12.6.  The framework in full: F1–F11

The definitions, propositions, and theorems summarised above are stated here in full, with their proofs and established-in references, before the two main theorems.

12.6.1.  Definitions (F1–F3)

12.6.1.1.  Definition F1 (PST partition-function spectral triple)

A PST partition-function spectral triple is a tuple $(\mathcal{A},\mathcal{H},D_{\mathrm{sub}},\mu,\beta)$ where $(\mathcal{A},\mathcal{H},D_{\mathrm{sub}})$ is the substrate spectral triple of §1.5 with $\mathcal{A}$ the Boolean $*$-algebra on $D$, $\mathcal{H}=\ell^{2}(\mathcal{P}(D),\mu)$, $D_{\mathrm{sub}}$ the canonical Boolean Dirac operator; $\mu=\bigotimes_{a\in D}\mathrm{Bern}(\tfrac{1}{2})$ is the Bernoulli substrate measure; and $\beta>0$ is the temperature parameter.

12.6.1.2.  Definition F2 (partition function of an observable)

For an observable $O\colon\mathcal{P}(D)\to\mathbb{R}$, the $\beta$-partition function of $O$ is

$$Z(O,\beta)\;:=\;\mathbb{E}_{\mu}\bigl{[}\exp(-\beta\,O(C))\bigr{]}.$$ (147)

12.6.1.3.  Definition F3 (Matched-scaling exponent)

For the substrate spectral triple, the matched-scaling exponent is

$$\beta\;:=\;2,$$ (148)

equal to the spectral range of a single binary distinction, $2=1-(-1)$: each exchange $\tau_{i}$ is a self-adjoint involution ($\tau_{i}^{2}=1$) with eigenvalues $\{+1,-1\}$, so $(1-\tau_{i})$ has nonzero eigenvalue $2$, forced with no freedom by P1’s binary alphabet (equivalently the one-bit Clifford $\mathrm{Cl}(1,0)$ range; Computation 100, 102). The numerically equal $\mathrm{KO}_{\mathrm{total}}\bmod 8=2$ (with $\mathrm{KO}_{\mathrm{total}}=4+6=10$, §1.5) shares the value but is a gauge- and spacetime-sector fact, never an input here; the name “binary-gap temperature” $\beta_{\mathrm{KO}}$ is retained below as a label only. This $\beta=2$ is also the calibration point of the Bernoulli-MGF route (the value, and only the value, for which $\mathbb{E}_{\mu}[\exp(-\beta\bar{X})]\to e^{-1}$): $e^{-1}$ is the signature of the binary gap.

12.6.2.  Propositions (F4–F7)

These restate ingredients already established elsewhere in the book; they are collected here so the framework is self-contained.

12.6.2.1.  Proposition F4 (PST EFT below $M_{*}$)

The PST effective field theory below the matched scale $M_{*}=4\pi m_{h}\sqrt{2/3}$ equals the Standard Model exactly: no PST modes propagate below $M_{*}$, so the SM Lagrangian (gauge $+$ Higgs $+$ matter) is the leading effective content of the theory below $M_{*}$.

Established as the leading-order Wilsonian EFT in §12.2, equation (127); the gauge $+$ Higgs $+$ matter content is derived as the projected spectral action in Chapter 13.

12.6.2.2.  Proposition F5 (Wilsonian matching at $M_{*}$)

The IR SM couplings at $M_{*}$ equal the UV PST couplings transformed by the standard Wilsonian matching condition, expressed as the substrate-side path-integral over modes with momenta in $[M_{*},\Lambda]$ where $\Lambda=\sqrt{D}$ is the matched-scaling substrate cutoff.

Established in standard EFT machinery; the substrate-side specification is given in F8 below.

12.6.2.3.  Proposition F6 (PST UV Higgs quartic)

The PST UV theory at $M_{*}$ has bare Higgs quartic coupling $\lambda_{\mathrm{PST}}^{\mathrm{bare}}=b=1/4$, fixed by the Landau–Ginzburg modal potential coefficient.

Established in §1.5; cf. equation (140).

12.6.2.4.  Proposition F7 (Substrate UV-IR scale separation)

At matched scaling $\Lambda=\sqrt{D}$, the substrate UV modes lie in the interval $[M_{*},\Lambda]$. The path-integral over these modes contributes the substrate vacuum-fluctuation factor to the SM EFT below $M_{*}$.

Established by matched-scaling A1 of §18.1.

12.6.3.  Theorems (F8–F11)

These carry the substrate-side content that is new to this framework.

12.6.3.1.  Theorem F8 (substrate Higgs Hamiltonian)

The substrate Higgs Hamiltonian $H_{\mathrm{Higgs}}\colon\mathcal{P}(D)\to\mathbb{R}$ is uniquely determined by the three structural constraints (i) additivity (Bernoulli factorisation), (ii) matched scaling (per-site energy $1/D$), and (iii) uniformity (P1 distinguishability symmetry); explicitly,

$$H_{\mathrm{Higgs}}(C)\;=\;\frac{1}{D}\sum_{a\in D}B_{a}(C)\;=\;\bar{X}(C).$$ (149)

No free parameters.

Proof: Computation 89; cf. equation (137).

12.6.3.2.  Theorem F9 (tempered Bernoulli partition function)

The partition function of $H_{\mathrm{Higgs}}$ at the matched-scaling binary-gap temperature $\beta_{\mathrm{KO}}$ satisfies

$$Z(H_{\mathrm{Higgs}},\beta_{\mathrm{KO}})\;=\;\mathbb{E}_{\mu}\bigl{[}\exp\bigl{(}-\beta_{\mathrm{KO}}\,\bar{X}\bigr{)}\bigr{]}\;\xrightarrow{D\to\infty}\;\mathrm{e}^{-1}.$$ (150)

Proof: standard Cramér–Bernoulli asymptotic [150]; $\mathbb{E}_{\mu}[\exp(c\bar{X})]\to\exp(c\,\mu_{\mathrm{site}})=\exp(c/2)$ delivers $\mathrm{e}^{-1}$ at $c=-\beta_{\mathrm{KO}}=-2$ (the $\mathrm{Cl}(1,0)$ binary-distinction gap, not Bott periodicity). Computation 88; cf. equations (135), (136).

12.6.3.3.  Theorem F10 (unique soft mode; total reduction)

Let the substrate action be $\Delta+V(\bar{X})$, where $\Delta=\sum_{i}(1-\tau_{i})$ is the ultralocal Boolean Laplacian (P1: independent bits) and $V$ is the Landau–Ginzburg potential, a function of the linear collective coordinate $\bar{X}=(1/D)\sum_{a}C_{a}$ alone (Theorem F8: $H_{\mathrm{Higgs}}=\bar{X}$). Then, past the LG threshold (P3), $\bar{X}$ is the unique soft mode: every non-collective substrate mode (the $S_{D}$ standard representation in the $|S|=1$ shell, and all $|S|\geq 2$ Walsh shells) retains the free binary-gap mass $\geq 2$ up to $O(1/D)$, while $V$ shifts only the $S_{D}$-singlet direction $\bar{X}$. Hence below $M_{*}$ the single surviving light field is the Higgs, the substrate$\to$Higgs reduction is total, and the field-strength of step 3 of the four-step reduction of bridge premise (B) is the full substrate embedding norm.

Proof (all-orders $S_{D}$-equivariant Morse–Bott lemma): because $\bar{X}$ is linear in the bits, the chain rule gives, at every order $n$, $\partial^{n}V/\partial C_{a_{1}}\cdots\partial C_{a_{n}}=V^{(n)}(\bar{X})/D^{n}$, the totally symmetric all-ones tensor, independent of the indices. Under $S_{D}$ the single-bit-fluctuation space $\mathbb{R}^{D}$ splits into the singlet line $\mathbb{R}|1\rangle$ (carrying $\bar{X}$, $|1\rangle=\sum_{a}e_{a}$) and the standard representation $W=\{v:\sum_{a}v_{a}=0\}$ (the non-collective modes). An all-ones tensor contracted in any slot with $v\in W$ gives $(\sum_{a}v_{a})(\cdots)=0$, so $V$ has zero matrix element on $W$ at every order: the non-collective modes are free fields with mass set by $\Delta$ alone (the binary gap $\geq 2$). $S_{D}$-equivariance (Schur’s lemma [158]) guarantees the singlet and $W$ never mix and that no $S_{D}$-invariant interaction can deform the $W$-block. Thus the Hessian, to all orders, is Morse–Bott: a soft critical direction (the order parameter $\bar{X}$, softened to criticality at threshold) and a uniformly gapped normal bundle $W$. The only coupling between bits beyond this is the $O(1/D)$ sum-constraint correlation, which vanishes as $D\to\infty$, so the gap is protected non-perturbatively in the substrate limit. (Computations 115, 116; formalised as the all-orders lemma in Computation 118, and closed exactly and non-perturbatively by sufficiency of the empirical mean in Computation 121, of which this lemma is the Gaussian shadow.) ∎

12.6.3.4.  Main theorem of the framework

12.6.3.5.  Theorem F11 ($Z^{2}\to\mathrm{e}^{-1}$ in the substrate)

In the substrate $D\to\infty$ limit, with the substrate-side field-strength factor closed (steps 1–4, including Theorem F10) and the Wilsonian matching $\lambda_{\mathrm{SM}}(M_{*})=b\cdot Z_{\phi}^{2}$ of Propositions F4–F7, the PST framework delivers the asymptotic identification

$$\lambda_{\mathrm{SM}}(M_{*})\;=\;b\cdot Z(H_{\mathrm{Higgs}},\beta_{\mathrm{KO}})\;\xrightarrow{D\to\infty}\;\frac{\mathrm{e}^{-1}}{4}\;=\;\frac{1}{4\mathrm{e}}\;\approx\;0.0920,$$ (151)

equivalent to

$$Z^{2}\;=\;\frac{\lambda_{\mathrm{SM}}(M_{*})}{b(M_{*})}\;\xrightarrow{D\to\infty}\;\mathrm{e}^{-1},$$ (152)

with the substrate-side fact $Z(H_{\mathrm{Higgs}},\beta_{\mathrm{KO}})=\mathrm{e}^{-1}$ structurally derived from P1 ($\mu_{\mathrm{site}}=1/2$), the binary-distinction gap $\beta_{\mathrm{KO}}=1-(-1)=2$, and the matched-scaling limit, with no free parameters.

Proof: combining propositions F4 (PST EFT $=$ SM below $M_{*}$), F5 (Wilsonian matching), F6 ($\lambda_{\mathrm{PST}}^{\mathrm{bare}}=b$), and F7 (UV scale separation) with theorems F8 ($H_{\mathrm{Higgs}}=\bar{X}$) and F9 (binary-gap-tempered partition function $\to\mathrm{e}^{-1}$), the four-step reduction of (B) (steps 1–3 and Theorem F10) yields equation (151), conditional on the Wilsonian matching $\lambda_{\mathrm{SM}}(M_{*})=b\cdot Z_{\phi}^{2}$ identifying the substrate field-strength with the SM-side coupling ratio. ∎

12.6.4.  Remark on what the framework achieves

Theorem F11, together with the four-step reduction of (B), derives the substrate-side value $Z_{\phi}^{2}\to\mathrm{e}^{-1}$ from P1–P3 in the substrate limit, in the language of a partition-function-level analogue of the Chamseddine–Connes correspondence [49, 50] adapted to the discrete substrate triple. Every substrate-side input is derived from PST postulates. The framework therefore:

• reduces the matching premise (B) to a field-strength renormalisation identity. The identity is well-typed (multiplicative, not the additive Jacobian of Computation 103) and its substrate-side value $\mathrm{e}^{-1}$ is derived within PST rather than inherited from an SM-side CC postulate, via the wave-function-renormalisation route (steps 1–3) and the unique-soft-mode theorem (F10);

• separates the substrate-side closure (F8, F9, F10, exact in the substrate limit) from the substrate$\to$SM-coupling matching (resting on the emergence premise F4 $+$ the RGE test, supported numerically at $1.2\%$), which remains the single open step; the matching’s internal renormalisation is fully derived (field-strength and vertex $Z_{\Gamma 4}=1$, Computation 119), the all-orders decoupling closed exactly and non-perturbatively by sufficiency of the empirical mean (Computation 121, eq. (146)), with only the standard $O(1/D)$ caveat on the value $\mathrm{e}^{-1}$ remaining on the substrate side;

• provides a unified mathematical statement of the $Z^{2}$ identification in the language of partition functions on substrate observables;

• documents what is and is not established at each step.