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000044 | \subsection{Relative APS and superconnection setup}
000045 | 
000046 | The geometric completion is formulated on a graded bundle
000047 | \[
000048 | \mathcal E_{\mathrm{rel}}=\mathcal E_{\mathrm{rel}}^+\oplus\mathcal E_{\mathrm{rel}}^-,
000049 | \]
000050 | with the geometric Dirac anchor $D_{\mathrm{geo}}$, the declared reference operator
000051 | $D_{\mathrm{ref}}$, and APS-type boundary conditions on the seam $\Seam$; see
000052 | Appendix~\ref{app:aps}. The geometric package is
000053 | \[
000054 | \mathbb A_{\Seam}:=(D_{\mathrm{geo}},D_{\mathrm{rel}},A_{\Sigma}^{\mathrm{geo}}),
000055 | \qquad
000056 | A_{\Sigma}^{\mathrm{geo}}:=\mathcal B_{\mathrm{rel}}\oplus\nabla_{\chigeo}.
000057 | \]
000058 | Here $D_{\mathrm{geo}}$ is the geometric Dirac anchor, while $D_{\mathrm{rel}}$ is the relative comparison operator entering the spectral subtraction formulas. The bundle data $A_{\Sigma}^{\mathrm{geo}}$ reorganize the primitive seed package $A_{\Sigma}^{\mathrm{seed}}$ introduced earlier in \Cref{eq:tfpt-master}.
000059 | \begin{theorem}[Geometric reconstruction from the Dirac anchor]
000060 | \label{thm:geometric-reconstruction-anchor}
000061 | \label{thm:geometric-reconstruction-from-Dgeo}
000062 | \label{thm:relative-geometric-reconstruction}
000063 | Assume
000064 | \[
000065 | Z(\mathcal A)\cong C^\infty(M),
000066 | \]
000067 | and the geometric anchor $D_{\mathrm{geo}}$ is of Dirac type with principal symbol
000068 | \[
000069 | \sigma(D_{\mathrm{geo}})(x,\xi)^2
000070 | =
000071 | g^{\mu\nu}(x)\xi_\mu\xi_\nu\,\mathbf 1.
000072 | \]
000073 | Then $D_{\mathrm{geo}}$ reconstructs the metric $g$, the spin structure, and the
000074 | Levi--Civita connection on the geometric branch. Inner fluctuations of the same anchored
000075 | spectral datum generate the gauge sector.
000076 | The operator $D_{\mathrm{rel}}$ is the reference-subtracted comparison operator on that
000077 | same branch and is not itself used as the local geometric anchor.
000078 | \end{theorem}
000079 | 
000080 | \begin{proof}[Proof sketch]
000081 | On the admissible branch the commutative center identifies the manifold algebra, while the
000082 | principal symbol of the Dirac anchor determines the quadratic form on cotangent vectors.
000083 | Standard Dirac-type reconstruction then recovers the spin bundle and Levi--Civita connection
000084 | from that same symbol data. The relative operator changes the comparison bookkeeping, not the
000085 | geometric source of the local symbol.
000086 | \end{proof}
000087 | 
000088 | \begin{theorem}[Retained branch from the admissibility selector]
000089 | \label{thm:retained-branch-from-Padm}
000090 | \label{thm:admissible-sector-on-reconstructed-geometry}
000091 | \label{thm:full-operator-reconstruction}
000092 | Let $D_{\mathrm{geo}}$ be the geometric anchor and let $\Padm$ be the retained
000093 | admissibility selector.
000094 | Then:
000095 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt]
000096 | \item $X=\mathrm{Spec}\,Z(\mathcal A)$ and $\Seam=\mathrm{Fix}(\inv)$;
000097 | \item the local spectral density of $D_{\mathrm{geo}}$ determines the geometric response
000098 |       $\chigeo$;
000099 | \item $\Padm$ defines the retained admissible sector for observables, transport,
000100 |       and positivity statements on that branch.
000101 | \end{enumerate}
000102 | No claim is made that the compressed operator $\Padm D_{\mathrm{rel}}\Padm$
000103 | carries the local principal symbol of the geometry.
000104 | \end{theorem}
000105 | 
000106 | \begin{corollary}[No residual geometric datum beyond the geometric anchor]
000107 | The geometric package is reconstructed from $D_{\mathrm{geo}}$ and its local spectral density.
000108 | The selector $\Padm$ acts downstream on the retained sector and is not part of the
000109 | local geometric anchor.
000110 | \end{corollary}
000111 | 
000112 | \begin{definition}[Local spectral scale]
000113 | \label{def:local-spectral-scale}
000114 | Write the geometric response as a local spectral scale
000115 | \[
000116 | \chigeo=\Lambda e^\sigma,
000117 | \qquad
000118 | D_{\sigma,\mathrm{geo}}:=e^{-\sigma/2}D_{\mathrm{geo}}e^{-\sigma/2}.
000119 | \]
000120 | The relative spectral action is then organized as
000121 | \[
000122 | S_{\mathrm{rel}}[\sigma,D]
000123 | :=
000124 | \tr f\!\left(\frac{D_{\sigma,\mathrm{geo}}+\mathcal B_{\mathrm{rel}}}{\Lambda}\right)
000125 | -\tr f\!\left(\frac{D_{\mathrm{ref}}}{\Lambda}\right)
000126 | +\frac{i\pi}{2}\Delta\eta_\Sigma.
000127 | \]
000128 | \end{definition}
000129 | 
000130 | \begin{corollary}[Constant-scale reduction of the geometric branch]
000131 | \label{cor:constant-scale-reduction}
000132 | If the local spectral scale is frozen to a constant $\sigma=\sigma_0$, then $\chigeo=\chigeozero:=\Lambda e^{\sigma_0}$ and the present geometric branch reduces to the constant-$\chigeo$ closure relations
000133 | \[
000134 | \Mpl^2=\frac{\chigeozero^2}{2\pi^2},
000135 | \qquad
000136 | \vgeo^2=Z_H\,\frac{\mu_\Phi^2(\chigeozero)}{\lambda_\Phi}.
000137 | \]
000138 | Thus the former constant-$\chigeo$ formulas are retained as the homogeneous branch of the local spectral-scale formulation rather than as a competing package.
000139 | \end{corollary}
000140 |