12.7. Status of the derivations

This section lists each derived result of the chapter with its present status, from fully established theorems to the items that remain conditional. The ledger comes first; the notes that follow give the reasoning behind each entry.

Result	Status
$\phi_{\mathrm{Higgs}}=\Pi(\psi)$	Identification (§15.2)
$M_{*}=4\pi m_{h}\sqrt{2/3}\approx 1.285$ TeV	Derived, scalar sector only; the full-SM extension is settled in the negative (Computation 120)
$Z^{2}\approx\mathrm{e}^{-1}$ , hence $\lambda_{\mathrm{SM}}(M_{*})=b\,\mathrm{e}^{-1}$	Reduced, not closed: the substrate side is derived from P1–P3; the SM-side matching rests on the emergence premise F4 plus the RGE test
A7 equicoercivity	Theorem given A1, A2, A4 and A3
$d_{0}$ and the projection coefficient	Not pinned by the postulates alone

The one entry that is not a theorem is the $Z^{2}$ row, and it is the one the chapter’s headline number depends on. The notes below say why in each case.

$\phi_{\mathrm{Higgs}}=\Pi(\psi)$ (§15.2).: The SM Higgs field and the modal order parameter are the same field at different scales, with $m_{h}=125$ GeV recovered via the LG quartic.
The modal scale $M_{*}$ .: $M_{*}=4\pi m_{h}\sqrt{2/3}\approx 1.285$ TeV (Computations 67, 69, 93, 97). The apparent zero-mode off-by-one of the boson–fermion cancellation (Computations 67, 69) is closed by the substrate-vs-emergent distinction (Computations 93, 97). The substrate Walsh zero mode $\chi_{\emptyset}$ identifies with the vacuum expectation value (VEV) direction (the constant function on $M$ ), structurally distinct from the emergent 4-component SM Higgs doublet generated by inner fluctuations on the spectral triple $M\times F$ . The resolution is therefore a VEV-vs-fluctuation split, with the projected zero mode being $\langle\phi\rangle$ rather than the physical $h$ . Including the zero mode in $N_{B}$ does not double-count $C_{H}=6\lambda=3/2$ , and the scalar self-mass relation $m_{h}^{2}=(3/2)\,M_{*}^{2}/(16\pi^{2})$ recovers as a parameter-free one-loop derivation of the scalar sector. The extension of the boson–fermion cancellation to the full SM (top/ $W$ / $Z$ sectors) requires three further ingredients: the configuration-to-particle map, cross-sector coupling universality at $M_{*}$ , and the inner-fluctuation sign assignment of §12.2. It is settled in the negative (Computation 120). No such EFT-layer cancellation exists within PST, consistent with the large negative Veltman residual of equation (125), and $M_{*}$ is irreducibly a scalar-sector result.
$Z^{2}\approx\mathrm{e}^{-1}$ .: The substrate-side factor $Z_{H}(\beta_{\mathrm{KO}})\to\mathrm{e}^{-1}$ is derived structurally from P1–P3, and its identification with the SM-side ratio $\lambda_{\mathrm{SM}}(M_{*})/b$ (bridge premise B) is reduced within PST to a field-strength-renormalisation identity whose internal renormalisation is fully derived (field-strength and vertex $Z_{\Gamma 4}=1$ ), so $\lambda_{\mathrm{SM}}(M_{*})=b\,\mathrm{e}^{-1}\approx 0.0920$ rests on the emergence premise F4 plus the empirical RGE test ( $1.2\%$ , $1.4\sigma$ , at three-loop precision), not a free parameter. The full account, with the spectral-action route, the Bernoulli-MGF derivation, the seven-step Wilsonian matching, and the partition-function-level Chamseddine–Connes framework, is canonical at §19.7.
A7 equicoercivity.: The Mosco-convergence theorem of §18.1 is stated on seven assumptions A1–A7. A7 is a theorem of §18.1.6 under A1, A2, A4 and the equidistribution condition A3, the rate hypothesis, the closure margin of emergent spacetime. The three closure subsections (a)–(c) deliver the rate hypothesis at proof-detail level: the explicit $\Pi_{n}^{\dagger}$ construction, the thin-support $d_{0}$ -corrections, and the Riemannian extension on $M=\mathbb{R}\times S^{3}$ (full detail in §18.2). Computations 70 and 83 corroborate the rate on the i.i.d. surrogate across single and seven-family test sets. The remaining work is explicit numerical-constant computation in the Sobolev embeddings (standard finite-element analysis, Brenner–Scott [102]). A3 is a single diffeomorphism-invariant equidistribution inequality: the leading-coefficient bound $\eta_{K,R}<d_{0}^{2}/4$ , not a mixing rate (Computation 129). Its fluctuation channel is closed unconditionally (McDiarmid, Computation 125). The leading-coefficient inequality itself is supplied by the maximum-entropy sublimation of P3, by typicality: the realised configuration is a typical one under $\mu$ , its emergence cloud equidistributed at the rate A3 states. That is a selection principle internal to P3, not a fourth postulate (no P4), and it is corroborated on the i.i.d. surrogate (Computations 70, 83).