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000039 | \subsection{Canonical spectral Einstein functional and geometric Hodge closure}
000040 | 
000041 | The local target action is written as the low-curvature expansion of a canonical spectral
000042 | Einstein functional:
000043 | \begin{equation}
000044 | \label{eq:chi-effective-action}
000045 | \begin{aligned}
000046 | \Gammaeff
000047 | =
000048 | \int d^4x\sqrt{-g}\,
000049 | \Big[
000050 | c_R\chigeo^2R
000051 | +Z_\chi(\partial\chigeo)^2
000052 | -\mu_\Phi^2(\chigeo)\,\Phi^\dagger\Phi
000053 | +\lambda_\Phi(\Phi^\dagger\Phi)^2
000054 | +\frac{1}{6M^2}R^2
000055 | +\cdots
000056 | \Big].
000057 | \end{aligned}
000058 | \end{equation}
000059 | The relative bosonic endomorphism used below is
000060 | \begin{equation}
000061 | \label{eq:relative-bosonic-endomorphism}
000062 | E_{\mathrm{rel}}
000063 | =
000064 | -\frac14 R\,\mathbf 1_{48}
000065 | -
000066 | (\Phi^\dagger\Phi)\,P_2
000067 | +
000068 | \frac12\gamma^{\mu\nu}\mathbb F_{\mu\nu},
000069 | \end{equation}
000070 | and the resulting low-curvature residue decomposition is summarized in
000071 | \Cref{tab:heat-kernel-map}.
000072 | 
000073 | \begin{table}[t]
000074 | \centering
000075 | \scriptsize
000076 | \setlength{\tabcolsep}{3pt}
000077 | \begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.15\linewidth}>{\raggedright\arraybackslash}p{0.22\linewidth}>{\raggedright\arraybackslash}p{0.22\linewidth}>{\raggedright\arraybackslash}p{0.14\linewidth}>{\raggedright\arraybackslash}p{0.14\linewidth}@{}}
000078 | \toprule
000079 | Residue term & Operator sector & Generated term & $\chigeo$ scaling & Role \\
000080 | \midrule
000081 | $a_0$ & relative bulk trace & vacuum-energy bookkeeping & $\chigeo^4$ & Hard \\
000082 | $a_2$ & seam-even bosonic bulk term & $\chigeo^2R$, $\chigeo^2\Phi^\dagger\Phi$ & $\chigeo^2$ & Closure \\
000083 | $a_4$ & local bosonic sector & $|D\Phi|^2$, $(\Phi^\dagger\Phi)^2$, $R\,\Phi^\dagger\Phi$, $\tr\mathbb F^2$, $R^2$ & $\chigeo^0$ & Closure \\
000084 | boundary/defect term & seam correction and parity data & local defect counterterms, seam-even selection & $\Delta_{\Seam}$ & Closure \\
000085 | $\eta$ term & APS spectral asymmetry & sign and spectral-flow control & scale free & Closure \\
000086 | \bottomrule
000087 | \end{tabular}
000088 | \caption{Low-curvature residue map for the canonical spectral Einstein functional.}
000089 | \label{tab:heat-kernel-map}
000090 | \end{table}
000091 | 
000092 | The resulting geometric logic is displayed in \Cref{fig:chi-bifurcation}. The canonical spectral
000093 | residue fixes the Einstein term first; the low-curvature expansion then yields the Higgs
000094 | normalization, the Higgs vacuum relation, and the retained curvature/gauge side terms. This is
000095 | why the local spectral-scale sector is the principal geometric closure channel of the manuscript
000096 | rather than just one more scalar parameter.