Appendix B Computation Scripts, Computations 103 to 136

Computation 103 ·  §12.5 ·  [NUMERICAL, superseded by Computations 110–116]
Bridge premise (B) attack: explicit path-integral Jacobian, honest negative result.

Attempts Computation 101’s sub-question (1): is the matched-scaling $\Pi$’s path-integral Jacobian equal to $\exp(-\beta_{\mathrm{KO}}H_{\mathrm{Higgs}}(C))$? Two natural interpretations both fail: (A) the linear-map determinant of $\Pi:V_{0}\oplus V_{1}\to\mathrm{SM}_{\mathrm{low}}$ is configuration-independent (constant); (B) the Cramér–Bernoulli LDP near the vacuum $\bar{X}=1/2$ gives a Gaussian weight in $(\bar{X}-1/2)$, not exponential in $\bar{X}$ (numerically verified $I(x)=x\ln(2x)+(1-x)\ln(2(1-x))\approx 2(x-1/2)^{2}$ near $1/2$). The multiplicative matching $\lambda_{\mathrm{SM}}(M_{*})=b\cdot Z_{H}(\beta_{\mathrm{KO}})$ is therefore not derivable from a path-integral change-of-variable. Confirms that the Jacobian route to (B) does not close. Sub-question (2) [$Z_{\Gamma 4}=1$] is satisfied at tree level (standard CC inner fluctuations). Net: this is a negative result for the measure-Jacobian reading specifically; it does not obstruct (B), because (B) is a wave-function renormalisation (multiplicative), not a coupling-matching Jacobian (additive). The Gaussian found here is precisely what the product-structure argument later excludes for the field-strength kernel (Computation 111), and (B) is subsequently reduced within PST to a field-strength-renormalisation identity (substrate side closed; SM-side matching open) by the wave-function route (Computations 110–116).

 

Computation 104 ·  §12.5 ·  [PROOF, partially retracted, superseded by Computations 110–119]
Bridge premise (B) factored: substrate side derived, SM side inherited from standard CC literature.

Attempts to extend Mosco convergence from Dirichlet forms (which §18.1 establishes) to spectral actions, with the goal of deriving bridge premise (B) at the same level Mosco delivers the SM kinetic term. The structural analysis shows (B) FACTORS into two parts: (B-substrate) $(1/2^{|D|})\,\mathrm{Tr}\,f(\Delta/|D|)\to e^{-1}$, which is CLOSED at proof-detail level from P1–P3 by Computation 100; and (B-SM) the SM-side spectral-action identification of the matched-scaling coefficient with $\lambda_{\mathrm{SM}}(M_{*})$, which is the standard Chamseddine–Connes spectral-action principle [49, 50] applied to PST’s substrate spectral triple. This is an ESTABLISHED FRAMEWORK with 30+ years of literature precedent, not a PST-specific postulate. PST extends standard CC by providing a precausal foundation BENEATH the spectral triple (P1–P3 deliver the substrate spectral triple itself, Computations 3, 19); it does not invent the SM-side framework. Whether the partition-function-level CC framework has standing in the spectral-action literature: yes, the framework PST uses is standard CC, with the partition-function-level adaptation being PST’s specific way of handling the discrete substrate where heat-kernel collapses (Computations 85–86). PST’s foundational claim refines to “P1–P3 + standard CC spectral-action framework”: the substrate-side contribution is the genuinely new PST result (Computation 100, derived from P1–P3), and the SM-side framework is inherited from established CC literature.

This inheritance framing is subsequently superseded: the matching (B) is reduced internally to PST, in the substrate limit, to a field-strength-renormalisation identity whose internal content is fully derived (field-strength and vertex, Computations 110–119), without inheriting an SM-side CC postulate; the foundational claim is then “P1–P3 $+$ matched scaling”, with the bridge resting on the emergence premise F4 (§12.6.2.1) $+$ the RGE test rather than an inherited framework.

 

Computation 105 ·  §12.5 ·  [NUMERICAL]
Two-layer one-loop diagram (no double counting of $M_{*}$ zero modes).

Addresses the concern of potential double-counting between the substrate $\chi_{\emptyset}$ singleton-cylinder order parameter and the emergent SM Higgs field $C_{H}$ from Mosco convergence. The PST partition function factorises as $Z_{\mathrm{PST}}(M_{*})=Z_{\mathrm{sub}}(M_{*},\Lambda{=}\sqrt{D})\cdot Z_{\mathrm{EFT}}(M_{*},\mathrm{IR})$, with the two layers integrating over disjoint momentum ranges $[M_{*}^{2},D]$ and $[m_{h}^{2},M_{*}^{2}]$ that meet only at the single matching scale $M_{*}$ (measure zero in both integrals). Three independent non-double-counting criteria are established: (i) disjoint momentum support; (ii) distinct dynamical variables ($\chi_{\emptyset}$ on $\mathcal{P}(D)$ vs $C_{H}$ on $M=\mathbb{R}\times S^{3}$); (iii) distinct counterterm structures (the matching is a boundary condition, not a loop integration). Numerical verification: the substrate full-trace partition function (Computation 100) decomposes exactly as $Z^{\mathrm{full}}_{\mathrm{sub}}(D)=Z^{\mathrm{IR}}_{\mathrm{sub}}(D,k_{*})+Z^{\mathrm{UV}}_{\mathrm{sub}}(D,k_{*})$ across the $M_{*}$-shell partition with $|Z^{\mathrm{full}}-(Z^{\mathrm{IR}}+Z^{\mathrm{UV}})|<10^{-14}$ for $D\in\{10,20,50,100\}$. Closes the double-counting question at the diagram level.

 

Computation 106 ·  §14.2 ·  [NUMERICAL, supersedes the earlier PROOF]
Colour does not preserve the generation sectors, and the earlier block-diagonality claim is withdrawn.

An earlier version claimed every $g\in\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}$ preserves each generation sector $G_{k}$, and concluded that tree-level flavour-changing neutral currents are absent as a structural theorem. That is withdrawn. Its part (1) asserted that $\mathrm{SU}(3)_{c}$ acts within a single Fano sub-algebra through $\hat{\tau}$; but $\mathrm{Stab}_{G_{2}}(\hat{\tau})$ fixes $\hat{\tau}$, and each $G_{k}$’s octonionic factor is one complex line, which cannot carry a colour triplet. Its numerical check verified a Kronecker identity: $\mathbb{1}_{\mathrm{gen}}\otimes X$ is block-diagonal in the generation index for every $X$, so the check could not fail. What this computation now does: builds $\mathrm{Der}(\mathbb{O})$ (dimension $14$), takes the subalgebra annihilating $\hat{\tau}$ (dimension $8$, the $\mathfrak{su}(3)$), exponentiates to genuine octonion automorphisms, and measures how much each generation sector leaks. The leakage is $O(1)$, and the subalgebra preserving all three Fano lines through $\hat{\tau}$ is two-dimensional, the maximal torus. Conclusion: parts (2) and (3) stand, $\mathrm{SU}(2)_{L}$ and $\mathrm{U}(1)_{Y}$ being generation-blind by the tensor-product structure; colour is not. Tree-level FCNC absence is not established, and would additionally require a statement about the mass eigenbasis that §14.3 places in the contingent layer.

 

Computation 107 ·  §12.2 ·  [NUMERICAL]
Full-SM extension of the $M_{*}$ boson-fermion cancellation: sector-blindness sub-result.

The substrate-level binomial cancellation $N_{B}=N_{F}=2^{D-1}$ is SECTOR-BLIND (invariant under any partition of the $2^{D}$ Walsh modes; numerically verified across $D\in\{4,\ldots,20\}$; superseded by Computation 120, which settles the extension in the negative). This sector-blindness does not transfer the cancellation to the SM top/$W$/$Z$ sectors: by the disjoint-layer structure of Computation 105 the substrate cancellation lives in the UV shell $[M_{*},\sqrt{D}]$ while the physical top/$W$/$Z$ loops run in the emergent EFT shell below $M_{*}$, so the substrate identity cannot cancel the EFT-layer loops. The “coupling universality at $M_{*}$” premise ($c_{S}=1$ for every Walsh mode) is contradicted by the observed Yukawa hierarchy, which the book assigns to $T(C)$-contingency (§14.3). The empirical SM Veltman residual at $M_{*}$ is $\approx-3$ (top-dominated, negative), not the $+m_{h}^{2}$ a clean cancellation would leave. The sector-blindness of the substrate count is a fact; it does not close the full-SM extension, which does not exist within PST (Computation 120), so $M_{*}$ is irreducibly a scalar-sector prediction.

 

Computation 108 ·  §12.2 ·  [PROOF]
Inner-fluctuation sign assignment for SM gauge bosons is BOSONIC (sub-result; does not close the full-SM extension).

Verifies that the substrate $|S|$-parity boson/fermion sign assignment (CAR realisation, §11.2) carries through to $A_{F}$ inner-fluctuation gauge bosons that are not themselves Walsh modes. Three independent readings agree: (R1) algebraic, Connes 1-forms $A=\sum_{i}a_{i}[D,b_{i}]$ for $a_{i},b_{i}\in A_{F}$ are operators of EVEN Clifford grade; (R2) substrate-side lift, $A_{F}$ is generated by $\mathrm{Cl}_{\mathrm{even}}(\mathcal{P}(D))$ under the CAR realisation, so every inner fluctuation lifts to an $|S|$-even substrate configuration; (R3) one-loop diagrammatic, the standard QFT vector-boson loop sign ($+$) is inherited unchanged by the CC inner-fluctuation construction. Dof count: bosonic inner-fluctuation dofs on $A_{F}$ (off-shell) are $\mathrm{U}(1)_{Y}$: 4, $\mathrm{SU}(2)_{L}$: 12, $\mathrm{SU}(3)_{c}$: 32, Higgs doublet: 4, total 52; the substrate $|S|$-even Walsh subspace has dimension $2^{D-1}$ accommodating these at $D\geq 6$. Status: the bosonic sign assignment is a correct sub-result, but it does not close the full-SM-extension question. The full-SM cancellation fails for the layer-disjointness reason of Computation 107 (the substrate cancellation does not reach the EFT-layer top/$W$/$Z$ loops), independent of the inner-fluctuation sign. That question was subsequently settled in the negative by Computation 120 (the full-SM extension does not exist within PST); this sign sub-result stands.

 

Computation 109 ·  §12.2 ·  [PROOF]
Higgs-quartic normalisation + $\mathrm{O}(4)$ one-loop self-energy: two technical verifications.

Verifies two technical findings on the Higgs-sector derivation at proof-detail level. (A) Normalisation: the book commits to the doublet convention, $\lambda_{\mathrm{PST}}=\tfrac{1}{4}$ is the coefficient of $(\Phi^{\dagger}\Phi)^{2}$, the same convention as the SM $V_{\mathrm{SM}}=\lambda_{\mathrm{SM}}(\Phi^{\dagger}\Phi)^{2}$, with $\langle\psi,\psi\rangle\leftrightarrow\Phi^{\dagger}\Phi$. Under the radial reduction $\langle\psi,\psi\rangle\to\psi^{2}$ the postulate $\mathcal{F}=\int[-\varepsilon\langle\psi,\psi\rangle+(1/4)\langle\psi,\psi\rangle^{2}+c|\nabla\psi|^{2}]\mathrm{d}\mu$ collapses to the scalar form $-\varepsilon\psi^{2}+(1/4)\psi^{4}$ (eq. (24)). The field-normalisation ratio $Z^{2}=\lambda_{\mathrm{SM}}(M_{*})/\lambda_{\mathrm{PST}}$ is then a ratio of two same-convention couplings. (B) Vertex contraction: explicit $\mathrm{O}(N)$ vertex contraction $V_{ijkl}=2\lambda(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$ closed into the one-loop seagull $(1/2)V_{ijkl}\delta_{kl}=(N+2)\lambda\,\delta_{ij}$ delivers the standard $\mathrm{O}(N)$ result $(N+2)\lambda$ rather than $N\lambda$. For $N=4$ the Higgs self-loop coefficient is $C_{H}=6\lambda_{\mathrm{PST}}=3/2$, giving $M_{*}=4\pi m_{h}\sqrt{2/3}\approx 1285$ GeV. Broken-phase cross-check: $3\lambda$ (physical Higgs loop) $+3\lambda$ (three Goldstones) $=6\lambda$ confirms the symmetric-phase result. This computation propagates corrections through eq. (128), the $M_{*}$ value, the naturalness coefficient, and the affected Computations 67, 69, 93, 97, 107, 108.

 

Computation 110 ·  §12.5 ·  [PROOF]
Bridge premise (B): wave-function-renormalisation route.

Recasts (B), following Computation 101, as a field-strength (wave-function) renormalisation $Z_{\phi}^{2}$, which acts multiplicatively on the quartic ($\lambda\to\lambda Z_{\phi}^{2}$), the form (B) requires, sidestepping the additive-matching obstruction Computation 103 found for the coupling-Jacobian reading. The route first sets up the discrete Wilson–Polchinski flow on the Boolean lattice with the P1-forced cutoff $f=\exp$ and matched scaling $\Lambda^{2}=D$, reducing to the low Walsh shells $V_{0}\oplus V_{1}$ where $\bar{X}$ lives (Computation 98). It then verifies that the field-strength factor is $Z_{\phi}^{2}=Z_{H}(\beta_{\mathrm{KO}})\to e^{-1}$, so $\lambda_{\mathrm{SM}}=b\,Z_{\phi}^{2}\to e^{-1}/4$. Sets up the route; the crux (the form and value of the kernel) is closed in Computations 111–116.

 

Computation 111 ·  §12.5 ·  [PROOF]
Bridge premise (B): the field-strength kernel is forced into $\exp(-\beta\bar{X})$ form.

The field-strength dressing must respect P1’s product measure (the matched-scaling cutoff factorises over bits, Computation 100), so the per-configuration kernel $K(C)=\prod_{a}g(C_{a})$ is a product over bits. A product kernel over Boolean bits is necessarily $\exp$ of a quantity linear in $\bar{X}$, i.e. $\propto\exp(-\beta\bar{X})$, the same form the substrate side evaluates to $e^{-1}$. The Gaussian $\exp(-\alpha(\bar{X}-1/2)^{2})$ that obstructs the Jacobian route (Computation 103) is thereby excluded: it correlates the bits, breaking the product structure (verified: bit–bit covariance $0$ under $\exp(-2\bar{X})$, nonzero under the Gaussian). The wave-function route is therefore unobstructed, in contrast to the Jacobian route.

 

Computation 112 ·  §12.5 ·  [PROOF]
Bridge premise (B): $\beta_{\mathrm{KO}}=2$ is the binary-distinction gap.

The load-bearing exponent is the spectral range of a single binary distinction, $2=1-(-1)$: each exchange $\tau_{i}$ is a self-adjoint involution with spectrum $\{+1,-1\}$, so $(1-\tau_{i})$ has nonzero eigenvalue $2$, forced by P1’s binary alphabet with no freedom (equivalently the one-bit Clifford $\mathrm{Cl}(1,0)$ range). $q$-ary comparison: only $q=2$ gives the self-adjoint involution P1 requires, and $e^{-1}$ is the signature of the binary gap ($Z_{H}\to e^{-\beta/2}$ lands on $e^{-1}$ only at $\beta=2$). Dissolves the “two 2’s” question: $\mathrm{KO}_{\mathrm{total}}\bmod 8=2$ shares the value but is a gauge/spacetime-sector fact, never an input.

 

Computation 113 ·  §12.5 ·  [PROOF]
Bridge premise (B): the $Z_{\phi}\leftrightarrow$ trace identification, total-reduction residual.

Tests the field-strength$\leftrightarrow$matched-scaling-trace identification ab initio. The order-parameter two-point residue (single $|S|=1$ shell) gives a single-mode factor, not the full $e^{-1}$, unless all substrate modes are folded in. The matched-scaling trace and the tempered order-parameter expectation $\mathbb{E}_{\mu}[\exp(-2\bar{X})]$ are verified exactly equal, so the identification holds iff the substrate$\to$Higgs reduction is total. Isolates the residual as one concrete property (total reduction), closed by Computations 114–116.

 

Computation 114 ·  §12.5 ·  [PROOF]
Bridge premise (B): the standard-EFT normalisation is excluded; embedding-norm reading.

Two field-theoretic normalisations of the reduction: (A) the total-reduction embedding norm of the symmetric Higgs vacuum $=\mathbb{E}_{\mu}[\exp(-\beta_{\mathrm{KO}}\bar{X})]\to e^{-1}$, verified exactly equal to the tempered vacuum amplitude; (B) the standard-EFT two-point residue $\mathrm{Var}_{\mu}(\bar{X})=1/(4D)\to 0$. The bridge needs (A); route (B), the Higgs as a light field whose two-point function is corrected by heavy loops, gives $\lambda_{\mathrm{SM}}\to 0$ and is excluded by the finiteness of the observed quartic. The Higgs is an emergent projected field, normalised by its embedding norm, not a fundamental light field.

 

Computation 115 ·  §12.5 ·  [PROOF]
Bridge premise (B): unique-soft-mode theorem, total reduction.

Past the LG threshold (P3) the order parameter $\bar{X}$ is the unique soft mode. Lemma: $\bar{X}=(1/D)\sum_{a}C_{a}$ is linear in the bits, so the LG Hessian is rank one (the all-ones matrix), softening exactly the $S_{D}$-singlet direction ($=\bar{X}$) and leaving the $D-1$ standard-rep modes and all $|S|\geq 2$ shells at the binary gap. Verified: $\mathrm{rank}=1$, one soft eigenvalue, $D-1$ pinned at the gap; the threshold sweep softens $\bar{X}$ alone. Establishes total reduction (Theorem F10, Gaussian order), so the bridge factor is the full embedding norm $e^{-1}$.

 

Computation 116 ·  §12.5 ·  [PROOF]
Bridge premise (B): all-orders decoupling theorem.

Upgrades the unique-soft-mode result to all orders. Because the LG potential is a function of the linear collective coordinate $\bar{X}$ alone, every derivative tensor of $V$ is the totally symmetric all-ones tensor $V^{(n)}/D^{n}$, with zero matrix element on the non-collective subspace at every order: the standard-rep and $|S|\geq 2$ modes are free fields, decoupled, pinned at the binary gap, with no $V$-vertex from which any correction could be built. Verified exactly with the full $\exp(-V)$: the $V$-induced bit correlation is $O(1/D)$, the standard channel sits at the free binary-gap value with $O(1/D)$ deviation, and only the singlet $\bar{X}$ responds to $V$. The spectral gap is protected non-perturbatively in the substrate limit; the residual is the standard $O(1/D)$ finite-size correction.

 

Computation 117 ·  §12.5 ·  [PROOF]
Bridge premise (B): the embedding-norm selection is structural, not empirical.

Removes the empirical crutch in the field-strength selection. The two-point-residue reading sets $Z_{\phi}^{2}=\mathrm{Var}_{\mu}(\bar{X})=1/(4D)\to 0$; that vanishing is the law of large numbers, $\bar{X}$ is an intensive average of the i.i.d. bits (P1), so it self-averages to a sharp value, and a quantity the LLN sends to zero cannot be a field-strength (it gives a free emergent Higgs, $\lambda_{\mathrm{SM}}\to 0$). The embedding norm is the $O(1)$ object $\to e^{-1}$, the only candidate that normalises a canonical emergent field. Since the emergent EFT below $M_{*}$ is the interacting Standard Model (F4), an interacting Higgs is required, hence $Z_{\phi}^{2}=O(1)$, hence the embedding-norm reading, forced by P1’s LLN and F4, not by the measured value $\lambda_{\mathrm{SM}}=0.0927$, which is demoted from an input to the selection to a test of the prediction $b\,e^{-1}$. Non-circular: F4 selects the reading, the substrate gives the value $e^{-1}$. Physically, below the LG threshold $\langle\bar{X}\rangle$ is a sharp classical condensate and the physical Higgs is the radial excitation, normalised by the embedding norm, not the order parameter’s vanishing susceptibility. 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 118 ·  §12.5 ·  [PROOF]
Bridge premise (B): F10 decoupling as a formal all-orders $S_{D}$-equivariant Morse–Bott lemma.

States and proves the F10 decoupling as a formal lemma. Because $F_{\mathrm{int}}=\Phi(\bar{X})$ factors through the linear collective coordinate, the chain rule gives $\partial^{n}F_{\mathrm{int}}=\Phi^{(n)}(\bar{X})/D^{n}$ times the all-ones tensor at every order. Under $S_{D}$ the fluctuation space splits into the singlet line (carrying $\bar{X}$) and the standard representation $W=\{v:\sum_{a}v_{a}=0\}$; an all-ones tensor contracted with any $v\in W$ gives $(\sum_{a}v_{a})(\cdots)=0$, so the interaction has zero matrix element on $W$ at every order and the $W$-Hessian is the bare Boolean Laplacian, bounded below by the binary gap $\geq 2$. Schur’s lemma ($S_{D}$-equivariance) forbids singlet–$W$ mixing and any invariant deformation of the $W$-block. This is the all-orders $S_{D}$-equivariant Morse–Bott structure: a soft critical direction (the order parameter) and a uniformly gapped normal bundle. Verified: derivative tensors index-independent at $n=2,3,4$; annihilation of $W$ to machine precision; the singlet softens $4\to 0$ under the LG curvature while $W$ stays pinned at $2$; a random $S_{D}$-invariant Hessian has $W$-eigenvalue independent of the singlet coupling. Residual: the $O(1/D)$ finite-size sum-constraint correction (Computation 116), vanishing as $D\to\infty$. SUBSUMED by Computation 121: the decoupling is closed exactly and non-perturbatively by sufficiency of the empirical mean (this Morse–Bott lemma is its de Moivre–Laplace shadow), with the $O(1/D)$ isolated to the value; Computation 122 shows F8’s linearity controls only that $O(1/D)$.

 

Computation 119 ·  §12.5 ·  [PROOF]
Bridge premise (B): vertex renormalisation $Z_{\Gamma 4}=1$ to all orders, completing the matching’s internal content.

The matching ratio $Z^{2}=\lambda_{\mathrm{SM}}/b$ decomposes into a field-normalisation piece (the embedding norm $\to e^{-1}$, Computation 117) and a vertex piece $Z_{\Gamma 4}$ (the four-point $1$PI correction). Computation 101 fixed $Z_{\Gamma 4}=1$ only at tree level; this computation closes it to all orders. The matching integrates out the substrate UV modes (the non-collective $S_{D}$ standard representation $W$ and the gapped $|S|\geq 2$ shells); by the Morse–Bott decoupling lemma (Computation 118), $V=\Phi(\bar{X})$ has zero matrix element on $W$ at every order, and the $n=4$ case is the Higgs quartic vertex, so the integrated-out modes do not couple to it and generate no vertex correction. Verified: the quartic all-ones tensor contracted on $W$ legs is machine-zero. Hence $Z_{\Gamma 4}=1$ to all orders and $Z^{2}=e^{-1}$ with no residual internal renormalisation. The only quartic loop corrections are the collective Higgs self-loops, the SM RGE running below $M_{*}$, which confirms $b\,e^{-1}=0.0920$ against the two-loop-run $0.0927$ at $0.8\%$ (quantified as a rigid $1.0\sigma$ match by Computation 123), not a $Z_{\Gamma 4}$ correction. Net: the matching’s internal renormalisation content is complete; the open content of (B) reduces to the emergence premise F4 plus the RGE test, not an internal step. 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 120 ·  §12.2 ·  [EMPIRICAL]
Full-SM extension of the $M_{*}$ cancellation settled in the negative.

Answers the open full-SM-extension question (does an EFT-layer top/$W$/$Z$ cancellation exist, and what partner spectrum would it need?): no, not within PST. The SM Veltman bracket $[6\lambda+(9/4)g^{2}+(3/4)g^{\prime 2}-6y_{t}^{2}]$ is $-3.5$ at the EW scale and $-3.3$ at $M_{*}$, top-dominated, with no zero crossing, $M_{*}$ is not a Veltman-natural full-theory scale. Cancelling the dominant $-6y_{t}^{2}$ requires bosonic top-partners (stop-like) at top strength near $M_{*}\approx 1.3$ TeV; such a sector is collider-disfavoured (LHC excludes scalar top-partners / vector-like quarks to $\sim 1.2$–$1.4$ TeV, i.e. at $M_{*}$), forbidden by the “no new light states” premise (§15.11), absent from PST’s spectrum (F4: emergent EFT $=$ SM, no superpartners), and unreachable by the substrate mechanism (Computation 105: layer-disjoint). Conclusion: $M_{*}=4\pi m_{h}\sqrt{2/3}$ is irreducibly a scalar-sector prediction, exactly as the book scopes it. This is a negative settlement, a definitive limitation on the scope of $M_{*}$, not a positive closure; it removes the item from the open register with $M_{*}$ correctly and finally scoped.

 

Computation 121 ·  §12.5 ·  [PROOF]
Bridge premise (B): the all-orders decoupling closed exactly by sufficiency of the empirical mean (subsumes Computation 118).

Closes the one residual rigour task carried by F10/Computation 118 (the all-orders $S_{D}$-equivariant decoupling) by an exact, non-perturbative identity. Theorem F8 makes $H_{\mathrm{Higgs}}=\bar{X}$ exactly the empirical mean of i.i.d. per-site bits, so the interaction $\Phi(\bar{X})$ is a function of a sufficient statistic. By Fisher–Neyman factorisation the configurations at fixed $\bar{X}=k/D$ are counted by the interaction-independent multiplicity $\binom{D}{k}$, and the substrate partition function factorises exactly at every finite $D$: $Z[\Phi]=\sum_{k}\binom{D}{k}2^{-D}e^{-\Phi(k/D)}$. The transverse (non-collective) modes enter only through the $\Phi$-independent multiplicity, so $Z_{\Gamma 4}=1$ (and every $n$-point $\Phi$-vertex correction) vanishes exactly, with no “higher-order” gap (the factorisation holds at all orders simultaneously). Verified: the factorisation to machine precision for linear/quartic/mixed $\Phi$; the quartic-probe vertex test ($Z_{\Gamma 4}=1$ to $3\times 10^{-16}$) with a transverse-coupled control that breaks sufficiency (gap $0.30$, the test has teeth); the $O(1/D)$ isolated to the value, the decoupling carrying no $D$-dependence; and the de Moivre–Laplace reduction showing Computation 118’s Morse–Bott lemma is the Gaussian shadow. Net: the substrate side of (B) carries only the standard $O(1/D)$ caveat on the value $\mathrm{e}^{-1}$; the single open content of (B) remains the SM-side matching (F4 + RGE; quantified as a rigid, $D$-independent $1.0\sigma$ test by Computation 123). 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 122 ·  §12.5 ·  [PROOF]
Bridge premise (B): what the substrate side needs of F8 linearity (decoupling $=$ symmetry; value $=$ vacuum normalisation).

Isolates the role of Theorem F8’s linearity ($H_{\mathrm{Higgs}}=\bar{X}$) and finds it bears far less weight than “the whole substrate side hangs on exact linearity”. (1) Uniformity alone (the $S_{D}$-symmetry that is P1) forces $H_{\mathrm{Higgs}}=g(\bar{X})$ for some $g$, because any symmetric function of binary bits depends only on the count; this is exact and is all that the all-orders decoupling (Computations 118, 121) needs. (2) Linearity enters only via additivity: a single-bit function is necessarily affine, so an additive Hamiltonian is exactly affine in $\bar{X}$ with no $O(1/D)$ remainder; the only route to a nonlinear $g$ is a cross-bit term, an interaction between P1-independent distinctions, additivity is P1 independence read at the level of the energy. (3) The leading value $\mathrm{e}^{-1}$ is robust to relaxing linearity: by the LLN $\bar{X}\to 1/2$, so $Z_{H}(\beta)\to e^{-\beta g(1/2)}$, depending on $g$ only through $g(1/2)$; with the binary gap $\beta=2$, $\mathrm{e}^{-1}$ follows from the single normalisation $g(1/2)=1/2$. Verified: four $g$’s (linear $+$ nonlinear deformations) all with $g(1/2)=1/2$ give $\mathrm{e}^{-1}$ to leading order, the difference $O(1/D)$; a slope sweep confirms value $=e^{-2g(1/2)}$ exactly in the limit. Net: the substrate-side closure of (B) does not hinge on $H_{\mathrm{Higgs}}$ being exactly linear, its structure is exact from symmetry, its value rests on a vacuum normalisation plus the binary-gap temperature, and linearity refines only the $O(1/D)$.

 

Computation 123 ·  §12.5 ·  [EMPIRICAL]
Bridge premise (B): two-loop SM-side error budget and the finite-$D$ rigidity of the $\mathrm{e}^{-1}$ matching.

Strengthens the SM-side matching (the single open content of (B)) from a “$0.8\%$ agreement” to a quantified, rigid, falsifiable test. (A) SM side: running the SM quartic up to $M_{*}=1285$ GeV at two loops gives $\lambda_{\mathrm{SM}}(M_{*})=0.0927\pm 0.0007$, the uncertainty dominated by the top and Higgs masses (parametric $\pm 0.00068$: $m_{t}\,\pm 0.00054$, $m_{h}\,\pm 0.00040$, $\alpha_{s}\,\pm 0.00010$; truncation $\pm 0.0002$), not by logarithmic running. The value converges monotonically with loop order ($0.09145\to 0.09234\to 0.09270$, steps shrinking), so the match is a genuine two-loop statement, not a low-order accident; the integration is validated by the vacuum instability scale. Against the substrate prediction $\mathrm{e}^{-1}/4=0.09197$ the gap is $1.0\sigma$, consistent. (B) Substrate side: the prediction is rigid. By eq. (134) the field-normalisation factor is $\mathrm{e}^{-1}$ exactly and $D$-independently (the matched-scaling eigenvalue collapse $\pm\sqrt{D}/\Lambda\to\pm 1$); in the partition-function route the $O(1/D)$ correction would need an implausible $D\approx 63$ to close the gap, against a substrate cardinality $|D|=V/a_{n}^{4}$ (site spacing $a_{n}\to 0$) larger by dozens of orders of magnitude, so there is no finite-$D$ freedom to absorb the residual. Superseded on the SM side by Computation 202: the anchor $0.0927$ is not reproduced by its source’s own formulas (they give $0.0924$ on the 2013 inputs) and the budget counted $m_{t}$ at $\pm 0.5$ GeV where 2013 had $\pm 0.76$; the current statement is $0.0930\pm 0.0008$, $1.4\sigma$ on the 2026 Review. Net, as this computation stood: $\lambda_{\mathrm{SM}}(M_{*})=\mathrm{e}^{-1}/4$ is a rigid, parameter-free prediction, consistent with the SM at $1.0\sigma$, with the residual living entirely in the $m_{t}/m_{h}$ inputs and sharply falsifiable as those (and the SM matching order) tighten.

 

Computation 124 ·  §7.8 ·  [PROOF]
Displacement covariance is positive-semidefinite; the Lorentzian signature is the $\tau_{\mathrm{null}}$ sign cut, not isotropy.

Backs the clause at (56). The empirical displacement covariance $C^{\mu\nu}=|D|^{-1}\sum_{a}(\rho(a)^{\mu}-x^{\mu})(\rho(a)^{\nu}-x^{\nu})$ is a sum of outer products, hence positive-semidefinite (all four eigenvalues $\to d_{0}^{2}/4>0$), whereas the Lorentzian $g=\mathrm{diag}(+1,-1,-1,-1)$ is indefinite, so $C$ cannot equal $(d_{0}^{2}/4)g$. Under $O(4)$ isotropy $C\to(d_{0}^{2}/4)\delta$ (eigenvalue ratio $\to 1$, off-diagonal $\to 0$ as $N\to\infty$): isotropy fixes only the positive Euclidean magnitude and its scale. The signature is the $\tau_{\mathrm{null}}$ sign cut of (30) (Computation 2): across a quantile sweep the timelike fraction tracks the cut exactly while $C$ stays PSD throughout. Confirms (56) is read in the post-signature metric.

 

Computation 125 ·  §18.1.6 ·  [SURROGATE]
Locality of the emergence map $\rho_{n}(a)$: fluctuation local, bias global (the A7 split).

Backs the locality verdict and the analytic content of the rate hypothesis on a deterministic-functional-of-i.i.d.-bits toy (ring sites; pair-local tension $T(\{a,b\})=\mathrm{chord}^{2}+\epsilon\,\beta_{a}\beta_{b}$; metric (30); positions by global MDS and by a locally-supported quasi-interpolation $\Pi^{\dagger}$). (1) Tension and metric are exactly edge-local (zero non-incident edges change under a single bit flip). (2) $\tau_{\mathrm{null}}$ is single-flip-stable ($|\delta\tau_{\mathrm{null}}|\leq 2\epsilon$, $\sim$ machine zero): an ensemble cut, not a per-draw order statistic. (3) The paper’s local-sum $A_{n}$ carries the $O(1/|D_{n}|)$ McDiarmid bounded difference (empirical slope $-0.73$, no position-independence assumed). (4) Under one bit flip the $\Pi^{\dagger}$ fluctuation stays within a bounded neighbourhood ($93\%$ within ring-distance $8$, zero past $12$) while the global MDS embedding spreads it ($19\%$, flat in distance). Confirms fluctuation local / bias global: edge-local metric does not imply local fine-placement, the embedding being globally rigid. Stands in for: the emergence map, by a deterministic functional of i.i.d. bits on ring sites.

 

Computation 126 ·  §4.1 ·  [NUMERICAL]
Forced/free decomposition of the tension functional $T$.

Backs the structural sharpening after (17), by direct enumeration of $\mathcal{P}(D)$. (1) The admissible magnitudes ((14) zero-locus $+$ (15) monotonicity) form a cone of dimension $2^{|D|}-(|D|+1)$, whose ratio to $2^{|D|}$ rises ($0.50\to 0.96$ for $|D|=3\to 8$): the form is not a finite-parameter family, $T_{\mathrm{ex}}$ one point of it. (2) Several distinct magnitudes (uniform/weighted edge counts, a rank potential) are simultaneously admissible. (3) A monotone reparametrisation preserves every threshold super-level set exactly (downstream reads only the ordering), while a different magnitude changes it (the contingent per-configuration content). (4) The forced content (zero-locus; $T(D)\neq 0$, so $\hat{\tau}$ is defined) is invariant across all admissible magnitudes. (5) The per-configuration count is $2^{|D|}=64$ at $|D|=6$, of which the $\sim 20$ observed flavour parameters consume $\sim 31\%$.

 

Computation 127 ·  §3.4 ·  [NUMERICAL]
P2’s irreducible primitive is non-balance of $v$, independent of (14).

Backs the reducibility verdict on P2 (the P2 non-degeneracy item of Ch. 19): almost all of P2 reduces to P1 $+$ (14), the surviving primitive being one non-degeneracy condition, the non-balance of the elementary directions $\bigl{|}\sum_{a\in D}v(a)\bigr{|}>0$. (1) A balanced assignment ($\sum v=0$, three antipodal $V_{7}$-pairs) has maximal edge count yet $|\sum v|\approx 10^{-16}$, so (14) secures the count but not the vector bias $T(D)$: non-balance is logically independent of (14), vindicating (16) as a separately-adopted constraint. (2) $G_{2}\subset SO(7)$ acts linearly, so $g(\sum v)=\sum g(v(a))$: over $2000$ random $SO(7)$ maps the balanced set stays balanced (drift $\sim 10^{-16}$) and norms are preserved, refuting any reduction of non-balance to the Hurwitz/$G_{2}$ algebra. (3) Among $2\times 10^{5}$ random unit assignments none lands within $10^{-3}$ of the balanced locus (codimension seven at generic balanced points; six on the all-parallel stratum, which at $|D|=2$ is the whole locus, Computation 152): non-balance is measure-zero but, since PST fixes no measure over the structural datum $v\colon D\to V_{7}$, not forced here. Computation 152 supplies the measure-free replacement.

 

Computation 128 ·  §1.5 ·  [PROOF]
The minimal KO split $10=4+6$ is forced given the inputs; the only underived substrate choice is the odd-$|D|$ parity.

Backs the minimal-KO-split verdict (§1.5), which Computation 148 then sharpens from a selection principle to a consequence of the two verified KO-dimensions. With the Lorentzian target $\equiv 2\pmod{8}$ (the Connes value) and the parity-selected substrate KO-6 (Computation 3), the spacetime KO $s$ satisfies $s+6\equiv 2\pmod{8}$, so $s\equiv 4$: among $s\in\{0,\dots,7\}$ the unique solution is $s=4$ (next $s=12$), making spacetime KO-4 the arithmetic complement, not an independent input (Computation 4 verifies the spin structure on an already-four-dimensional $M=\mathbb{R}\times S^{3}$). The minimal total $\equiv 2\pmod{8}$ with $s\geq 1$ is $10=4+6$ (next $18=12+6$), so the split is forced given the substrate six; minimality is non-vacuous only at the lift $s=4$ over $12$, fixed by the Mosco limit. Varying the internal $f$ flips the split, so the odd-$|D|$ parity that fixes $f=6$ is the single load-bearing underived choice, not the mod-8 arithmetic.

 

Computation 129 ·  §18.1.6 ·  [SURROGATE]
A spectral gap reaches the relaxation exponent, not the stationary-mean coefficient: the A7 bias residual is immune to mixing rates.

Backs the A7 spectral-gap verdict (§18.1.6). The rate-hypothesis residual binds at the leading coefficient $\eta_{K,R}<d_{0}^{2}/4$, not an exponent. (1) A one-parameter family of reversible 2-state chains with fixed spectral gap $g=\tfrac{1}{2}$ has its stationary mean swept across a $0.8$-wide range with the gap identical to machine precision: a relaxation rate does not determine the stationary mean. (2) For $A_{n}(u)$ with a controlled equidistribution defect $\varepsilon$, the systematic mean deviation is flat in $|D_{n}|$ (matching the predicted $\varepsilon G$) while the statistical fluctuation decays as $|D_{n}|^{-1/2}$ (std halving per $4\times$ sample). So a covariance-decay / spectral-gap estimate reaches only the fluctuation exponent; the systematic-mean coefficient is decoupled (the stationary distribution is the kernel eigenvector), and the A7 bias half is foreclosed to mixing / spectral-gap methods. Stands in for: any mixing route to the A7 residual, by two-state chains with a fixed gap; a counterexample, not a model of the cloud.

 

Computation 130 ·  §18.1.6 ·  [SURROGATE]
The A7 bias residual $c_{u}$ is a cell-volume equidistribution defect; the matched-scaling count does not force it.

Backs the A7 deterministic-frontier reduction (§18.1.6). The quantity

$$c_{u}\;=\;\frac{|D_{n}|^{-1}\sum_{a}|\nabla u(\rho_{n}(a))|^{2}}{|K|^{-1}\int_{K}|\nabla u|^{2}}$$ (284)

is the uniform-weight nodal quadrature defect of the emergence cloud, converging to $1$ iff the points cell-volume-equidistribute (not the Delaunay finite-element interpolation error). (1) A deterministic volume-equidistributed cloud gives $c_{u}\to 1$. (2) A matched-count cloud with a central void in a low-$|\nabla u|^{2}$ region retains a persistent $|c_{u}-1|\approx 0.094$, flat across a $64\times$ count increase and matching the analytic void bias to $10^{-4}$: refinement (the A1 count) does not close it. (3) The defect grows monotonically with the void radius. So the A7 bias half is conditional on cell-volume equidistribution of the emergence cloud, the deterministic sibling of the A2 weak-discrepancy (its rate A3 supplied by P3’s maximum-entropy sublimation), not on the point count. Stands in for: a non-equidistributed emergence cloud, by a cloud with a void at the same count.

 

Computation 131 ·  §18.1.6 ·  [SURROGATE]
The weak A7 discrepancy is blind to the null-cone vanishing-separation set (codim $\geq 1$); only the codim-$0$ node-density binds.

Backs the A2-core characterisation. $c_{u}$ is a single-node average, so the Lorentzian vanishing-separation locus (a codim-$\geq 1$ pair relation) cannot move it. On a $3$-cube with $s^{2}=-\Delta x_{0}^{2}+\Delta x_{1}^{2}+\Delta x_{2}^{2}$: (1) injecting exact null pairs drives the strong Lorentzian minimum separation to $0$; (2) the weak defect $|c_{u}-1|$ is unchanged by that injection (the paired nodes sit at bulk-typical positions), staying $\approx 0$; (3) only a codim-$0$ central void biases $c_{u}$, growing with the void volume. So strong separation fails on the null cone while the weak $H^{1}$-tested discrepancy survives, and the A7 residual is purely the codim-$0$ cell-volume equidistribution of the emergence cloud. Stands in for: the emergence cloud under the signed-tension metric, by $3$-cube clouds with injected null pairs.

 

Computation 132 ·  §18.1.6 ·  [SURROGATE]
The tension functional is blind to bulk node density: a void costs nothing in T-terms.

Backs the final foreclosure of the A7 internal-route chain. $T$ reaches physics through three channels, all functions of the distinction data ($v\colon D\to V_{7}$ and $\delta$), not the placement $\rho$: the pairwise tensions, the magnitude ordering of $|T(C)|$, and the global direction $\hat{\tau}$. (1) Holding the distinction data fixed, all three channels are identical to machine precision across a uniform and a void placement: $T$ takes no spatial-placement argument. (2) The single-node discrepancy $c_{u}$ reads $\rho$, so it sees the void, growing with the void radius. (3) The placement-independent toy $T$-energy changes by exactly $0$ across void radii while $c_{u}$ rises. So a positive-volume void costs nothing in $T$-terms, and the codim-$0$ cell-volume equidistribution at the core of A7 is forced by no internal lever. Stands in for: the placement $\rho$, by fixed distinction data placed uniformly and with a void.

 

Computation 133 ·  §18.1.6 ·  [SURROGATE]
A metric-preserving void biases only the flat node-quadrature: a coordinate artefact, not a density axiom.

Backs the coordinate-artefact verdict. The tension data fixes the intrinsic metric only up to diffeomorphism; bulk node density is a distinct diffeomorphism-class invariant it does not pin. Two coordinate charts of the same intrinsic 1D manifold with $N$ geodesically-uniform nodes: a uniform chart, and a void chart $x^{\prime}=\mathrm{CDF}^{-1}(x)$ with a central density dip and metric $g_{2}=n^{2}$. (1) Every pairwise geodesic distance is identical to machine precision ($\sim 10^{-16}$): the void chart is a genuine diffeomorphic representative (same intrinsic geometry, $T$-realisable). (2) The coordinate uniform-weight node-quadrature $c_{u}$, which the bias half requires to approach $1$, is unbiased in the uniform chart but biased by $\sim 0.10$ in the void chart. (3) The bias grows with the void depth. So the flat $c_{u}$ is chart-dependent; restoring the void chart’s own metric factors makes the defect void-blind (Computation 134), and a void with its co-transforming metric is one gauge orbit carrying no independent density freedom. Stands in for: the coordinate freedom of $\rho$, by two charts of one one-dimensional manifold.

 

Computation 134 ·  §18.1.6 ·  [SURROGATE]
Computation 133’s void bias is a dropped-metric-factor artefact; the genuine A7 residual is the fixed-$g$ rate.

Corrects Computation 133. On its void chart ($x^{\prime}=\mathrm{CDF}^{-1}(x)$, metric $g_{2}=n^{2}$): (1) the flat node-quadrature $c_{u}=\mathrm{mean}\,|u^{\prime}|^{2}/\int|u^{\prime}|^{2}$ is biased $\approx 0.10$; (2) the metric-covariant $c_{u}$ (intrinsic gradient $|u^{\prime}|^{2}/g_{2}$, volume $\sqrt{g_{2}}\,\mathrm{d}x^{\prime}=n\,\mathrm{d}x^{\prime}$) equals $1$ to machine precision, void-blind, so the flat bias is purely the dropped $g^{-1}$ and $\sqrt{g}$ factors; (3) a genuine fixed-metric density modulation (nodes of density $n$ on $g=1$, not a diffeomorphism) does bias the covariant $c_{u}$. So the metric-preserving void is pure gauge (no bulk-density axiom, no new postulate), and the genuine content is the diffeomorphism-invariant leading-coefficient inequality $\eta_{K,R}<d_{0}^{2}/4$ for the emergence-determined $\rho_{n}$, the equidistribution condition A3 supplied by P3’s maximum-entropy sublimation and corroborated on the i.i.d. surrogate. Stands in for: the same two-chart construction as Computation 133, with the metric factors restored.

 

Computation 135 ·  §10.2 ·  [NUMERICAL]
Born rule via Gleason on $L^{2}(\mathcal{P}(D),\mu)$.

Verifies Gleason’s dimension hypothesis ($\dim H=2^{|D|}\geq 3$) and demonstrates the Born form $P(C)=|\psi(C)|^{2}\mu(C)$ on the substrate Hilbert space for $|D|=2,3,4$: total probability one, frame-function additivity in an arbitrary orthonormal basis, and non-contextuality of shared projectors. Confirms the $|\psi|^{2}$ rule is the Gleason frame function of the pure state, not an added postulate.

 

Computation 136 ·  Ch. 13 ·  [PROOF]
Gauge anomaly cancellation and the $N_{c}=3$ condition.

Evaluates all six triangle-anomaly coefficients for one Standard-Model generation as exact functions of the colour rank $N_{c}$. All vanish at $N_{c}=3$; the $[\mathrm{U}(1)_{Y}]^{3}$ coefficient is $-2N_{c}+6$ and the $[\mathrm{SU}(2)]^{2}\,\mathrm{U}(1)_{Y}$ coefficient is $(N_{c}-3)/6$, so anomaly freedom (given the SM hypercharges) forces $N_{c}=3$, the dimension of the $M_{3}(\mathbb{C})$ colour factor. A consistency check, not a derivation of the hypercharges.