TFPT scientific source excerpt Original repository path: documents/tfpt_2_standard_model/sections/010-section-the-explicit-mass-gap-closes-dynamics-and-cluste.tex Original file SHA-256: 3a31f272d999f1e0b027f25a877068576c6cbdc83150c7b5c0f8a6e17bed1904 Inclusive original lines: 273-322 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. 000273 | \section{The ambient closure: one spectral-action measure, IR-complete by decoupling} 000274 | % ===================================================================== 000275 | The two formerly-hardest items --- the \emph{ambient} (unprojected) QFT measure beyond the selector 000276 | $P_{\mathrm{adm}}$, and \emph{full quantum gravity} beyond the $R^2$ scalaron --- are not two separate 000277 | problems. They are \emph{one} object: the relative spectral-action measure 000278 | \[ 000279 | S=\operatorname{Tr}f\!\Bigl(\tfrac{D_{\mathrm{rel}}+A_\Sigma}{\chi}\Bigr) 000280 | -\operatorname{Tr}f\!\Bigl(\tfrac{D_{\mathrm{ref}}}{\chi}\Bigr)+\tfrac{i\pi}{2}\Delta\eta_\Sigma , 000281 | \] 000282 | with $D_{\mathrm{rel}}$ the matter/gauge Dirac operator and the gravitational data in the same trace. 000283 | Three facts reduce both open chunks to a single, named residual. 000284 | 000285 | \begin{proposition}[Ambient closure by decoupling] 000286 | \label{prop:ambient} 000287 | \emph{(i) Well-defined.} The cutoff $f$ makes $\operatorname{Tr}f$ finite (heat kernel), and the reference 000288 | subtraction makes $S$ \emph{relative}; the ambient measure is therefore UV-finite, not a formal path 000289 | integral. \emph{(ii) Gapped in both sectors.} The matter sector has the explicit transfer gap 000290 | $\Delta=6\log\tfrac32$ (Lemma~\ref{lem:gap}); the gravity sector has the scalaron mass 000291 | $M=\cthree^{7/2}\Mbar=3.06\times10^{13}$ GeV. Non-admissible matter states (those with 000292 | $\Delta_{\mathrm{adm}}>0$, i.e.\ outside $\ker\Delta_{\mathrm{adm}}$) and higher-curvature gravitational 000293 | modes are massive. \emph{(iii) Wilsonian decoupling.} Massive modes integrate out, so the IR measure 000294 | factorises, 000295 | \[ 000296 | S_{\mathrm{IR}}=\underbrace{(\text{admissible matter QFT})}_{\text{Prop.~\ref{prop:qftmeasure}}} 000297 | \times\underbrace{(R^2\ \text{gravity})}_{\text{Lemma~\ref{lem:trunc}}} 000298 | \;+\;O\!\bigl((2/3)^{6n}\bigr)\;+\;O\!\bigl((R/M^2)^{n-2}\bigr), 000299 | \] 000300 | both correction series convergent. Hence the unprojected ambient theory carries \emph{no new IR physics} 000301 | beyond the admissible matter QFT and $R^2$ gravity. 000302 | \end{proposition} 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