TFPT scientific source excerpt
Original repository path: documents/tfpt_2_standard_model/sections/010-section-the-explicit-mass-gap-closes-dynamics-and-cluste.tex
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000303 | 
000304 | \par\smallskip\noindent\textbf{The cutoff and the Dirac of $S_{\mathrm{rel}}$ are now fixed by the seam state
000305 | (Modular Spectral Closure, \veri{v258\_dirac\_covariance\_induction.py}, \veri{v259\_modular\_cutoff\_kappa.py}).}
000306 | The relative spectral action above is not a free template. Its Dirac operator $D_{\mathrm{rel}}$ is the
000307 | \emph{modular/covariance induction} of the boundary seam KMS state --- $[D_F]=[D_\Sigma]\widehat\otimes_{(E_8)_1}
000308 | [\mathcal K_{\mathrm{car}}]$, the finite implementation being $\log((1{-}C_F)C_F^{-1})$ with $C_F=P_F C_\Sigma
000309 | P_F$ (\veri{v258\_dirac\_covariance\_induction.py}) --- and the cutoff $f$ \emph{is} that same KMS weight, so by
000310 | Tomita--Takesaki/Bisognano--Wichmann ($\beta{=}1$, $2\pi{=}1/(4\cthree)$) its moments collapse to $f_2/f_0=1$
000311 | exactly (\veri{v259\_modular\_cutoff\_kappa.py}), removing the last scheme freedom and turning $\kappa$ into a
000312 | finite-triple trace ratio. The Seeley--DeWitt expansion of this $S_{\mathrm{rel}}$ over the explicit $96$-dim
000313 | finite triple is written out (\veri{v255\_spectral\_action\_expansion.py}), and the whole assembly --- seam,
000314 | carrier-$16$ and $E_8$ on one Kummer/K3 --- is certified as one relative object reduced to the single premise
000315 | \texttt{QGEO.SYM.01} (\veri{v261\_modular\_spectral\_closure.py}). So this ``ambient closure'' delivers the
000316 | \emph{boundary relative object}; the \emph{real $4$D interacting} ambient measure $\mathrm{QG}_{\mathrm{amb}}$ is
000317 | now a \mC\ redundancy (\vref{v369}, conditional on \texttt{SEAM.EQUIV.01}${+}$Bisognano--Wichmann), with only the
000318 | cited continuum-existence theorems staying \mO\ (\vref{v336}).
000319 | 
000320 | \noindent\textbf{Regimes, all TFPT-fixed.} Gravity has three clean windows set by the seam scales:
000321 | $E<M\!\sim\!3\times10^{13}$ GeV is pure Einstein ($a_2$), $M<E<\Mbar$ is $R^2$ Starobinsky ($a_4$, the
000322 | scalaron propagates), and $E\sim\Mbar$ is the full non-perturbative trace. The Seeley--DeWitt tower
000323 | $a_{2n}$ enters with $(R/M^2)^{n-2}$; on the inflationary attractor $R/M^2\sim1/(2N_\star)\approx0.009$,
000324 | so the $R^3$ term in the \emph{action} is $\sim1\%$ and higher terms are Planck-suppressed. (This action-level
000325 | $\sim1\%$ is distinct from the corrections to the \emph{observables} $(n_s,A_s)$, which enter at
000326 | $O(1/N_\star^2)\approx3\times10^{-4}$ after the attractor suppression --- two different estimates, not a