12.8.  Summary

This chapter builds the quantum fields out of the substrate and goes after one number, the Higgs quartic. The gradient term of P3, evaluated at the vacuum, gave the geometry; evaluated away from it, the same term gives the spectrum of excitations, and that branch projects into quantum field theory (§12.1). The interacting vacuum is stable at one loop and reproduces the Standard Model as its Wilsonian EFT below $M_{*}$, with the Higgs mass nominally relating to the modal scale via $m_{h}=M_{*}\sqrt{6}/(8\pi)$ at the LG quartic $\lambda_{\mathrm{PST}}=\tfrac{1}{4}$. The numerical value $M_{*}=4\pi m_{h}\sqrt{2/3}\approx 1285\;\mathrm{GeV}$ is a parameter-free one-loop derivation of the scalar sector, fixed by the single measured input $m_{h}$ (the full-SM extension of the cancellation is settled in the negative, Computation 120), with the substrate-vs-emergent zero-mode distinction (Computations 93, 97) placing $N_{B}$ (including the substrate Walsh zero mode) and $C_{H}$ (the emergent Higgs doublet) in structurally distinct layers of the projection chain. The coherence length $d_{0}$ and the projection coefficient are Landau–Ginzburg/condensate quantities whose magnitudes are not pinned by the postulates alone. The binary-gap-tempered partition function closes the substrate side of the bridge premise, $Z_{H}(\beta_{\mathrm{KO}})\to\mathrm{e}^{-1}$ from P1–P3 alone (§12.5); the matching of that factor to the measured quartic rests on the emergence premise F4 and is tested rather than proved, at $1.4\sigma$ on current inputs (Computation 202). The framework F1–F11 states what the chapter assumes and what it proves, and the status ledger records which entries are theorems and which is the one step reduced rather than closed.