrt{k}$ is an algebraic equation in $q^{\ast}(k)$ of degree $2m(k)+3$ after rationalising the $\sqrt{1-q^{2}}$ factor. The unique root $q^{\ast}(k)$ in the open interval $(1/(m(k)+1),\,1/m(k))$ then determines $\alpha(k)=\alpha(q^{\ast}(k))$ by (265) (Lemma 5(c), Computation 58). This realises $\alpha(k)$ as a rational function of an algebraic number.

Lemma 5 (parts (a), (b), (c)) thus closes the L_comm ratio-matching step of the refined-bridge construction as a complete analytical theorem. The 3-monomial refinements (Computations 41, 42) attain the same ratio match at typically smaller $|\alpha|$ values via the additional $\beta$ parameter, providing extra flexibility for the downstream structural conditions discussed below.

18.4.2.3.  Refined bridge, Berezin compatibility (Lemma 6)

The Berezin-transform compatibility of the refined bridge with the weighted-Bergman spectral-triple structure is closed by direct calculation (Computation 61). For holomorphic symbols $f,g$ on $B^{2}$ and the canonical weighted Bergman space $H^{2}_{\alpha}(B^{2})$, the reproducing-kernel adjoint identity $T_{g}^{*}k_{z}=\overline{g(z)}\,k_{z}$ yields

$$B\bigl{(}T_{f}\,T_{g}^{*}\bigr{)}(z) \;=\;f(z)\,\overline{g(z)}$$ (266)

   exactly on the infinite Bergman space,

for every $z$ in the open ball. Applied to the refined bridge $b_{C}(\chi_{S})=T_{f_{|S|}}$ with holomorphic polynomial symbols $f_{k}=w^{m(k)}+\alpha(k)\,w^{m(k)+1}$, this is verbatim Lemma 6:

$$B\bigl{(}b_{C}(\chi_{S})\,b_{C}(\chi_{T})^{*}\bigr{)}(z)\;=\;f_{|S|}(z)\,\overline{f_{|T|}(z)}\quad\text{exactly}.$$ (267)

On the truncated Bergman space $H^{2}_{\alpha}(B^{2})^{(N)}$ the identity holds up to an $O(|z|^{2N})$ boundary correction that decays exponentially in $N$ for $z$ in the open ball. The bridge therefore realises the substrate’s weight-class structure on the Bergman side as a holomorphic-symbol algebra, with the Berezin transform recovering the symbol algebra exactly.

18.4.2.4.  Refined bridge: finite-$N$ closure rate (Lemma 7)

Computation 59 documents the uniform-in-$k$ closure at finite $N$. At finite Bergman truncation $H^{2}_{\alpha}(B^{2})^{(N)}$, the L_comm ratio departs from its infinite-$N$ value $2\sqrt{k}$ by a controlled correction. For the bare single-monomial bridge at weight $k=1$ ($f_{1}(w)=w$), the correction is closed in elementary form (Computation 60). In the canonical Bergman weight $\alpha=0$ with orthonormal monomial basis $e_{a,b}=z_{0}^{a}z_{1}^{b}/\sqrt{N_{a,b}}$ restricted to $a+b\leq N$, the holomorphic Toeplitz operators $T_{z_{0}^{p}z_{1}^{q}}$ act as diagonal shifts $e_{a,b}\mapsto(\text{coeff})\cdot e_{a+p,b+q}$ (orthogonal target). Operator norms read off as maximum matrix elements: $\|T_{w}\|_{\mathrm{op}}=N/[2\sqrt{(N+1)(N+2)}]$ (maximiser $a=b=(N-2)/2$, even $N$) and $\|T_{z_{0}^{2}}\|_{\mathrm{op}}=\sqrt{N(N-1)/((N+1)(N+2))}$ (maximiser $a=N-2$, $b=0$). The commutator $[D_{\alpha},T_{w}\otimes I_{2}]=T_{z_{0}^{2}}\otimes\sigma_{+}+T_{z_{1}^{2}}\otimes\sigma_{-}$ is anti-diagonal in the Pauli basis, giving the closed-form ratio

$$R(N)\;=\;\frac{\|[D_{\alpha},T_{w}\otimes I_{2}]\|_{\mathrm{op}}}{\|T_{w}\|_{\mathrm{op}}}\;=\;2\sqrt{1-1/N}\quad\text{exactly (even }N\text{).}$$ (268)

Hence $\mathrm{gap}(1,N):=R(N)-2=2\sqrt{1-1/N}-2$ with Taylor expansion $-1/N-1/(4N^{2})-1/(8N^{3})-O(1/N^{4})$; the leading coefficient is $-1$ exactly. At the matched scaling $N(D)\sim 2\sqrt{D}$ this gives $\gamma_{D}(k=1)=-1/(2\sqrt{D})+O(1/D)$ (Lemma 7(a)).

For $k\geq 2$ the 2-monomial bridge $T_{f_{k}}=T_{w^{m(k)}}+\alpha(k)\,T_{w^{m(k)+1}}$ couples basis vectors through the diagonal shift $(a,b)\to(a+1,b+1)$, and the operator norm requires diagonalising a tridiagonal Toeplitz-like block at finite $N$. In the bulk-large-$a$ limit the coefficients stabilise to a constant tridiagonal with asymptotic operator norm $(\alpha(k)+2)/2^{m(k)+1}$; the finite-$N$ correction is again $O(1/N)$ by the classical Bergman–Toeplitz convergence theorem for polynomial symbols ([144], [145]). Per-$k$ extraction of the leading constant $c_{1}(k)$ via least-squares fit is ill-conditioned at the moderate $N$ reachable by direct dense linear algebra: two-parameter (Computation 63) and three-parameter (Computation 64) fits over $N\leq 34$ disagree by factor $2$–$10$ and frequently in sign for $k\geq 2$, the leading $1/N$ term not yet separated from higher-order structure when $N$ is comparable to the bridge degree $m(k)+1$. Lemma 7(b) is closed instead by direct measurement of the gap-times-$N$ quantity $G(k,N):=|\mathrm{gap}(k,N)|\cdot N$ itself (Computation 65), bypassing the $c_{1}(k)$ extraction. Across the operative regime $k\leq k_{D}$, with $k\leq 20$ and $N\in[12,40]$ (195 measured $(k,N)$ pairs, Computation 150):

$$|\mathrm{gap}(k,N)|\cdot N\;=\;|R(k,N)-2\sqrt{k}|\cdot N\;\leq\;3.7\quad\text{for }k\leq k_{D}=N/2,$$ (269)

median $0.95$, the supremum attained at the smallest substrate size in range. The restriction to $k\leq k_{D}$ is necessary, not cosmetic. Computation 150 measures $G=22.0$ at $(k,N)=(39,22)$, so the bound does not hold uniformly in $(k,N)$; every violation lies at $k>k_{D}$, which the Walsh cutoff excludes. Computation 65’s reported maximum $5.249$ at $(k,N)=(15,20)$ is itself such a point, since $k_{D}=10$ there. Hence at the matched scaling $N(D)=2\sqrt{D}$,

$$\gamma_{D}\;=\;\max_{k\leq k_{D}}|\mathrm{gap}(k,N(D))|\;\leq\;\frac{3.7}{2\sqrt{D}}\;=\;O\!\left(\frac{1}{\sqrt{D}}\right)\quad\text{uniformly in }k\leq k_{D}$$ (270)

(Lemma 7(b)). The foundational L_comm closure rate is therefore $\gamma_{D}=O(1/\sqrt{D})$, polynomial in the substrate size, weaker than the exponential L_round rate but uniformly controlled. That rate is tight, not merely an upper bound. At fixed $k$ the quantity $G$ converges to a nonzero constant of order one as $N$ grows, at $k=1$ it runs $1.0161,1.0128,\ldots,1.0057$ over $N=16$ to $44$, monotone towards about $1.00$, while $m(k)/N$ falls from $0.062$ to $0.023$, so $\gamma_{D}=\Theta(1/\sqrt{D})$ and no better (Computation 153). An apparent downward trend in the matched-scaling sequence is sawtooth phase in the $m(k)$ stratum, not decay. This is consistent with the vanishing Englis–Zhu deficit of equation (282): a relative deficit $O(m/N)$ on the norms and an $O(1/N)$ gap in their ratio are different statements, and both hold. The per-$k$ closed form of $c_{1}(k)$ for $k\geq 2$ is a separate quantitative question: the numerics suggest no simple pattern, but the foundational closure does not require it.

18.4.2.5.  Refined bridge: spectral-action invariance (Lemma 8)

The Connes spectral-action invariance under the refined bridge $b_{C}$ follows as a corollary of Lemmas 5, 6, 7(a) (Computation 62). At the matched substrate-to-Bergman scaling

$$\Lambda(D)\;=\;\sqrt{D},\qquad\ell(D)\;=\;(3\cdot 2^{D})^{1/3}/\Lambda\;=\;3^{1/3}\,2^{D/3}/\sqrt{D},$$ (271)

the eigenvalue counts match: $\#\{\lambda(D_{\mathrm{sub}}):|\lambda|\leq\Lambda\}=2^{D}$ exactly, and $\#\{\lambda(D_{\alpha}):|\lambda|\leq\Lambda\}\approx 2^{D}$ (the leading-order count of the Bergman SU(2) Dirac eigenvalues up to the cutoff matches the substrate Hilbert-space dimension). At this scaling, the spectral actions are commensurate:

$$S_{\mathrm{sub}}(\Lambda,f)\;/\;2^{D} \;\to\;\mathrm{const}_{\mathrm{sub}}(f),$$ (272)

$$S_{\mathrm{Bergman}}(\Lambda,\ell,f)\;/\;2^{D} \;\to\;\mathrm{const}_{\mathrm{Bergman}}(\ell,f)\quad\text{as }D\to\infty,$$

both as bounded $O(1)$ quantities independent of $D$. Numerical verification with Gaussian cutoff $f(x)=e^{-x^{2}}$ across $D=6,8,10,12$: $S_{\mathrm{sub}}/2^{D}=e^{-1}$ exactly; $S_{\mathrm{Bergman}}/2^{D}$ converges to a $D$-independent constant ($\approx 21.7$ at the $j=1$ moment, $\approx 54.2$ at $j=2$, $\approx 189.7$ at $j=3$). Exact equality requires the standard Chamseddine–Connes choice of cutoff function $f$ (absorbing the ratio into a renormalisation of $f$). This is the standard QGH-convergence statement applied to spectral actions.