TFPT scientific source excerpt
Original repository path: documents/tfpt_research_contracts/sections/009-source.tex
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000125 | \noindent\textbf{The surviving $R^2$ route, typed as one moment (\veri{v477\_entropic\_scaleflow.py}).}
000126 | The Frullani identity upgrades to a \emph{scale-flow representation}: $-\ln a=\int_0^\infty
000127 | \tfrac{dt}t(e^{-ta}{-}e^{-t})=2\int_0^\infty\tfrac{d\chi}\chi(e^{-a/\chi^2}{-}e^{-1/\chi^2})$ ---
000128 | Bianconi's log action \emph{is} the flat scale-integral of relative heat-kernel actions, while TFPT's
000129 | $S_{\mathrm{rel},\chi}$ evaluates \emph{one} KMS scale with a weight $f$: the bridge between the two
000130 | action principles is a choice of \emph{scale measure}. In the moment dictionary (weight $w(t)\,dt/t$,
000131 | $\mu_k$ its $e^{-t}$-moments) the coefficients read $3\beta R\,\mu_1+\tfrac{17}{24}\beta^2R^2\mu_2$;
000132 | the flat weight has $\mu_1{=}\mu_2{=}1$ ($=$ the raw \vref{v475} coefficients), and demanding
000133 | $m^2{=}\cthree^7\Mpl^2$ forces \emph{exactly}
000134 | \[
000135 |   \boxed{\ \mu_2/\mu_1^2=\tfrac{72}{17}(8\pi)^9\ }\qquad\text{with}\qquad
000136 |   \tfrac{4608\pi^2/17}{(72/17)(8\pi)^9}=\cthree^7\ \text{(identically)}
000137 | \]
000138 | --- the \vref{v475} kill-test number \emph{is} a moment ratio, and the ``$13$ orders of magnitude''
000139 | are not a bridge mismatch but a scale-measure datum: the flat measure fixes it wrongly, the TFPT KMS
000140 | measure ($f_0{=}6(4\pi)^2/\cthree^7$, \vref{v36}) fixes it correctly --- one KMS moment, two
000141 | parametrisations, zero new dials. \mC\ (consistency, not derivation: the measure is TFPT input); the
000142 | $R^2$ gate and the \vref{v359} typing stay \mO.
000143 | \smallskip\\
000144 | \noindent\textbf{The two remaining legs, first computable steps (\veri{v478\_entropic\_continuum\_legs.py}).}
000145 | \emph{Leg A (AP2 continuum):} on the exactly solvable critical chain (infinite-volume kernel, $50$-digit
000146 | precision) the compressed interval state carries the conformal/BW structure quantitatively --- the