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000001 | % Auto-extracted from ../tfpt-42.tex for the TFPT 4.5 clean split.
000002 | % Source body assigned by the TFPT 4.5 clean split.
000003 | 
000004 | % ---- Geometric reconstruction, Planck closure, EW and pole matching: source lines 3752-5571 ----
000005 | \subsection{From bridge to geometric completion}
000006 | 
000007 | \begin{principle}[Closure split]
000008 | Once the carrier packet and family closure architecture are fixed, the focused main-text closure
000009 | problem splits into a geometric program $\Ageom$ and a transport program $\Atrans$. Record
000010 | algebra and prediction semantics are now tied directly to the admissible gap in the main theorem
000011 | stack, while E$_8$ scale grammar and horizon extrapolations are deferred to appendix-level
000012 | continuations.
000013 | \end{principle}
000014 | 
000015 | \[
000016 | D_{\mathrm{geo}}:=D_{\mathrm{ref}}+D_{\mathrm{rel}}=D.
000017 | \]
000018 | The two main-text programs are packaged as
000019 | \[
000020 | \Ageom
000021 | :=
000022 | \langle E_3\oplus E_2,\ D_{\mathrm{geo}},\ \mathcal B_{\mathrm{rel}},\ \wspin\rangle,
000023 | \qquad
000024 | \Atrans
000025 | :=
000026 | \langle U_6,\ T,\ D_y,\ \mathcal Y_y^{(\varepsilon)},\ \Ccyc\rangle.
000027 | \]
000028 | This split keeps the carrier theorem, the geometric reconstruction problem, and the record-level extensions from being flattened into one undifferentiated claim. In particular, the metric field is no longer treated here as an independent entry of $\Ageom$; it is reconstructed from the geometric Dirac anchor introduced next, while $D_{\mathrm{rel}}$ remains the comparison operator for spectral subtraction.
000029 | 
000030 | \subsection{Relative APS and superconnection setup}
000031 | 
000032 | The geometric completion is formulated on a graded bundle
000033 | \[
000034 | \mathcal E_{\mathrm{rel}}=\mathcal E_{\mathrm{rel}}^+\oplus\mathcal E_{\mathrm{rel}}^-,
000035 | \]
000036 | with the geometric Dirac anchor $D_{\mathrm{geo}}$, the declared reference operator
000037 | $D_{\mathrm{ref}}$, and APS-type boundary conditions on the seam $\Seam$; see
000038 | Appendix~\TFPTxref{app:aps}. The geometric package is
000039 | \[
000040 | \mathbb A_{\Seam}:=(D_{\mathrm{geo}},D_{\mathrm{rel}},A_{\Sigma}^{\mathrm{geo}}),
000041 | \qquad
000042 | A_{\Sigma}^{\mathrm{geo}}:=\mathcal B_{\mathrm{rel}}\oplus\nabla_{\chigeo}.
000043 | \]
000044 | 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 \TFPTxref{eq:tfpt-master}.
000045 | \begin{theorem}[Geometric reconstruction from the Dirac anchor]
000046 | \label{thm:geometric-reconstruction-anchor}
000047 | \label{thm:geometric-reconstruction-from-Dgeo}
000048 | \label{thm:relative-geometric-reconstruction}
000049 | Assume
000050 | \[
000051 | Z(\mathcal A)\cong C^\infty(M),
000052 | \]
000053 | and the geometric anchor $D_{\mathrm{geo}}$ is of Dirac type with principal symbol
000054 | \[
000055 | \sigma(D_{\mathrm{geo}})(x,\xi)^2
000056 | =
000057 | g^{\mu\nu}(x)\xi_\mu\xi_\nu\,\mathbf 1.
000058 | \]
000059 | Then $D_{\mathrm{geo}}$ reconstructs the metric $g$, the spin structure, and the
000060 | Levi--Civita connection on the geometric branch. Inner fluctuations of the same anchored
000061 | spectral datum generate the gauge sector.
000062 | The operator $D_{\mathrm{rel}}$ is the reference-subtracted comparison operator on that
000063 | same branch and is not itself used as the local geometric anchor.
000064 | \end{theorem}
000065 | 
000066 | \begin{proof}[Proof sketch]
000067 | On the admissible branch the commutative center identifies the manifold algebra, while the
000068 | principal symbol of the Dirac anchor determines the quadratic form on cotangent vectors.
000069 | Standard Dirac-type reconstruction then recovers the spin bundle and Levi--Civita connection
000070 | from that same symbol data. The relative operator changes the comparison bookkeeping, not the
000071 | geometric source of the local symbol.
000072 | \end{proof}
000073 | 
000074 | \begin{theorem}[Retained branch from the admissibility selector]
000075 | \label{thm:retained-branch-from-Padm}
000076 | \label{thm:admissible-sector-on-reconstructed-geometry}
000077 | \label{thm:full-operator-reconstruction}
000078 | Let $D_{\mathrm{geo}}$ be the geometric anchor and let $\Padm$ be the retained
000079 | admissibility selector.
000080 | Then:
000081 | \begin{enumerate}[label=(\roman*), leftmargin=1.5em, topsep=3pt, itemsep=2pt]
000082 | \item $\Xbulk=\mathrm{Spec}\,Z(\mathcal A)$ and $\Seam=\mathrm{Fix}(\taudbl)$;
000083 | \item the local spectral density of $D_{\mathrm{geo}}$ determines the geometric response
000084 |       $\chigeo$;
000085 | \item $\Padm$ defines the retained admissible sector for observables, transport,
000086 |       and positivity statements on that branch.
000087 | \end{enumerate}
000088 | No claim is made that the compressed operator $\Padm D_{\mathrm{rel}}\Padm$
000089 | carries the local principal symbol of the geometry.
000090 | \end{theorem}
000091 | 
000092 | \begin{corollary}[No residual geometric datum beyond the geometric anchor]
000093 | The geometric package is reconstructed from $D_{\mathrm{geo}}$ and its local spectral density.
000094 | The selector $\Padm$ acts downstream on the retained sector and is not part of the
000095 | local geometric anchor.
000096 | \end{corollary}
000097 | 
000098 | \begin{definition}[Local spectral scale and canonical relative spectral residues]
000099 | \label{def:local-spectral-scale}
000100 | \label{def:canonical-residues}
000101 | Write the geometric response as a local spectral scale
000102 | \[
000103 | \chigeo=\Lambda e^\sigma,
000104 | \qquad
000105 | D_{\sigma,\mathrm{geo}}:=e^{-\sigma/2}D_{\mathrm{geo}}e^{-\sigma/2}.
000106 | \]
000107 | \[
000108 | \mathcal Z_{\mathrm{rel}}(s;\sigma)
000109 | :=
000110 | \operatorname{Tr}_{\mathrm{rel}}\!\left(
000111 | |D_{\sigma,\mathrm{geo}}+\mathcal B_{\mathrm{rel}}|^{-2s}
000112 | -|D_{\mathrm{ref}}|^{-2s}
000113 | \right)
000114 | +\frac{i\pi}{2}\Delta\eta_\Sigma.
000115 | \]
000116 | The canonical Einstein residue and order-zero residue are
000117 | \[
000118 | \mathcal E_D
000119 | :=
000120 | \operatorname*{Res}_{s=1}\mathcal Z_{\mathrm{rel}}(s;\sigma),
000121 | \qquad
000122 | \Gamma_D^{(4)}
000123 | :=
000124 | \operatorname{FP}_{s=0}\mathcal Z_{\mathrm{rel}}(s;\sigma).
000125 | \]
000126 | The gravitational branch is then read from the profile-free residue pair rather than from a freely
000127 | chosen spectral profile.
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
000133 | $\chigeo=\chigeozero:=\Lambda e^{\sigma_0}$ and the present geometric branch reduces to the
000134 | constant-$\chigeo$ closure relations extracted from
000135 | $\Gamma_{\mathrm{grav}}:=-6\chigeo^2\mathcal E_D+\Gamma_D^{(4)}$:
000136 | \[
000137 | \Mpl^2=\frac{\chigeozero^2}{2\pi^2},
000138 | \qquad
000139 | \vgeo^2=Z_H\,\frac{\mu_\Phi^2(\chigeozero)}{\lambda_\Phi}.
000140 | \]
000141 | Once the unique stationary root of \TFPTxref{thm:absolute-spectral-planck-closure} is imposed, the
000142 | Einstein coefficient is equivalently the boundary-normalized readout
000143 | \[
000144 | \frac{\Mpl^2}{\lambda_\Sigma^2}=\frac{\rho_\star}{2\pi^2}.
000145 | \]
000146 | Thus the former constant-$\chigeo$ formulas are retained as the homogeneous branch of the local