18.3.  Dirac-operator convergence: a single analytic estimate

The Mosco argument above establishes strong resolvent convergence of the Boolean Laplacian to $\Delta_{g}$; the Connes spectral-triple framework assumption requires the stronger statement that the rescaled substrate spectral triple converges to the canonical Dirac triple

$$\bigl{(}C^{\infty}(\mathbb{R}\times S^{3}),\;L^{2}(\mathcal{S}),\;D_{\mathbb{R}\times S^{3}},\;J,\;\gamma\bigr{)}$$ (255)

in an appropriate substrate-to-continuum metric. The closure target for PST is the relaxed norm-equivalence criterion of equation (257) below (§18.3.5), in the distance framework of [116], strictly weaker than Latrémolière’s spectral propinquity [117, 118] (operator-level convergence of Dirac data). The relaxation is structurally forced by the foundational object’s fixed internal Hilbert space $H_{F}=\mathbb{C}^{8}$ (derived in §13.7). Computation 9 decomposes this convergence into three ingredients and reports the status of each. The structural prerequisites are established; the residual is a single, sharply-located analytic estimate of approximation-theoretic character.

18.3.1.  The three ingredients of the Dirac estimate

1. Mosco / strong-resolvent convergence of the Boolean Dirichlet forms to the $S^{3}$ Laplace–Beltrami form. Established in §7.8; this is the form-level convergence on which the substrate’s spectral-triple identification rests.

2. Unique spin structure on $M=\mathbb{R}\times S^{3}$. The second Stiefel–Whitney class [172] $w_{2}(M)$ vanishes and $H^{1}(S^{3};\mathbb{Z}_{2})=0$, so $M$ admits a unique spin structure and the canonical Dirac operator $D_{M}$ is uniquely determined (Computation 4). Friedrich’s eigenvalue formula [113, 114] gives the $S^{3}$ Dirac spectrum $\pm(n+\tfrac{3}{2})$ with multiplicity $2(n+1)(n+2)$.

3. Boolean–canonical-anticommutation-relations (CAR) Clifford-action convergence. This is the residual: the discrete CAR Clifford action must converge to the continuum Clifford action so that the first-order operators are compatible at the substrate-to-continuum metric, not merely their squares. The PST closure target is the relaxed norm-equivalence form below (equation (257); §18.3.5).

18.3.2.  Ingredient 3, what is exact at finite substrate size: the Clifford algebra, the multiplicities and the bridge map

At every finite substrate size $D$ the Boolean–CAR generators $\chi_{a},\chi^{\prime}_{a}$ ($a=1,\dots,D$) satisfy the $\mathrm{Cl}(0,2D)$ anticommutation $\{\chi_{a},\chi_{b}\}=2\delta_{ab}=\{\chi^{\prime}_{a},\chi^{\prime}_{b}\}$, $\{\chi_{a},\chi^{\prime}_{b}\}=0$ exactly (Computation 9, section 2 of the script, machine-precision check up to $D=6$); the Walsh-mode [165] binomial multiplicities $\binom{D}{k}$ flow under the Landau–Ginzburg (LG) induced reweighting (Chapters 4 and 7) to the $(k+1)^{2}$ spherical-harmonic multiplicities at every $k$ in the thermodynamic limit (Computation 9, section 3 of the script, exhibited at $D=6$, 8, 10, 12); the bridge map $b_{n}$ from Walsh-truncation polynomials to spherical-harmonic polynomials is constructed explicitly (Computation 9, section 7 of the script); and the low-mode commutator gap vanishes exactly on Walsh-weight $\leq 1$ test functions at every $D$ ($\|[\chi^{(D)}_{a},f]-b([\gamma_{a},f])\|=0$, verified numerically for $D=3,4,5,6$ in Computation 9, section 8 of the script).

18.3.3.  Ingredient 3, what remains: the commutator residual as a Sobolev-tail estimate

The full first-order convergence reduces, given the four items above, to a uniform-in-$D$ Sobolev-tail estimate on the dense polynomial subalgebra of $C^{\infty}(S^{3})$:

$$\bigl{\|}[\chi^{(D)}_{a},f_{D}]-b_{D}([\gamma_{a},f])\bigr{\|}_{\mathrm{op}} \;\leq\;C\,D^{-s/2}\,\|f\|_{H^{s+1}(S^{3})},$$ (256)

$$\qquad f\in H^{s+1}(S^{3}),$$

where $\chi^{(D)}_{a}$ is the discrete CAR Clifford generator at substrate size $D$, $f_{D}$ is the Walsh-truncation of $f$ at weight $k_{D}=D^{1/2}$, $b_{D}$ is the bridge map of Computation 9, section 7 of the script, and $\gamma_{a}$ is the continuum Clifford action. Computation 9, section 11 of the script gives a six-step proof structure for this estimate. Steps 1–4 (spherical-harmonic decomposition, Walsh truncation, tail-norm bound, Clifford-derivation lift) and step 6 (triangle inequality + Walsh–Hadamard unitarity [165]) are standard Sobolev / Clifford-derivation manipulations on compact spin manifolds; the numerical check in Computation 9, section 13 of the script confirms the tail-rate scaling. Step 5, the discrete-derivation defect bound at rate $D^{-1/2}$, is the substantive analytic content.

18.3.4.  Step 5, the discrete-derivation defect: why no citation closes it, and what does

A direct inspection of Latrémolière 2018 [117, 118] (Computation 9, section 15 of the script) confirms that step 5 is not closed by routine citation under the spectral-propinquity framing: spectral-propinquity convergence is established case-by-case for each finite-dimensional approximation scheme. The two sphere-approximation precedents in the literature (Rieffel’s matrix-algebra $\to S^{2}$ convergence (Berezin/Kähler scheme, in the quantum Gromov–Hausdorff (QGH) distance, [115]) and Bhattacharyya–Singla’s Toeplitz-algebra [144] $\to S^{2k+1}$ convergence (Bergman scheme, also in the quantum Gromov–Hausdorff distance, [119])) both use complex-projective / coherent-state machinery that does not directly port to the Boolean cube’s real Dirichlet-form structure. Diaconis–Shahshahani’s hypercube-random-walk convergence theorem [120] gives the appropriate measure-level convergence rate $(\log D)/D$; lifting this to operator-level convergence in the spectral-propinquity framework is not the route PST commits to. The closure target (§18.3.5) is the relaxed norm-equivalence criterion of equation (257), for which the Fréchet smooth Lip-norm machinery from L_round closure (Computations 20–23) transfers; the empirical $\gamma_{D}=0.007$ at $D=8$, $N=20$ at the Walsh-weight cutoff (Computation 36) is the finite-truncation verification, with the analytical closure achieved via the refined symmetric-monomial bridge (Computations 39–42).

The Connes real spectral-triple identification has six checkable pieces. Five are verified at the structural level with no obstruction.

• Mosco resolvent convergence of the Boolean Laplacian (§7.8).

• Unique spin structure (Computation 4).

• Exact discrete Clifford algebra at every $D$ (Computation 9, section 2 of the script).

• Exact multiplicity flow (Computation 9, section 3 of the script).

• Exact low-mode commutator gap (Computation 9, section 8 of the script).

The sixth piece, the Sobolev-tail bound of equation (256), is the residual. The residual splits cleanly into two sub-statements, an observation we record explicitly (see [116]). The first is a quantum Gromov–Hausdorff section round-trip lemma analogous to Bhattacharyya–Singla 2022 Lemma 3.2. The second is the lift of that round-trip from quantum Gromov–Hausdorff to spectral propinquity, the genuinely Dirac-aware statement of equation (256).

18.3.4.1.  Section round-trip in Gromov–Hausdorff distance

The first sub-statement, the section round-trip, is the L_round side, and it closes rigorously. With $\pi_{D}:A_{D}\to C(S^{3})$ the Walsh-coefficient truncation at weight $|S|\leq k_{D}:=\lfloor\sqrt{D}\rfloor$ and $\sigma_{D}$ the trivial section, the Fréchet smooth Lip-norm $L_{F}$ of equation (259) satisfies the round-trip bound (258) at the rate $\gamma_{D}=O(D^{1/4}e^{-2\sqrt{D}})$, by the closed-form block decomposition (260) and the exact off-diagonal norm $\|P_{+}T_{\mathrm{tail}}P_{-}\|_{\mathrm{op}}=2^{D-1}-D$ of equation (261) (Computations 20, 22, 23, 50–52, 54, 55); polynomial Walsh-weight Lip-norms fail because the constant-coefficient tail has operator norm $2^{D}$ while any polynomial weight grows only as $D^{\alpha}$. Combined with the bridge construction of [116], this gives convergence $(\mathrm{Lip}(A_{D}),L_{F})\to(C(S^{3}),L)$ in the quantum Gromov–Hausdorff distance at exponential rate. The derivation in full, with the lemma chain and every singular-value computation, is §18.4.

18.3.5.  Dirac-aware lift to substrate-to-continuum metric

With the section round-trip closed (rigorously, via the block decomposition), the residual is the lift of the quantum-Gromov–Hausdorff convergence (Lip-norm only) to a Dirac-aware substrate-to-continuum metric. The original target was Latrémolière’s spectral propinquity ([117, 118]) at the level of equation (256), requiring full operator equality of bridge commutators. The actual closure target for PST (below) is the relaxed norm-equivalence criterion equation (257) in the [116] distance framework, strictly weaker than spectral propinquity but PST-appropriate as explained below. Either target requires the spinor extension of the bridge map $b_{D}$ implemented in Computation 32; the route’s first probes, the explicit substrate–Toeplitz bridge at $D=4$, its holomorphic restriction, the optimal-$\alpha$ diagnostic and the first spinor extension, are Computations 28–31, kept in Appendix B as the record of how the non-tensor coupling was arrived at.

The discrete-side audit (Computation 21, Computation 24) confirms the algebraic structure in two ways. The single-commutator Dirac Lip-norm satisfies $\|[D,\chi_{S}]\|_{\mathrm{op}}=2\sqrt{|S|}$ exactly, matching the Friedrich Dirac eigenvalue scaling $\sqrt{l(l+2)}\approx\sqrt{l}$ on round $S^{3}$ at degree $l=|S|$ ([113]; tabulated and cross-checked against Friedrich’s eigenspinor decomposition in Computation 25). And $[\chi_{a},\chi_{S}]$ is Hilbert–Schmidt [173] orthogonal to the entire Walsh subalgebra at every $D$ and every weight. The bridge defect therefore lives entirely in the non-abelian Pauli sector of the discrete CAR algebra, perpendicular to the abelian Walsh sector.

The bridge construction is narrowed by two structural results. Computation 26 establishes a structural no-go for the entire bijective single-mode class: the discrete-side commutator op-norm is exactly $\|[\chi_{a}^{\mathrm{Cliff}},\chi_{S}]\|_{\mathrm{op}}=2$ (independent of $D$ and of $|S|$), while the continuum-side norm scales as $\sqrt{l(l+1)}$ via the SU(2) Casimir; the L_round scaling on the same bridge fixes the overall weight-graded scalar, leaving no internal freedom to reconcile the two scales. Computation 27 implements the full BS 2022 Toeplitz framework on $H^{2}_{\alpha}(B^{2})$ explicitly. The Toeplitz commutator $[T_{\bar{z}_{1}^{k}},T_{z_{1}^{k}}]$ is diagonal on the monomial basis with an explicit entry $\Lambda(J,k,\alpha)$; at $k=1$ its operator norm equals $1/(\alpha+3)$ to machine precision (verified against the analytical formula). The Toeplitz operator norm $\|T_{z_{1}^{k}}\|_{\mathrm{op},\alpha}$ approaches $1$ and is $\alpha$-independent at leading order. The ratio $R(k,\alpha)=\|\mathrm{comm}\|/\|T_{z^{k}}\|$ is therefore $\alpha$-dependent, providing precisely the non-proportional $\alpha$-scaling needed. The BS 2022 closure mechanism is thus explicitly verified on $H^{2}_{\alpha}(B^{2})$: $\alpha=\alpha(D)$ supplies the internal degree of freedom needed to satisfy both fidelity and gap rate.

With the BS 2022 Toeplitz framework on $H^{2}_{\alpha}(B^{2})$ explicitly verified (Computation 27), the explicit bridge identifying the PST substrate $\mathrm{Cl}(0,2D)$ algebra with the Toeplitz algebra is the non-tensor coupling of Computation 32: the round-$S^{3}$ Dirac $D=\sum_{a}\sigma_{a}\otimes J_{a}$ on $H^{2}_{\alpha}(B^{2})\otimes\mathbb{C}^{2}$, where $J_{a}$ are the SU(2) generators acting on the Bergman monomial basis as raising/lowering operators. The framework is operational: the Hermitian Dirac [184] is verified, the su(2) algebra holds to machine precision, and the eigenvalue range approximates Friedrich’s $\pm(n+3/2)$ $S^{3}$ spectrum. The L_comm commutator $\|[D,b_{\mathrm{holo}}(\chi_{S})\otimes I]\|_{\mathrm{op}}$ is non-zero, with normalised ratios $1.02$, $1.83$–$2.29$ and $2.45$–$3.04$ at $|S|=1,2,3$, approaching the substrate target $2\sqrt{|S|}$. Mosco averaging across same-weight Walsh modes, combined with $\alpha$-rescaling (Computation 33), produces ratios that monotonically approach $1$ with weight: $0.60$, $0.71$ and $0.87$ at $k=1,2,3$. Scaling to $D=6$, $N=12$ pushes the $k=3$ ratio to $0.96$, within $4\%$ of target (Computation 34), with a clean structural crossover at $k\sim 3$ reflecting the $3+1$ structure of the round $S^{3}$ Dirac.

These successes measure $\|\mathrm{bridge}\,\mathrm{commutator}\|$ against the substrate norm $2\sqrt{|S|}$, however, which is weaker than the genuine L_comm criterion. The genuine criterion is the homomorphism gap $\|\mathrm{bridge}([D_{\mathrm{sub}},\chi_{S}])-[D_{\alpha},\mathrm{bridge}(\chi_{S})]\|_{\mathrm{op}}$, tested in Computation 35 with the natural multiplicative extension $\mathrm{bridge}(\chi_{a}^{\mathrm{Cliff}}):=J_{a}\otimes\sigma_{a}$. The result is structurally negative: the homomorphism gap is large ($\mathrm{gap}/\|\mathrm{LHS}\|\in[0.68,1.01]$) and does not shrink with $(D,N)$, uniformly across $(4,5)$, $(4,10)$, $(6,12)$. The diagnosis is a dimensional mismatch. The substrate has $D$ anti-commuting Clifford generators $\chi_{a}^{\mathrm{Cliff}}$ while the $S^{3}$ spinor fibre $\mathbb{C}^{2}$ carries only three Pauli matrices; the cyclic assignment forced upon $D>3$ identifies $\mathrm{bridge}(\chi_{a}^{\mathrm{Cliff}})=\mathrm{bridge}(\chi_{b}^{\mathrm{Cliff}})$ for $a\equiv b\pmod{3}$, which is incompatible with a homomorphism since the substrate generators anti-commute.

The L_comm closure target is therefore the relaxed norm-equivalence criterion:

$$\bigl{|}\,\|[D_{\alpha},\mathrm{bridge}(\chi_{S})]\|_{\mathrm{op}}$$ (257)

$$\qquad-\|[D_{\mathrm{sub}},\chi_{S}]\|_{\mathrm{op}}\,\bigr{|}$$

$$\;\leq\;\gamma_{D}\cdot\|\chi_{S}\|_{\mathrm{op}},$$

$$\qquad\gamma_{D}\to 0\text{ at exponential rate},$$

on the Walsh-weight-bounded subalgebra $|S|\leq k_{D}=\lfloor\sqrt{D}\rfloor$. The choice of this relaxed target is structurally forced: the alternative, enlarging the bridge spinor fibre to $\mathbb{C}^{2^{D}}$ to make the operator equality $\mathrm{bridge}([D_{\mathrm{sub}},\chi_{S}])=[D_{\alpha},\mathrm{bridge}(\chi_{S})]$ achievable, is incompatible with the foundational object’s fixed internal Hilbert space $H_{F}=\mathbb{C}^{8}$ (§13.7). The spectral-triple framework insists on a single Hilbert space per triple, and an L_comm-specific $\mathbb{C}^{2^{D}}$ spinor structure that grows with the substrate size has no home alongside the foundational $\mathbb{C}^{8}$.

18.3.6.  Why the relaxed criterion suffices for PST

Every load-bearing derivation in the book uses the substrate’s spectral-triple structure at finite $D$: Newton’s $G$ via the spectral action $\mathrm{Tr}\,f(D/\Lambda)$ (§8.5); the gauge group $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ via the unimodular unitaries of $A_{F}$ (Chapter 13); gauge bosons and Higgs as inner fluctuations of $D$ (Chapter 13); and the real K-theory (KO) dimension assignment (Computation 3). In every case the continuum limit is the endpoint of the construction, not the input. The spectral action depends on the spectral data (eigenvalues and traces) of $D$, which the relaxed criterion preserves under equation (257); full operator equality of $\mathrm{bridge}([D_{\mathrm{sub}},\chi_{S}])$ with $[D_{\alpha},\mathrm{bridge}(\chi_{S})]$ is not invoked. Closure under (257) therefore suffices for the Standard Model derivation chain. The inner-fluctuation one-form $A=\sum_{i}a_{i}[D,b_{i}]$ that generates the gauge bosons and Higgs is an operator-level object, but it is built on the continuum spectral triple $M\times F$ that equation (257) delivers as its endpoint. It lives over the continuum algebra $C^{\infty}(M)\otimes A_{F}$, with $A_{F}$ fixed at $\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$, and is not part of the substrate-to-continuum bridge data the relaxed criterion controls. The criterion supplies the continuum Dirac operator and its spectral data, on which the (exact, post-limit) inner fluctuations are then constructed in the standard way, so the inner-fluctuation case consumes only the spectral data that (257) preserves.

The distance framework is [116]’s, the same machinery already used to lift L_round closure to quantum Gromov–Hausdorff convergence. Computations 33, 40, 42 establish equation (257) at finite-$N$ matched truncation, at Bergman dimension $N\sim 2.5\,D$ (the working truncation of those computations; the asymptotic matched scaling $N(D)=2\sqrt{D}$ of the refined-bridge lemma chain below is a different convention, chosen for the infinite-$N$ limit, and the two are not to be compared). At the Walsh-weight cutoff $k_{D}=\lfloor\sqrt{D}\rfloor$, the gap $\gamma_{D}=|1-\mathrm{ratio}|$ at $N=20$ reaches $\gamma_{D}=0.007$ at $D=8$ (Computation 36), within 0.7% of the substrate target, with empirical fit $\gamma_{D}\sim 19.3\cdot e^{-0.93D}$ across $D=4,6,8$.

18.3.6.1.  Caveat on the rate

Computations 37–38 isolate the infinite-Bergman-truncation contribution to $\gamma_{D}$ via a closed-form sup-norm criterion on $\partial B^{2}$ (§18.3.5). The infinite-$N$ analytical gap at $k_{D}=2$ across $D=4,6,8$ is $\gamma_{D}^{(\infty)}=0.389,0.193,0.048$, fitting an exponential at rate $c\approx 0.527$, not $0.93$. The remainder of the apparent finite-$N=20$ rate ($0.93-0.527\approx 0.4$) is contributed by the truncation correction $\gamma_{D}^{(\infty)}-\gamma_{D}^{(N)}$, which is specific to the matched truncation $N(D)\sim 2.5\,D$. In addition, at the cutoff transition $k_{D}=2\to 3$ (between $D=9$ and $D=10$), the analytical gap jumps from $0.048$ to $0.444$ (Computation 38 at $D=10$), so the infinite-$N$ closure is not uniform in $D$ across cutoff transitions. The exponential closure verified in Computation 36 is therefore best read as a matched-truncation L_comm criterion at the working $N(D)\sim 2.5\,D$ scaling, not as a spectral-propinquity statement on the infinite Bergman space.

18.3.7.  Refined bridge: the Lemma 5–8 closure of L_comm

The infinite-$N$ closure of equation (257) requires a bridge whose substrate-to-Bergman ratio matches $2\sqrt{k}$ uniformly across $k\leq k_{D}$, smoothing the cutoff discontinuity across the Bergman cutoff transitions. Computations 39–42 construct the symmetric-monomial bridge $b_{C}(\chi_{S}):=T_{f_{|S|}}$ with $f_{k}(z_{0},z_{1})=(z_{0}z_{1})^{m(k)-1}+\beta(k)(z_{0}z_{1})^{m(k)}+\alpha(k)(z_{0}z_{1})^{m(k)+1}$, whose gap $\gamma(k)$ is $D$-independent, eliminating the cutoff discontinuity and making the Mosco averaging trivial. Four lemmas close the construction analytically, and we state each as a result; the full derivations are in §18.4.

Lemma 5 (ratio match; Computations 56–58). Because the bridge symbol depends only on $w=z_{0}z_{1}$, the constrained sup over $\partial B^{2}$ reduces to a 1D optimisation, giving the closed-form single-monomial ratio (263), which equals $2=2\sqrt{1}$ at $m=1$ exactly. Since the ratio is strictly increasing in $m$ and asymptotically $\sim m+1/2$, for every $k$ the target $2\sqrt{k}$ is bracketed by consecutive monomial ratios, and Bolzano’s theorem supplies a unique closing $\alpha(k)$ with $R_{m(k)}(\alpha(k))=2\sqrt{k}$ exactly (eq. (264)), realised as a rational function of an algebraic root (265).

Lemma 6 (Berezin compatibility; Computation 61). The reproducing-kernel adjoint identity gives $B(T_{f}T_{g}^{*})(z)=f(z)\overline{g(z)}$ exactly on the infinite Bergman space (eq. (266)), so the refined bridge realises the substrate’s weight-class structure as a holomorphic-symbol algebra, up to an $O(|z|^{2N})$ boundary correction that decays exponentially in $N$.

Lemma 7 (finite-$N$ closure rate; Computations 60, 65). At weight $k=1$ the gap is closed in elementary form, $R(N)=2\sqrt{1-1/N}$ exactly (eq. (268)), giving leading correction $-1/N$. For $k\geq 2$ the per-$k$ leading constant is ill-conditioned at moderate $N$ (diagnosed in Computations 63, 64), so the foundational bound was established by direct measurement of $|\mathrm{gap}(k,N)|\cdot N\leq 3.7$ across the operative regime, and is now derived at its maximiser and in its large-$N$ limit (Computations 213, 214), $k\leq k_{D}$ (eq. (269), median $0.95$, Computation 150; certified in exact arithmetic over $k\leq 20$ by Computation 203, and the single-monomial part of the problem closed there for every $m$), yielding the L_comm closure rate $\gamma_{D}=O(1/\sqrt{D})$ uniformly in $k\leq k_{D}$ (eq. (270)), polynomial in the substrate size.

Lemma 8 (spectral-action invariance; Computation 62). At the matched substrate-to-Bergman scaling (271) the eigenvalue counts match ($2^{D}$ exactly on the substrate side) and the spectral actions are commensurate: $S_{\mathrm{sub}}/2^{D}=e^{-1}$ exactly, while $S_{\mathrm{Bergman}}/2^{D}$ converges to a $D$-independent constant, the standard QGH-convergence statement applied to spectral actions.

The L_comm closure programme is a complete theorem at the relaxed-criterion level, rigorous in all parts except the uniform-in-$k$ bound of Lemma 7(b), which is derived at its maximiser and in its large-$N$ limit and scanned between (Computations 213, 214). A proof over the whole regime is the one remaining quantitative refinement (Chapter 19). The relaxed criterion is sufficient for the downstream spectral-action and gauge-group recoveries (§Why the relaxed criterion suffices for PST above). The closed sub-results are:

• the ratio-matching theorem Lemma 5(a, b, c) (Computations 56–58);

• the Berezin-transform identity Lemma 6 (Computation 61, equation (266));

• the closed-form finite-$N$ gap Lemma 7(a) (Computation 60, equation (268));

• the uniform-in-$k$ direct-gap bound Lemma 7(b) (Computation 65, equation (269)), established by measurement across $k\in[1,16]$ and $N\in[12,32]$ there, since extended to $k\leq 80$ and $N\leq 600$ and derived at its maximiser (Computations 203, 214);

• and the spectral-action invariance corollary Lemma 8 (Computation 62, equation (271)).

The foundational closure rests on the direct measurement of $|\mathrm{gap}(k,N)|\cdot N$ across the relevant $(k,N)$ range (Computation 65), giving the L_comm closure rate $\gamma_{D}=O(1/\sqrt{D})$ uniformly in $k\leq k_{D}$. See also Appendix B rows 22–65.