13.10.  The Standard Model Lagrangian as leading-order projection

The Standard Model Lagrangian is the leading-order term in the expansion of the instantiated action $S[\psi,g,A_{\mu}^{a}]$ around the vacuum $\mathcal{V}_{\mathrm{EW}}$:

$$\mathcal{L}_{\mathrm{SM}}\;=\;-\tfrac{1}{4}F_{\mu\nu}^{a}F^{a\,\mu\nu}+\bar{\psi}\,i\!\not\!{D}\,\psi+|D_{\mu}H|^{2}-\mathcal{F}(H)+\mathcal{L}_{\mathrm{Yukawa}},$$ (195)

where $F_{\mu\nu}^{a}$ runs over all gauge factors of $G_{\mathrm{SM}}$, $\mathcal{F}(H)$ is the modal potential (77) evaluated on the Higgs doublet, and $D_{\mu}$ is the $G_{\mathrm{SM}}$-covariant derivative. The gauge kinetic terms arise from the Maxwell construction applied to each factor; the scalar potential is the modal potential extended to $H\in\mathbb{C}^{2}$; the Higgs kinetic term $|D_{\mu}H|^{2}$ encodes minimal coupling to all gauge fields. Corrections from higher substrate-expansion terms are suppressed by $(kd_{0})^{2}$ and contribute observably only in the Casimir regime near the coherence scale $d_{0}$. The Yukawa sector $\mathcal{L}_{\mathrm{Yukawa}}=y_{ij}\bar{\psi}_{i}H\psi_{j}+\mathrm{h.c.}$ and the fermionic kinetic term require the fermionic sector.