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000001 | \subsection{Canonical spectral Einstein functional and geometric Hodge closure}
000002 | 
000003 | The local target action is written as the low-curvature expansion of a canonical spectral
000004 | Einstein functional:
000005 | \begin{equation}
000006 | \label{eq:chi-effective-action}
000007 | \begin{aligned}
000008 | \Gammaeff
000009 | =
000010 | \int d^4x\sqrt{-g}\,
000011 | \Big[
000012 | c_R\chigeo^2R
000013 | +Z_\chi(\partial\chigeo)^2
000014 | -\mu_\Phi^2(\chigeo)\,\Phi^\dagger\Phi
000015 | +\lambda_\Phi(\Phi^\dagger\Phi)^2
000016 | +\frac{1}{6M^2}R^2
000017 | +\cdots
000018 | \Big].
000019 | \end{aligned}
000020 | \end{equation}
000021 | The relative bosonic endomorphism used below is
000022 | \begin{equation}
000023 | \label{eq:relative-bosonic-endomorphism}
000024 | E_{\mathrm{rel}}
000025 | =
000026 | -\frac14 R\,\mathbf 1_{48}
000027 | -
000028 | (\Phi^\dagger\Phi)\,P_2
000029 | +
000030 | \frac12\gamma^{\mu\nu}\mathbb F_{\mu\nu},
000031 | \end{equation}
000032 | and the resulting low-curvature residue decomposition is summarized in
000033 | \Cref{tab:heat-kernel-map}.
000034 | 
000035 | \begin{table}[t]
000036 | \centering
000037 | \scriptsize
000038 | \setlength{\tabcolsep}{3pt}
000039 | \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}@{}}
000040 | \toprule
000041 | Residue term & Operator sector & Generated term & $\chigeo$ scaling & Role \\
000042 | \midrule
000043 | $a_0$ & relative bulk trace & vacuum-energy bookkeeping & $\chigeo^4$ & Hard \\
000044 | $a_2$ & seam-even bosonic bulk term & $\chigeo^2R$, $\chigeo^2\Phi^\dagger\Phi$ & $\chigeo^2$ & Closure \\
000045 | $a_4$ & local bosonic sector & $|D\Phi|^2$, $(\Phi^\dagger\Phi)^2$, $R\,\Phi^\dagger\Phi$, $\tr\mathbb F^2$, $R^2$ & $\chigeo^0$ & Closure \\
000046 | boundary/defect term & seam correction and parity data & local defect counterterms, seam-even selection & $\Delta_{\Seam}$ & Closure \\
000047 | $\eta$ term & APS spectral asymmetry & sign and spectral-flow control & scale free & Closure \\
000048 | \bottomrule
000049 | \end{tabular}
000050 | \caption{Low-curvature residue map for the canonical spectral Einstein functional.}
000051 | \label{tab:heat-kernel-map}
000052 | \end{table}
000053 | 
000054 | The resulting geometric logic is displayed in \Cref{fig:chi-bifurcation}. The canonical spectral
000055 | residue fixes the Einstein term first; the low-curvature expansion then yields the Higgs
000056 | normalization, the Higgs vacuum relation, and the retained curvature/gauge side terms. This is
000057 | why the local spectral-scale sector is the principal geometric closure channel of the manuscript
000058 | rather than just one more scalar parameter.
000059 | 
000060 | \begin{figure}[t]
000061 | \centering
000062 | \scriptsize
000063 | \begin{tikzpicture}[
000064 |     >=Latex,
000065 |     node distance=8mm and 11mm,
000066 |     a2/.style={draw=blue!60!black, fill=blue!6, rounded corners, align=center, text width=2.4cm, minimum height=0.9cm},
000067 |     a4/.style={draw=purple!70!black, fill=purple!7, rounded corners, align=center, text width=2.4cm, minimum height=0.9cm},
000068 |     flow/.style={draw=orange!75!black, fill=orange!8, rounded corners, align=center, text width=2.35cm, minimum height=0.95cm},
000069 |     outbox/.style={draw=green!50!black, fill=green!8, rounded corners, align=center, text width=2.6cm, minimum height=0.95cm}
000070 | ]
000071 | \node[a2] (a2node) {canonical spectral residue\\Einstein seed};
000072 | \node[a4, right=24mm of a2node] (a4node) {low-curvature expansion\\Higgs / curvature terms};
000073 | 
000074 | \node[flow, below=of a2node] (ein0) {$\chigeo^2R$\\Einstein term first};
000075 | \node[flow, below=of a4node] (kin0) {$|D\Phi|^2,\ (\Phi^\dagger\Phi)^2$\\$\tr\mathbb F^2,\ R^2$};
000076 | \node[flow, below=of $(ein0)!0.5!(kin0)$, yshift=-3mm] (chi) {same seam-even $\chigeo$ channel\\canonical spectral functional};
000077 | 
000078 | \node[outbox, below left=of chi] (mpl) {$\Mpl^2=\chigeozero^2/(2\pi^2)$\\Einstein branch};
000079 | \node[outbox, below=of chi] (zh) {$Z_H=1/(16\pi^2)$\\canonical Higgs normalization};
000080 | \node[outbox, below right=of chi] (vev) {$\vgeo^2=Z_H\,\mu_\Phi^2(\chigeozero)/\lambda_\Phi$\\Higgs vacuum branch};
000081 | \node[flow, right=18mm of vev] (curv) {$R^2/M^2,\ \tr\mathbb F^2$\\higher-curvature / gauge branch};
000082 | 
000083 | \draw[->, thick] (a2node) -- (ein0);
000084 | \draw[->, thick] (a4node) -- (kin0);
000085 | \draw[->, thick] (ein0) -- (chi);
000086 | \draw[->, thick] (kin0) -- (chi);
000087 | \draw[->, thick] (chi) -- (mpl);
000088 | \draw[->, thick] (chi) -- (zh);
000089 | \draw[->, thick] (chi) -- (vev);
000090 | \draw[->, thick] (kin0.east) -- (curv.west);
000091 | \draw[->, thick] (zh) -- (vev);
000092 | \end{tikzpicture}
000093 | \caption{Canonical spectral flow of the local spectral-scale channel. The spectral residue fixes the
000094 | Einstein term first; the low-curvature expansion then yields Higgs normalization, the Higgs
000095 | vacuum branch, and the retained $R^2$/gauge side terms.}
000096 | \label{fig:chi-bifurcation}
000097 | \end{figure}
000098 | 
000099 | \begin{theorem}[Canonical spectral Einstein functional and scalaron seed]
000100 | \label{prop:canonical-bosonic-normalization}
000101 | \label{thm:canonical-einstein-residue}
000102 | In four dimensions the seam-even relative residue expansion gives
000103 | \[
000104 | \mathcal E_D
000105 | =
000106 | -\int d^4x\sqrt g
000107 | \left(
000108 | \frac{1}{24\pi^2}R
000109 | +
000110 | \frac{1}{48\pi^2}\Phi^\dagger\Phi
000111 | \right)
000112 | +
000113 | \mathcal E_\Sigma,
000114 | \]
000115 | and
000116 | \[
000117 | \begin{aligned}
000118 | \Gamma_D^{(4)}
000119 | ={}&
000120 | \int d^4x\sqrt g
000121 | \Big(
000122 | \frac{1}{16\pi^2}|D\Phi|^2
000123 | +\frac{1}{16\pi^2}(\Phi^\dagger\Phi)^2
000124 | +\frac{1}{96\pi^2}R\,\Phi^\dagger\Phi
000125 | \\
000126 | &\qquad
000127 | +\frac{1}{192\pi^2}\tr\mathbb F^2
000128 | +\frac{1}{6M^2}R^2
000129 | +\cdots
000130 | \Big)
000131 | +\mathcal A_\Sigma.
000132 | \end{aligned}
000133 | \]
000134 | The seam-even scale channel $\chigeo$ is the unique scalar geometric mode on the retained branch;
000135 | together with the retained $R^2$ term it forms a positive scalaron block, so no independent
000136 | conformal negative mode survives as a separate physical degree of freedom.
000137 | Therefore
000138 | \[
000139 | \Gamma_{\mathrm{grav}}
000140 | :=
000141 | -6\chigeo^2\mathcal E_D+\Gamma_D^{(4)}
000142 | \]
000143 | has low-curvature form
000144 | \[
000145 | \Gamma_{\mathrm{grav}}
000146 | =
000147 | \int d^4x\sqrt g
000148 | \left(
000149 | \frac{\Mpl^2}{2}R
000150 | +
000151 | \frac{\Mpl^2}{12M^2}R^2
000152 | +
000153 | |D\Phi|^2
000154 | -V(\Phi,\chigeo)
000155 | -\sum_a \frac{1}{4g_a^2}\tr F_a^2
000156 | +\cdots
000157 | \right),
000158 | \]
000159 | with
000160 | \[
000161 | \Mpl^2=\frac{\chigeozero^2}{2\pi^2},
000162 | \qquad
000163 | Z_H=\frac{1}{16\pi^2},
000164 | \qquad
000165 | \Phi_{\mathrm{can}}:=\sqrt{Z_H}\,\Phi=\frac{\Phi}{4\pi},
000166 | \]
000167 | and therefore
000168 | \begin{equation}
000169 | \label{eq:induced-einstein}
000170 | \Mpl^2=\frac{\chigeozero^2}{2\pi^2},
000171 | \qquad
000172 | \vgeo^2=Z_H\,\frac{\mu_\Phi^2(\chigeozero)}{\lambda_\Phi}.
000173 | \end{equation}
000174 | After \Cref{thm:absolute-spectral-planck-closure}, the first identity is equivalently
000175 | \[