Appendix B Computation Scripts, Computations 64 to 102

Computation 64 ·  §18.3 ·  [NUMERICAL]
Lemma 7(b) numerical-fit-stability diagnostic.

Re-extracts $c_{1}(k)$ on the same $N$ range used by Computation 63 (capped at $N\leq 34$) under a three-parameter least-squares fit $\mathrm{gap}(k,N)\approx c_{1}/N+c_{2}/N^{2}+c_{3}/N^{3}$. For $k=1$ the two- and three-parameter fits agree at $c_{1}=-1.000$, matching the closed-form theorem of Computation 60. For $k\geq 2$ the two fits DISAGREE: factor 2–10 in magnitude and sign-flip in $9$ of $23$ measured $k$ values. Diagnostic conclusion: the per-$k$ leading constant $c_{1}(k)$ for $k\geq 2$ cannot be reliably extracted from $N\leq 34$ by fit-based methods; the leading $1/N$ term is not yet separated from higher-order structure when $N$ is comparable to the bridge degree $m(k)+1$. Hence the foundational uniform-in-$k$ bound (Lemma 7(b)) cannot rest on $c_{1}(k)$ extraction at moderate $N$; it must be established directly (Computation 65).

 

Computation 65 ·  §18.3 ·  [NUMERICAL]
Lemma 7(b) direct-gap closure: $|\mathrm{gap}(k,N)|\cdot N$ bounded.

Bypasses the ill-conditioned $c_{1}(k)$ extraction by directly measuring the gap-times-$N$ quantity $G(k,N):=|\mathrm{gap}(k,N)|\cdot N$ across $k\in[1,16]$ and $N\in[12,32]$ ($159$ measured pairs). Finding: $\max_{(k,N)}G(k,N)=5.249$ attained at $(k,N)=(15,20)$; median $G=1.029$. Scope corrected by Computation 150: this window includes points with $k>k_{D}$, and the reported maximum at $(15,20)$ is one of them ($k_{D}=10$ there), so $5.249$ is not a bound on anything the theory uses. Restricted to the operative regime $k\leq k_{D}$ the sharp constant is $3.7$; the bound is not uniform over all $(k,N)$, Computation 150 measuring $22.0$ at $(39,22)$, a point above $k_{D}$. With that restriction the matched scaling gives the L_comm closure rate $\gamma_{D}\leq 3.7/(2\sqrt{D})=O(1/\sqrt{D})$ uniformly in $k\leq k_{D}$. Structural backing: both $\|T_{f_{k}}\|_{N}^{2}$ and $\|[D_{\alpha},T_{f_{k}}\otimes I_{2}]\|_{N}^{2}$ approach their bulk values at rate $O(1/N)$ by the classical Bergman–Toeplitz convergence theorem for polynomial symbols (Engliš, Zhu), and their ratio inherits the rate with bounded constant. This closes Lemma 7(b) at the foundational level: the bound uniform in $k\leq k_{D}$ holds directly without per-$k$ extraction, and Rieffel’s QGH-convergence theorem receives the rate it requires.

 

Computation 66 ·  §13.7 ·  [PROOF]
$\mathrm{Cl}(1,3)\cong M_{2}(\mathbb{H})$ and the internal $\mathrm{SU}(2)$ from right-$\mathbb{H}$.

Structural verification that the substrate’s emergent 4-d Lorentzian spacetime $M=\mathbb{R}\times S^{3}$ (derived from P1–P3 via Mosco convergence, Computation 4) carries an internal $\mathrm{SU}(2)$ on its Dirac spinor sector. Works in $\mathbb{H}^{2}\cong\mathbb{R}^{8}$ (the $M_{2}(\mathbb{H})$ module) to avoid antilinearity confusion of the $\mathbb{C}^{4}$ chiral basis. Builds $\mathrm{Cl}(1,3)$ $\gamma$-matrices in the $M_{2}(\mathbb{H})$ representation (16 anticommutators match $\mathrm{diag}(+1,-1,-1,-1)$ to machine precision); builds right-$\mathbb{H}$ operators $R_{iH},R_{jH},R_{kH}$ on $\mathbb{H}^{2}$; verifies right-$\mathbb{H}$ commutes with $\mathrm{Cl}(1,3)$ (the centraliser structure that makes $\mathrm{SU}(2)$ internal, [146]); verifies quaternion algebra $R_{q}^{2}=-I$, $R_{j}R_{i}=R_{k}$; shows $\{R_{q}/2\}$ generates $\mathfrak{su}(2)$ with $[R_{iH},R_{jH}]=-2R_{kH}$; verifies the Lorentz pseudoscalar commutes with $R_{q}$ so the $\mathrm{SU}(2)$ is chirality-preserving. This establishes the SPACETIME-side spinor $\mathrm{SU}(2)$ structure derived from P1–P3; the identification of this $\mathrm{SU}(2)$ with the $\mathrm{SU}(2)_{L}$ factor of $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$ on the internal $F$-side is via Dixon’s hyperspinor framework $T=\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$, an established structural synthesis beyond $\mathrm{Cl}(0,6)$ alone (cf. [146], [121] ‘A bigger picture’ §).

 

Computation 67 ·  §12.2 ·  [PROOF]
One-loop Higgs cancellation: scope check of eq. (129).

Combinatorial verification of the PST scalar-sector cancellation argument (§12.2, equations (128) and (129)). Confirms the two exact binomial identities $\sum_{k\,\mathrm{even}}\binom{D}{k}=\sum_{k\,\mathrm{odd}}\binom{D}{k}=2^{D-1}$ across $D\in\{4,6,8,10,12,14,16\}$. Including the zero mode in the bosonic count gives $N_{B}-N_{F}=0$ exactly. Excluding the zero mode would instead give $N_{B}^{\mathrm{(nonzero)}}-N_{F}=-1$ for every $D$, contributing a residual $-M_{*}^{2}/(16\pi^{2})$ to eq. (129) rather than $0$. Added to the self-loop $+(3/2)M_{*}^{2}/(16\pi^{2})$ of eq. (128), this gives total $\delta m_{h}^{2}=M_{*}^{2}/(32\pi^{2})$, conflicting with the predicted $m_{h}=M_{*}\sqrt{6}/(8\pi)$. The off-by-one is resolved by the substrate-vs-emergent layer distinction the body now adopts (Computations 93, 97): the substrate Walsh zero mode $\chi_{\emptyset}$ maps under $\Pi$ to the constant function on $M$ (the vacuum direction), and is structurally distinct from the emergent 4-component SM Higgs doublet generated by inner fluctuations on $M\times F$. Counting the substrate zero mode in $N_{B}^{\mathrm{full}}$ therefore does not double-count the emergent doublet, which is carried separately by $C_{H}=6\lambda=3/2$ in eq. (128). With $N_{B}^{\mathrm{full}}-N_{F}=0$ from the substrate trace and the $+(3/2)M_{*}^{2}/(16\pi^{2})$ self-loop supplying the Higgs mass, total $\delta m_{h}^{2}=(3/2)M_{*}^{2}/(16\pi^{2})$, recovering $m_{h}=M_{*}\sqrt{6}/(8\pi)$ and confirming $M_{*}\approx 1.285$ TeV as a parameter-free one-loop derivation. The three mechanisms once entertained here, (a) the Higgs doublet’s three eaten Goldstones at the EW scale, (b) reinterpreting the zero mode as part of the cancellation sum, and (c) a complementation/parity pairing of the non-zero Boolean modes, are superseded by this resolution and are walked through and set aside in Computation 69.

 

Computation 68 ·  Chapter 15 ·  [NUMERICAL]
Casimir-coefficient $\xi$ kernel-shape sensitivity.

Tests the sensitivity of the parameter-free Casimir-correction coefficient $\xi=(15/\pi^{2})\cdot M_{4}/(d_{0}^{2}M_{2})$ (eq. (213)) to the assumed projection-kernel shape, by evaluating the second and fourth moments $M_{2},M_{4}$ for six kernel families: Gaussian (book canonical), squared Lorentzian, single-sided exponential, sech-squared, compact parabolic, and compact quartic-bump. Findings: the $d^{-6}$ power-law exponent is preserved across every kernel (categorical prediction, kernel-independent). The coefficient $\xi$ varies from $2.0$ (compact quartic-bump) to $21.0$ (sech²) across kernels with exponential or compact-support tails (factor $\sim 10$ range); the squared Lorentzian gives $\xi=378.7$ from its $1/x^{4}$ heavy tail (physically implausible for a discreteness scale). The diagnostic $d_{0}$ bound from nanometre-Casimir data scales as $1/\sqrt{\xi}$, giving the bound range $d_{0}\lesssim 1.5$ nm (sech²) to $d_{0}\lesssim 5$ nm (compact quartic) at the current $0.19\%$ data, with the Gaussian value $2.3$ nm sitting near the middle. Order of magnitude (few nm) is robust; precise numerical bound is kernel-conditional with factor-of-$\sim 3$ spread. Honest reading: “$d_{0}\lesssim 1.5$–$5$ nm depending on the projection-kernel choice; the Gaussian (canonical) choice gives $2.3$ nm.” The derived value $d_{0}=2.2$ am (Computation 201) lies nine orders below every bound in the range.

 

Computation 69 ·  §12.2 ·  [PROOF]
Higgs cancellation: detailed analysis of the three Computation 67 resolutions.

Carries out the explicit walkthrough of all three resolutions listed in Computation 67 for the off-by-one in eq. (129). (a) Higgs-doublet eaten Goldstones: the resolution is at the EW scale (broken phase), while the cancellation argument is at $M_{*}\gg v$ (symmetric phase), dimensionally a category error; not constructible without an explicit RG-matching argument the book does not provide. (b) Zero mode included in the cancellation sum: gives $N_{B}-N_{F}=0$ exactly, but no leftover $+M_{*}^{2}/(16\pi^{2})$ to be the Higgs self-mass, total $\delta m_{h}^{2}=0$, $m_{h}=0$ at one loop; a no-op resolution. (c) Complementation $S\to S^{c}$ pairing: for even $D$, $|S|$ and $|S^{c}|$ have the same parity so complementation does not cross-pair bosons with fermions; for odd $D$ the pairing exists structurally but the zero-mode / full-set pair is broken if the zero mode is set aside as the Higgs/order-parameter direction. None of these three resolutions, as constructible from current PST postulates without ad hoc structure, closes the off-by-one. The fourth resolution does: the substrate Walsh zero mode $\chi_{\emptyset}$ maps under $\Pi$ to the constant function on $M$ (vacuum direction), structurally distinct from the emergent 4-component SM Higgs doublet generated by inner fluctuations on $M\times F$ (Computations 93, 97). Including the zero mode in $N_{B}$ does not double-count $C_{H}=6\lambda=3/2$, and $M_{*}=4\pi m_{h}\sqrt{2/3}$ stands as a parameter-free one-loop derivation.

 

Computation 70 ·  §18.1.6 ·  [SURROGATE]
$K$-fixed Berry–Esseen rate for A7.

Verifies the Berry–Esseen-type convergence rate underlying the A7 reduction lemma: the empirical Boolean Dirichlet form converges to the continuum $L^{2}$-gradient at CLT slope $\approx-0.467$ on a single compact-support test family in $K=[-2,2]^{3}\subset\mathbb{R}^{3}$. Three test functions (Gaussian, $C^{\infty}$ bump, polynomial-cosine). Establishes the $K$-fixed Berry–Esseen rate; Computation 83 extends to $K$-uniform. Stands in for: the emergence-determined $\rho_{n}$, by i.i.d. uniform sites with the identity embedding.

 

Computation 71 ·  §12.2.2 ·  [NUMERICAL]
$Z^{2}\approx e^{-1}$ candidate identification via substrate spectral action.

Tests whether the one-loop near-coincidence $Z^{2}:=\lambda_{\mathrm{SM}}(M_{*})/b(M_{*})\approx e^{-1}$ (0.8% deviation) lifts to a structural identification via the substrate spectral-action ratio $S_{\mathrm{sub}}/2^{D}=e^{-1}$ (exact, $D$-independent at matched scaling $\Lambda=\sqrt{D}$). Verifies the substrate side exactly for $D\in\{4,6,8,10,12,14,16\}$; computes one-loop SM RGE running $\lambda_{\mathrm{SM}}(v)\to\lambda_{\mathrm{SM}}(M_{*})$ across $\{\lambda,y_{t},g_{1},g_{2},g_{s}\}$; reports $\lambda_{\mathrm{SM}}(M_{*})\approx 0.087$. Identifies the spectral-action Higgs-quartic identity as the open structural conjecture. Forward reference: the SM-side value quoted here is the anchor of its day; Computation 202 restates it on current inputs at three loops, $0.0930\pm 0.0008$ on the 2026 Review, $1.4\sigma$ from $\mathrm{e}^{-1}/4$, which is what the book states from v32.153 (v32.137 printed $1.6\sigma$ on the 2024 Review).

 

Computation 72 ·  §12.2.2 ·  [NUMERICAL]
Higgs perturbation and $\phi^{4}$ coefficient at matched scaling.

Carries out the symbolic Taylor expansion of the substrate spectral action under a Higgs-like perturbation $D_{\phi}=D_{\mathrm{sub}}+y\,\phi\,\Sigma_{X}$ at matched scaling, extracts the $\phi^{4}$ coefficient $e^{-1}\cdot y^{4}/(2D^{2})$, and reduces the $Z^{2}$ identification to the single-scalar condition $y/\Lambda=2^{-1/4}=\mu_{\mathrm{site}}^{1/4}$ (the fourth root of the per-site Bernoulli probability of P1). Identifies the open structural content: deriving $y/\Lambda=2^{-1/4}$ from CC inner fluctuation.

 

Computation 73 ·  Chapter 15 ·  [NUMERICAL]
Casimir $\xi$ from the substrate Boolean kernel.

Sharpens Computation 68’s kernel-shape sensitivity by deriving $\xi=(15/\pi^{2})M_{4}/(d_{0}^{2}M_{2})$ from the substrate’s actual projection kernel (the binomial Boolean kernel) at finite $|D|$. Computes the moments $M_{2},M_{4}$ of the Boolean kernel as binomial-sum identities; identifies the leading non-Gaussian correction; extrapolates to the matched-scaling limit. Confirms the Gaussian kernel as the natural matched-scaling limit; documents finite-$|D|$ corrections at the $O(1/D)$ level.

 

Computation 74 ·  §12.2.2 ·  [NUMERICAL, superseded by Computation 100]
First-cut CC inner-fluctuation for $Z^{2}=e^{-1}$.

Tests the simplest natural single-generator CC inner fluctuation $A=i\phi[D_{\mathrm{sub}},\omega]/2$ with $\omega=e_{1}e_{2}$ (quaternion volume element). Computes the perturbed Dirac $D_{\phi}^{2}=(D+2\phi^{2})I$ and the $\phi^{4}$ coefficient $2e^{-1}/D^{2}$. Matching Computation 72 target gives $y=\sqrt{2}$ ($D$-independent) hence $y/\Lambda=\sqrt{2/D}$, which DECREASES with $D$. Target is $2^{-1/4}$ ($D$-independent). Conclusion: natural single-generator inner fluctuation does not close $Z^{2}=e^{-1}$. Identifies three possible resolutions (numerical coincidence, multi-generator $A$, alternative $\Lambda(D)$).

 

Computation 75 ·  §16.7 ·  [NUMERICAL]
Rate integral $j_{\tau}$ as Bernoulli large-deviation.

Sharpens the rate-integral closure (book eq. (229)) by identifying $j_{\tau}$ as a standard large-deviation integral under the Bernoulli measure. $j_{\tau}\sim\exp(-|D|\cdot I(\tau/\sqrt{|D|}))$ with $I$ the Cramér rate function. Computes saddle-point asymptotic for a representative $T(C)$ form; documents the rate integral’s structural form. Closes the rate-problem framework for downstream applications (cosmological constant, Computation 82).

 

Computation 76 ·  §14.3 ·  [PROOF]
Postulate-level $V_{7}$-direction constraint as a Yukawa-hierarchy source.

Tests whether constraining the direction of $v(a)\in V_{7}$ to the three quaternionic sub-algebras $Q_{k}$ of $\mathbb{O}$ containing $\hat{\tau}$ lifts the structural-scope theorem. Result: maximal $G_{2}$-symmetry preserves $S_{3}$-symmetry across the three $Q_{k}$, so this single-direction constraint alone does not lift the structural-scope theorem. The threshold-induced dynamical mechanism is taken up in Computation 77.

 

Computation 77 ·  §14.3 ·  [NUMERICAL]
Dynamical Yukawa hierarchy via the directed modal threshold per $Q_{k}$.

The directed P3 threshold-crossing rate per quaternionic sub-algebra $Q_{k}$ gives a hierarchy of generation Yukawa couplings via Boltzmann weighting $\exp(-\beta E_{k})$. Reproduces the SM up-type Yukawa hierarchy with $\beta\approx 28.24$ and $(p_{1},p_{2},p_{3})\approx(0.52,0.35,0.13)$. Identifies the two inputs requiring structural derivation: $\beta$ and the $(p_{k})$ pattern; structural candidates for each are taken up in Computations 78, 79, 80.

 

Computation 78 ·  §14.3 ·  [NUMERICAL]
Structural candidates for $\beta$ in the Boltzmann mechanism.

Tests structural candidates for the Boltzmann inverse-temperature $\beta$ from Computation 77’s fit. Candidates: $\dim(\mathrm{SO}(8))=28$, $\dim(G_{2})=14$, $4\dim(V_{7})=28$, $\dim(\mathrm{Spin}(7))=21$. Numerical match: $\dim(\mathrm{SO}(8))=28$ at $0.85\%$ from Computation 77 best-fit $28.24$. Motivation: $\mathrm{Spin}(8)$ is the rotation group of $\mathbb{O}=\mathbb{R}^{8}$, with triality permuting three $8$-dim reps and three PST generations.

 

Computation 79 ·  §14.3 ·  [NUMERICAL]
$\beta=28.25=\dim(\mathrm{SO}(8))+b_{\mathrm{PST}}$ and $p_{2}/p_{3}=e$.

Sharpens Computation 78’s $\beta\approx 28$ to $\beta=\dim(\mathrm{SO}(8))+b_{\mathrm{PST}}=28+1/4=28.25$, matching Computation 77 best-fit to $0.04\%$. Identifies $p_{2}/p_{3}=e$ in Computation 77’s $(p_{2},p_{3})$ ratio as structurally suggestive of the same $e^{-1}$ that drives $Z^{2}$. Locks in $\beta=28.25$ structurally; $p_{1}$ remains as the one residual free parameter.

 

Computation 80 ·  §14.3 ·  [PROOF]
$p_{1}$ closure via $\mathrm{Spin}(8)\to\mathrm{SU}(3)_{c}$ symmetry chain.

Examines whether $p_{1}$ in the dynamical mechanism of Computations 77 and 79 can be pinned by the substrate symmetry chain $\mathrm{Spin}(8)\supset\mathrm{Spin}(7)\supset G_{2}\supset\mathrm{SU}(3)_{c}$. Result: $\mathrm{SU}(3)_{c}$ stabiliser structure forces $p_{1}$ to a $T(C)$-contingent value rather than a structural constant. Reclassifies the Yukawa hierarchy as configuration-content (consistent with the structural-scope theorem of §14.3).

 

Computation 81 ·  §12.2.2 ·  [NUMERICAL, superseded by the partition-function route, Computations 87–92/100]
Full CC inner fluctuation $D+A+\varepsilon^{\prime}JAJ^{-1}$ for $Z^{2}$.

Extends Computation 74’s single-generator inner fluctuation to include the $JAJ^{-1}$ reality term with KO-2 sign $\varepsilon^{\prime}=+1$, and tests multi-component $A$. Result: $y/\Lambda=\sqrt{8/D}$ (vs Computation 74’s $\sqrt{2/D}$, factor $2$ from reality doubling), still $D$-dependent with wrong scaling. Confirms Computation 74’s verdict that single- and few-generator inner fluctuations do not close $Z^{2}=e^{-1}$. Identifies the structural obstacle: the discrete spectrum’s collapse-to-cutoff behaviour.

 

Computation 82 ·  §16.5 ·  [NUMERICAL]
$\Lambda$ magnitude: order-of-magnitude reclassification as $T(C)$-contingent.

Order-of-magnitude analysis for the cosmological-constant magnitude using Computation 75’s Cramér framework. $\rho_{\mathrm{inst}}=j_{\tau}\cdot\Delta\mathcal{F}\cdot(|D|/V_{M})$ with matched-scaling $|D|/V_{M}\sim 1/d_{0}^{4}$. Target $j_{\tau}$ for $\rho_{\mathrm{inst}}=\rho_{\Lambda,\mathrm{obs}}$ comes out to $\sim 10^{-37}$ per substrate configuration at $d_{0}=2.3$ nm (Computation 188 reproduces it); the required rate-function value $I(\alpha)\sim 10^{-110}$ is tiny. Conclusion: $\Lambda$ magnitude depends on the explicit $T(C)$ form plus threshold-scaling $\tau(D)$, both $T(C)$-contingent. Reclassifies $\Lambda$ magnitude from open derivation to structural contingent (joins Yukawa hierarchy, CKM, $d_{0}$). Sign + EoS $w=-1$ remain structural theorems.

 

Computation 83 ·  §18.1.6 ·  [SURROGATE]
$K$-uniform Berry–Esseen rate verification for A7.

Extends Computation 70’s $K$-fixed verification to $K$-uniform: seven test-function families with varying support locations, radii, and shapes (Gaussian, $C^{\infty}$ bump, polynomial-cosine). Mean slope $-0.458\pm 0.085$ across families; $5/7$ families within fitting tolerance of $-0.500$; two failing families have support radius $\sim d_{0}$ (Taylor-linearisation breakdown). $K$-uniform rate holds asymptotically in the natural physical regime (support radius $\gg d_{0}$). Together with Computation 70 + §18.1.6 Closures (a)–(c), closes the reduced problem at proof-detail level for the i.i.d. surrogate. Stands in for: the emergence-determined $\rho_{n}$, by the same i.i.d. uniform cloud as Computation 70.

 

Computation 84 ·  §12.2.2 ·  [NUMERICAL, superseded by the partition-function route, Computations 87–92/100]
Multi-generator CC inner-fluctuation route for the $Z^{2}$ identification.

Extends Computation 74’s single-generator inner fluctuation to $K$ coherent quaternion-pair generators $\omega_{k}=e_{2k-1}e_{2k}$. Computes $D_{\phi}^{2}=D+2K\phi^{2}$ and the $\phi^{4}$ coefficient. Matching target gives $y/\Lambda=\sqrt{2K/D}$, requires $K=D/(2\sqrt{2})$ (non-integer) for $D$-independent target $2^{-1/4}$. Bernoulli-activated extension: $y/\Lambda=\sqrt{p}$ requires $p=1/\sqrt{2}$, not derivable from $\mu_{\mathrm{site}}=1/2$. The multi-generator inner-fluctuation route does not deliver the target at a structurally natural value of $K$ or $p$; the partition-function-level identification of Computations 87–92 takes a different substrate-side route.

 

Computation 85 ·  §12.2.2 ·  [NUMERICAL, superseded by the partition-function route, Computations 87–92/100]
A_F-internal Higgs inner fluctuation for the $Z^{2}$ identification.

Places the Higgs in the canonical $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$ finite-internal factor (Connes–Chamseddine–Marcolli SM construction) rather than the Clifford volume element. The CCM moment-extraction $\lambda\sim(a/\pi^{2})f_{0}$ reduces to the same identification problem at the moment-extraction level; no new computational test. Identifies the structural obstacle made explicit in Computation 86: the heat-kernel formalism presupposes a continuous spectrum, which the discrete substrate triple lacks. The partition-function-level identification of Computations 87–92 provides the substrate-side route that does not depend on the heat-kernel formalism.

 

Computation 86 ·  §12.2.2 ·  [PROOF, superseded by the partition-function route, Computations 87–92/100]
Seeley–DeWitt heat-kernel analysis on the discrete substrate triple.

Examines the Seeley–DeWitt heat-kernel expansion $\mathrm{Tr}\,f(D^{2}/\Lambda^{2})\sim\sum_{n}f_{n}\Lambda^{d-2n}a_{n}$ for the substrate spectral triple. For the PST substrate at matched scaling: all eigenvalues at $\pm\sqrt{D}$, so $a_{0}=2^{D}$ (non-zero), $a_{n}=0$ for $n\geq 1$ (no continuous-spectrum curvature contributions). The moment structure that delivers $\lambda$ in continuum CCM is structurally absent. Standard CCM heat-kernel machinery is therefore structurally obstructed for the discrete substrate triple, ruling out the multi-generator CC route flagged by Computation 74. Computations 87–92 take the partition-function-level identification, which does not depend on the heat-kernel formalism.

 

Computation 87 ·  §12.3 ·  [NUMERICAL]
Bernoulli Laplace-transform identification: bypassing the CC inner-fluctuation route.

Bypasses CC inner fluctuation via the asymptotic Bernoulli moment-generating function. $\mathbb{E}_{\mu}[\exp(c\bar{X})]=((1+e^{c/D})/2)^{D}\to\exp(c/2)$ as $D\to\infty$. Hits $e^{-1}$ at $c=-2$. Numerical convergence verified at $D=10^{4}$ with error $\sim 10^{-5}$. Multiple structural readings of $c=-2$ (e.g. $-1/\mu_{\mathrm{site}}$, number of substrate states per bit) compatible; none uniquely forced. Computation 88 fixes the load-bearing reading as the binary-distinction $\mathrm{Cl}(1,0)$ gap $2=1-(-1)$.

 

Computation 88 ·  §12.3 ·  [PROOF]
Binary-gap-tempered Bernoulli MGF identification of $\mathrm{e}^{-1}$ at the matched scaling.

The scale $c=-2$ is structurally forced by the binary-distinction gap: each substrate exchange $\tau_{i}$ is a self-adjoint involution with eigenvalues $\{+1,-1\}$, so the one-bit Clifford $\mathrm{Cl}(1,0)$ fixes $\beta_{\mathrm{KO}}=1-(-1)=2$ with no freedom (P1’s binary alphabet; Computations 100 and 102). Setting $c=-\beta_{\mathrm{KO}}=-2$ gives $e^{-1}=\mathbb{E}_{\mu}[\exp(-\beta_{\mathrm{KO}}\bar{X})]$ asymptotically. The total KO-dimension $\mathrm{KO}_{\mathrm{total}}=4+6=10\equiv 2\pmod{8}$ shares this value but is structurally independent (an additive residue in $\mathbb{Z}/8\mathbb{Z}$, not a spectral width); the coincidence is retained only as the “binary-gap-tempered” label, never as an input. All three ingredients structurally derived ($\mu_{\mathrm{site}}$ from P1, the binary-distinction gap $2$ from P1, asymptotic limit from Cramér–Bernoulli). Second independent substrate-side derivation of $e^{-1}$; complements Computation 62’s spectral-action route.

 

Computation 89 ·  §12.3 ·  [PROOF]
$Z^{2}$ bridge step 1: substrate Higgs Hamiltonian.

Derives the substrate Higgs Hamiltonian $H_{\mathrm{Higgs}}(C)$ structurally from three constraints: (i) additivity (Bernoulli sites independent $\Rightarrow$ $H=\sum_{a}h_{a}(B_{a})$), (ii) matched scaling (per-site energy $1/D$), (iii) uniformity (P1 symmetry $\Rightarrow$ $h_{a}$ uniform). Together force $H_{\mathrm{Higgs}}(C)=(1/D)\sum_{a}B_{a}=\bar{X}(C)$, no free parameters. The binary-gap-tempered partition function is then $Z_{H}(\beta_{\mathrm{KO}})=\mathbb{E}_{\mu}[\exp(-2\bar{X})]\to e^{-1}$ asymptotically. Substrate side of bridge closed structurally. Numerical verification at $D=10,100,1000$.

 

Computation 90 ·  §12.3 ·  [EMPIRICAL]
$Z^{2}$ bridge step 2: three SM-side candidate formulations.

Reduces the SM-side bridge to a choice among three concrete candidate formulations: (I) Wilsonian effective-coupling RG, integrate substrate fluctuations between $M_{*}$ and $\Lambda$; (II) spectral-action variation, naive form gives $e^{-2}$ rather than $e^{-1}$, fails; (III) substrate-measure invariance, $V_{\mathrm{eff}}(C)/T_{\mathrm{KO}}=\beta_{\mathrm{KO}}\bar{X}(C)$ at matched scaling. Numerical: predicted $\lambda_{\mathrm{SM}}(M_{*})=e^{-1}/4\approx 0.0920$ vs observed $0.0927$ (Buttazzo 2013), ratio $1.008$. Formulation (II) does not close in naive form; (I) and (III) are taken up as concrete research directions in subsequent work. Forward reference: the SM-side value quoted here is the anchor of its day; Computation 202 restates it on current inputs at three loops, $0.0930\pm 0.0008$ on the 2026 Review, $1.4\sigma$ from $\mathrm{e}^{-1}/4$, which is what the book states from v32.153 (v32.137 printed $1.6\sigma$ on the 2024 Review).

 

Computation 91 ·  §12.3 ·  [EMPIRICAL, superseded by Computation 123]
$Z^{2}$ empirical-precision test: two-loop SM RGE.

Tests the empirical tightness of $Z^{2}\approx e^{-1}$ via SM RGE precision. Simplified one-loop running gives $\lambda_{\mathrm{SM}}(M_{*})\approx 0.087$ (5% below target); compact two-loop $\approx 0.085$ (7% below). The book’s $0.8\%$ near-coincidence claim depends on full Buttazzo-level SM RGE precision (two-loop running with explicit threshold corrections at $m_{t}$). Sensitivity analysis: variation of $m_{h}\pm 1$ GeV, $m_{t}\pm 2$ GeV gives $Z^{2}/e^{-1}$ in range $0.89$–$0.97$ at simplified one-loop. All readings consistent with structural identity within SM RGE truncation uncertainty. Substrate-side derivation (Computation 89) remains exact and RGE-truncation-independent. Superseded by Computation 123, which performs the proper two-loop running with a full input-uncertainty budget, $\lambda_{\mathrm{SM}}(M_{*})=0.0927\pm 0.0007$ ($m_{t}/m_{h}$-dominated), a rigid $1.0\sigma$ match to $\mathrm{e}^{-1}/4=0.09197$ that converges monotonically with loop order, confirming the simplified $0.085$–$0.087$ readings here were truncation artefacts, exactly as anticipated; and in turn by Computation 202, which computes the number at three loops on current inputs.

 

Computation 92 ·  §12.3 ·  [PROOF]
$Z^{2}$ bridge formalisation: seven-step structural proof sketch.

Reduces the $Z^{2}=e^{-1}$ closure to a single Wilsonian-matching premise via a seven-step structural derivation: (1) PST EFT below $M_{*}$ = pure SM (eq. (127)); (2) Wilsonian matching at $M_{*}$; (3) bare $\lambda_{\mathrm{PST}}^{\mathrm{bare}}=b=1/4$ (§1.5); (4) substrate UV at $\Lambda=\sqrt{D}$ contributes vacuum fluctuations; (5) fluctuations integrated over $\mu$ give $Z_{H}(\beta)=\mathbb{E}_{\mu}[\exp(-\beta H_{\mathrm{Higgs}})]$ with $H_{\mathrm{Higgs}}=\bar{X}$ (Computation 89); (6) $\beta_{\mathrm{KO}}=2$ forced by the binary-distinction one-bit Clifford $\mathrm{Cl}(1,0)$ gap $1-(-1)$ (Computations 88, 100, 112), the KO-mod-8 coincidence retained only as a label, never as an input; (7) $\lambda_{\mathrm{SM}}(M_{*})=b\cdot Z_{H}(\beta_{\mathrm{KO}})\to e^{-1}/4$ structurally, matching observed $0.0927$ at $0.8\%$. Steps (1)–(4), (6) established in existing PST framework; step (5) by Computation 89; step (7) follows. Elevates $Z^{2}$ closure status from “open conjecture” to “structural proof sketch established.” The seven steps are subsequently formalised in the partition-function-level Chamseddine–Connes correspondence framework of §12.5; the substrate-side content of Bridge Premise (B) is closed at proof-detail level from P1–P3 by Computation 100 (matched-scaling cutoff identification via tensor-product factorisation forced by P1’s Bernoulli product measure) and Computation 98 (mode-shell residual via the discrete Wilson block-spin RG fixed-point theorem); the matching identification of the substrate factor with the SM-side coupling ratio (bridge premise B itself) is subsequently reduced within PST to a field-strength-renormalisation identity (substrate side closed; SM-side matching open) by the wave-function-renormalisation route and the unique-soft-mode theorem (Computations 110–116). Forward reference: the SM-side value quoted here is the anchor of its day; Computation 202 restates it on current inputs at three loops, $0.0930\pm 0.0008$ on the 2026 Review, $1.4\sigma$ from $\mathrm{e}^{-1}/4$, which is what the book states from v32.153 (v32.137 printed $1.6\sigma$ on the 2024 Review).

 

Computation 93 ·  §12.2 ·  [PROOF]
$M_{*}=4\pi m_{h}\sqrt{2/3}$ off-by-one: fourth-resolution candidate via substrate-vs-emergent distinction.

Re-examines resolution (b) of Computation 69 under a foundational reading: the substrate-side Boolean zero mode $S=\emptyset$ is the substrate’s constant-mode profile, not the emergent SM Higgs doublet (captured separately by $C_{H}=6\lambda_{\mathrm{PST}}=3/2$ from inner fluctuations on $M\times F$). Under this substrate-vs-emergent distinction, including the zero mode in $N_{B}$ does not double-count the doublet, and total $\delta m_{h}^{2}=(M_{*}^{2}/16\pi^{2})(C_{H}+N_{B}^{\mathrm{full}}-N_{F})=(3/2)M_{*}^{2}/(16\pi^{2})$, recovering $m_{h}=M_{*}\sqrt{6}/(8\pi)$ as parameter-free one-loop derivation. Identifies the fourth-resolution candidate; full validation via projection-chain analysis follows in Computation 97.

 

Computation 94 ·  §12.5 ·  [NUMERICAL, superseded by Computations 95, 100]
Bridge Premise (B) fluctuation-matching $T_{\mathrm{KO}}$ (research-direction sketch).

Attempts to derive Bridge Premise (B) via formulation III with fluctuation-matching prescription $\langle(\bar{X}-1/2)^{2}\rangle_{\mu}=\langle h^{2}\rangle_{\mathrm{thermal},T}$, giving the matched-scaling temperature $T_{\mathrm{KO}}=\mu^{2}(M_{*})/(4D)$. Combined with $\beta_{\mathrm{KO}}=2$ delivers the testable prediction $\mu^{2}(M_{*})=2D\cdot M_{*}^{2}$. Acknowledged as dimensionally suspect ($\bar{X}$ dimensionless vs $h$ with GeV units; the prediction implies hierarchy violation $\mu\gtrsim M_{*}$ for $D\gtrsim$ few). Held as research-direction sketch rather than a closure; superseded by Computation 95 reduction and ultimately Computation 100 closure.

 

Computation 95 ·  §12.5 ·  [PROOF]
Bridge Premise (B) reduced to structural claim (***) via substrate-Wilsonian RG.

Reduces Bridge Premise (B) to a single sharper structural claim (***): the substrate Wilsonian RG flow generator equals the Boltzmann partition function $\exp(-\beta_{\mathrm{KO}}\cdot H_{\mathrm{Higgs}}(C))$ on substrate configurations, the partition-function-level analogue of the heat-kernel RG flow generator $\exp(-tD^{2})$ for the continuum spectral triple. If (***) is granted, the bridge identification follows structurally: $\lambda_{\mathrm{SM}}(M_{*})=b\cdot Z_{H}(\beta_{\mathrm{KO}})=e^{-1}/4\approx 0.0920$. Provides the residual-gap framing subsequently closed by Computations 98 (mode-shell) and 100 (matched-scaling cutoff).

 

Computation 96 ·  §12.5 ·  [NUMERICAL]
Polchinski-discrete derivation of claim (***): partial advance.

Polchinski-style derivation attempted on the discrete substrate. The partition-function form is structurally established via configuration-Hamiltonian Boltzmann averaging $Z_{D}(\beta_{\mathrm{KO}})=((1+e^{-\beta_{\mathrm{KO}}/D})/2)^{D}\to e^{-1}$ as $D\to\infty$. Identifies three rigorous-proof items as residual gap: (a) formal exact RG flow equation on discrete substrate triple, (b) uniqueness of Boltzmann form at matched scaling, (c) cutoff-function identification with asymptotic $D\to\infty$ limit. Partial advance; (a) and (b) closed by Computation 98, (c) by Computation 100.

 

Computation 97 ·  §12.2 ·  [PROOF]
$M_{*}$ off-by-one: projection-chain validation CLOSES the item.

Validates Computation 93’s substrate-vs-emergent distinction via explicit PST projection-chain analysis $\psi\to\Pi(\psi)\to H_{\mathrm{doublet}}$. The substrate Walsh zero mode $\chi_{\emptyset}$ maps under $\Pi$ to the constant function on $M=\mathbb{R}\times S^{3}$ (vacuum direction), structurally distinct from the emergent SM Higgs doublet (4 components from inner fluctuations on $A_{F}=\mathbb{C}\oplus\mathbb{H}\oplus M_{3}(\mathbb{C})$). Including the zero mode in $N_{B}$ does not double-count $C_{H}=6\lambda=3/2$. Total $\delta m_{h}^{2}=(3/2)M_{*}^{2}/(16\pi^{2})$, restoring $m_{h}=M_{*}\sqrt{6}/(8\pi)$ as parameter-free one-loop derivation. The off-by-one of Computations 67 and 69 is closed at proof-detail level; the “one-loop-complete / parameter-free” reading of $M_{*}=4\pi m_{h}\sqrt{2/3}$ is restored.

 

Computation 98 ·  §12.5 ·  [PROOF]
Bridge Premise (B) items (a)+(b) CLOSED via mode-shell block-spin RG.

Closes items (a), (b) of Computation 96 at proof-detail level via explicit discrete Wilson block-spin RG. Key observation: $H_{\mathrm{Higgs}}=\bar{X}$ is supported on the lowest two Walsh-mode shells $V_{0}\oplus V_{1}$ only, with explicit expansion $\bar{X}=(1/2)\chi_{\emptyset}-(1/(2D))\sum_{|S|=1}\chi_{S}$. Because $\bar{X}$ has no $|S|\geq 2$ content, the block-spin RG step from level $m+1$ to $m$ factorises trivially: $\bar{X}(\psi_{\leq m}+\psi_{m+1})=\bar{X}(\psi_{\leq m})$ for every $m\geq 1$, so the Boltzmann action $\beta_{\mathrm{KO}}\bar{X}$ is a TRIVIAL FIXED POINT (Theorem 1). Uniqueness follows from Computation 89 boundary condition. Residual item (c) remains as the matched-scaling identification (subsequently closed by Computation 100).

 

Computation 99 ·  §12.5 ·  [NUMERICAL, superseded by Computation 100]
Item (c) via thermal-QFT framing (EXPLORATORY, superseded by Computation 100).

Exploratory closure of item (c) via thermal-field-theory framing: postulates $\beta_{\mathrm{KO}}=\mathrm{KO}\bmod 8$ as substrate’s intrinsic inverse temperature in the partition-function-level CC framework. Standard thermal-QFT IR coupling formula $\lambda_{\mathrm{eff}}=\lambda_{\mathrm{bare}}\cdot Z(\beta)$ delivers $\lambda_{\mathrm{SM}}(M_{*})=b\cdot Z_{H}(\beta_{\mathrm{KO}})=b\cdot e^{-1}$. The “KO-thermal principle” is at the same level as the spectral-action principle in standard CC, a framework postulate beyond P1–P3. Held as exploratory, not as a P1–P3-only closure: PST’s claim is that all structure derives from P1–P3 alone, so a CC-framework postulate weakens that claim. SUPERSEDED by Computation 100, which closes (c) from P1–P3 alone via tensor-product factorisation, eliminating the thermal-postulate requirement.

 

Computation 100 ·  §12.5 ·  [PROOF]
Substrate-side spectral-action cutoff identification closed from P1–P3 via tensor product.

Closes the substrate-side content of item (c) of Computation 96 from P1–P3. P1’s Bernoulli independence forces tensor product $L^{2}(\mathcal{P}(D),\mu)=\bigotimes_{i}L^{2}(\{0,1\},\mu_{i})$. The Boolean Laplacian $\Delta=\sum_{i}(1-\tau_{i})$ with one-bit Clifford $\mathrm{Cl}(1,0)$ gives each $(1-\tau_{i})$ eigenvalues $\{0,2\}$. For $\mathrm{Tr}\,f(\Delta/\Lambda^{2})$ to be CONSISTENT with the tensor product structure (i.e., to factor as a product of one-bit traces over independent bits) $f$ must satisfy $f(x+y)=f(x)f(y)$, unique smooth solution $f(x)=\exp(-x)$. Forced by P1’s product structure, not postulated. At matched scaling $\Lambda^{2}=D$: $(1/2^{D})\,\mathrm{Tr}\,e^{-\Delta/D}=((1+e^{-2/D})/2)^{D}\to e^{-1}$ by binomial product convergence. The factor “2” in $\exp(-2/D)$ comes from one-bit Clifford structure, not KO mod 8 thermal interpretation (numerically equal but structurally independent). Computation 99’s KO-thermal postulate SUPERSEDED. The matching identification of this substrate-side factor with the SM-side coupling ratio $\lambda_{\mathrm{SM}}(M_{*})/b$ (bridge premise (B) itself) is subsequently reduced within PST to a field-strength-renormalisation identity (substrate side closed; SM-side matching open) by the wave-function-renormalisation route (Computations 110–116); $\lambda_{\mathrm{SM}}(M_{*})=b\cdot e^{-1}=0.0920$ matches observed $0.0927$ at $0.8\%$ Buttazzo-RGE precision ($5$–$7\%$ at one-loop, Computation 91). Forward reference: the SM-side value quoted here is the anchor of its day; Computation 202 restates it on current inputs at three loops, $0.0930\pm 0.0008$ on the 2026 Review, $1.4\sigma$ from $\mathrm{e}^{-1}/4$, which is what the book states from v32.153 (v32.137 printed $1.6\sigma$ on the 2024 Review).

 

Computation 101 ·  §12.5 ·  [PROOF]
Bridge premise (B) sharpened: $Z^{2}$ as wave-function renormalisation $Z_{\phi}^{2}$.

Addresses the structural concern that standard Wilsonian threshold matching of a quartic is additive in loop corrections, not multiplicative. Reinterprets the multiplicative matching $\lambda_{\mathrm{SM}}(M_{*})=b\cdot Z_{H}(\beta_{\mathrm{KO}})$ as a wave-function renormalisation of the substrate modal field $\psi$ to the SM Higgs $\phi$ at matched scaling, with the substrate-side partition function $Z_{H}(\beta_{\mathrm{KO}})\to e^{-1}$ playing the role of $Z_{\phi}^{2}$ via standard QFT decomposition $Z^{2}=Z_{\Gamma 4}/Z_{\phi}^{2}$. Reduces (B) to two well-posed sub-questions: (1) is the matched-scaling $\Pi$’s path-integral Jacobian equal to $\exp(-\beta_{\mathrm{KO}}H_{\mathrm{Higgs}}(C))$? (2) why is the vertex renormalisation trivial ($Z_{\Gamma 4}=1$)? Both are tractable extensions of Computation 73’s matched-scaling machinery; closing both delivers (B) automatically via standard renormalisation. Status: structural reduction of (B)’s open content from a non-standard hypothesis to two standard renormalisation questions. Both sub-questions are subsequently closed: the multiplicative (not additive) reading is the correct one (the additive-Jacobian route is excluded, Computation 103) and the vertex renormalisation $Z_{\Gamma 4}=1$ holds to all orders (Computation 119); (B) is thereby reduced (Computations 110–119).

 

Computation 102 ·  §12.5 ·  [PROOF]
Item 1.2 resolution: the two “2”s are structurally independent.

Addresses the concern that the identification $c=-(\mathrm{KO}_{\mathrm{total}}\bmod 8)=-2$ in the Bernoulli MGF is asserted, not motivated. Computation 100 already derived the “2” from one-bit Clifford structure $\mathrm{Cl}(1,0)$ (eigenvalue range of $(1-\tau_{i})$), independent of KO mod 8. This computation makes the structural-independence explicit: $\mathrm{KO}_{\mathrm{total}}\bmod 8=2$ (from spacetime + internal split $4+6=10$) and one-bit Clifford eigenvalue range $=2$ (from $\tau_{i}^{2}=1$) trace to different parts of the substrate construction and are numerically coincident but structurally unrelated. The honest framing is that Computation 100’s derivation uses ONLY one-bit Clifford structure; KO mod 8 = 2 is a parallel substrate fact, not an input to the matched-scaling exponent. Resolves item 1.2 as a derivation question; the remaining open content (whether partition-function-level CC has standing in the spectral-action literature) is an external-validation / literature-search task, not new research.