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000001 | \subsection*{(2) Gravity --- the classical covariant field equation carries no free dimensionless Newton coupling under the named QFT/entanglement premises \mC; the $R^2$ inflation layer up to one truncation \mE/\mC}
000002 | The metric-sector \emph{field equation} itself carries no free dimensionless Newton coupling under those premises: fixed-\emph{volume} entanglement
000003 | equilibrium (Jacobson 2015) forces the \emph{full} covariant Einstein equation
000004 | $G_{ab}{+}\Lambda g_{ab}=\cthree^{-1}T_{ab}$ with \emph{both} coefficients fixed
000005 | ($8\pi{=}1/\cthree$, $\Lambda$ from $\alpha$; \vref{v358}/\vref{v359}). What \emph{this} subsection details is
000006 | the complementary \emph{spectral-action / inflation} layer --- the $R+R^2$ readout --- whose closure is
000007 | \emph{not} a single hypothesis but a chain of three standard building blocks plus \emph{one} controlled
000008 | remainder:
000009 | 
000010 | \begin{center}
000011 | \begin{tikzpicture}[font=\footnotesize,
000012 |   st/.style={rectangle,rounded corners,draw=tfgreen!70!black,fill=tfgreen!8,thick,align=center,inner sep=3pt,minimum height=9mm},
000013 |   rest/.style={rectangle,rounded corners,draw=tfgold!80!black,fill=tfgold!12,very thick,align=center,inner sep=3pt,minimum height=9mm},
000014 |   ar/.style={-{Stealth[length=2mm]},thick,tfgray}]
000015 |   \node[st] (a) at (0,0) {spectral action\\(heat kernel $a_4$)\\$\to R{+}R^2{+}\operatorname{tr}\mathbb F^2$};
000016 |   \node[st] (b) at (3.7,0) {BRST/Hodge\\differential kills\\gauge directions};
000017 |   \node[st] (c) at (7.2,0) {FRW reduction\\$\to$ Starobinsky\\potential};
000018 |   \node[rest] (d) at (10.8,0) {\textbf{remainder:}\\low-curvature\\truncation\\$R^n\!\to\!R^2$};
000019 |   \draw[ar] (a)--(b); \draw[ar] (b)--(c); \draw[ar] (c)--(d);
000020 | \end{tikzpicture}
000021 | \end{center}
000022 | 
000023 | \begin{itemize}[leftmargin=1.4em]
000024 | \item \textbf{Spectral action:} the Seeley--DeWitt coefficients give Einstein--Hilbert
000025 |       ($a_2\!\sim\!-R/3$), $R^2$ ($a_4|_{R^2}\!\sim\!R^2/72$, coefficient $\tfrac1{6M^2}$) and gauge terms.
000026 |       Standard heat-kernel geometry (Gilkey), not a TFPT specificity; the scalaron ratio
000027 |       $M^2/\Mbar^2=6(4\pi)^2/f_0$ is fixed by the cutoff moment ($\cthree^7$). \mE\
000028 |       \veri{v36\_spectral\_action\_g2.py}
000029 | \item \textbf{Hodge/BRST closure:} the geometric BRST/Hodge differential quotients diffeomorphisms $+$
000030 |       local-frame gauge; the physical sector is its cohomology. A theorem as soon as the elliptic
000031 |       complex on the \emph{compactified} normal slice $S^2$ (APS boundary conditions) is
000032 |       Fredholm --- again standard index theory. \mE\ (given Fredholm)
000033 | \item \textbf{FRW reduction:} minisuperspace symmetry reduction onto the scale factor $+$ scalaron;
000034 |       yields the Starobinsky potential. Standard. \mE
000035 | \item \textbf{The only remainder:} the low-curvature \emph{truncation} --- higher curvature terms
000036 |       $R^n$ ($n\!\ge\!3$) reduce to the local $R^2$. A \emph{controlled approximation} with explicit
000037 |       domain of validity (curvature $\ll M^2$), \emph{not} a free assumption.
000038 | \end{itemize}
000039 | 
000040 | \par\smallskip\noindent\textbf{Sharpening (2): the truncation is explicitly bounded.}
000041 | $M_{\mathrm{Pl}}$ and $A_s=\tfrac{N_\star^2}{24\pi^2}c_3^7$ are determined \emph{exactly in the
000042 | low-curvature $R^2$ regime}. The truncation remainder is \emph{not} uncontrolled: on the Starobinsky
000043 | attractor during observable inflation $R/M^2\sim\tfrac1{2N_\star}$, i.e.\ an expansion parameter
000044 | $\approx0.009$ at $N_\star\!\approx\!55$. The $R^{n\ge3}$ corrections to $(n_s,A_s)$ are therefore
000045 | $O(1/N_\star^2)\approx3\times10^{-4}$ ($\sim0.03\%$) --- explicitly bounded and tiny. The leading
000046 | predictions $n_s=1-2/N_\star$ and $r=12/N_\star^2$ (the band over $N_\star\in[50,60]$) hit Planck. The
000047 | status is thus not ``conditional on an unproved Hodge theorem'' but ``exact up to an
000048 | $O(1/N_\star^2)$ truncation''. \textbf{Remaining path:} the Fredholm property of the seam elliptic
000049 | complex (APS, standard index theory). \mE\ in the regime.
000050 |