TFPT scientific source excerpt
Original repository path: documents/tfpt_research_contracts/sections/008-source.tex
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000311 | \begin{keybox}[The measure as one-loop fluctuations around the \emph{parameter-free} saddle (\veri{v365\_qg\_oneloop\_saddle.py}, \texttt{QGAMB.SADDLE.01})]
000312 | With the saddle now parameter-free (\vref{v358}/\vref{v359}), the boundary measure of Tier B organises as a
000313 | Gaussian one-loop fluctuation determinant whose ingredients are atom-fixed. \textbf{Fixed quadratic form \mE:}
000314 | the fluctuation inverse-propagator stiffness is the saddle coefficient $1/\cthree=8\pi$ --- no free dimensionless
000315 | dial, so the one-loop problem is \emph{well-posed} (it was not before \vref{v358}). \textbf{Gap-controlled
000316 | convergence \mE:} the fluctuation operator obeys
000317 | $M\ge\Delta_{\mathrm{eff}}=6\log\tfrac32-\tfrac{31}{4\pi^2}\approx1.648>0$ (\vref{v76}/\vref{v337}), so the Gaussian
000318 | integral converges mode-by-mode with no zero/negative modes besides the saddle directions; on the OS spectrum
000319 | $\{0,6\log\tfrac32,6\log3\}$ the regularised one-loop log-det $\operatorname{Tr}'\log M=6\log\tfrac92$ is finite.
000320 | \textbf{Conformal direction \mC:} the one wrong-sign mode ($c_{\mathrm{conf}}(4)=-\tfrac32$) is made convergent by
000321 | the GHP contour ($\int e^{-|c|\phi^2}=\sqrt{2\pi/3}$) with IDG dressing (\vref{v334}). \textbf{Tightness \mE:}
000322 | $\chi=1/(1-(2/3)^6)=729/665<\infty$ (\vref{v330}), so the one-loop measure has a projective limit on the
000323 | admissible sector, with \emph{zero} new dimensionless dials (\vref{v364}). \textbf{Residual:} the full
000324 | \emph{non-perturbative} projective limit (G6 on the ambient sector) is now discharged as a \mC\ redundancy
000325 | (\vref{v369}, keybox below) --- this sharpens C7 from
000326 | ``diffuse full QG'' to a one-loop-controlled Gaussian problem around a parameter-free saddle and then to a
000327 | certification object. \mE/\mC
000328 | \end{keybox}
000329 | 
000330 | \begin{keybox}[The holographic route: the ambient measure is boundary-\emph{redundant}, not missing dynamics (\veri{v369\_qgamb\_redundancy.py}, \texttt{QGAMB.REDUNDANCY.01})]