TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--05_metrology_source/sections/001-subsection-from-bridge-to-geometric-completion.tex
Original file SHA-256: f03609effe2abf6629a0057fa024c29fba0c59176072588599bfa1646f48939e
Inclusive original lines: 108-146
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

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