%=============================================================================
% APPENDIX SM: THE "SCALARON MASS" IS NOT A DFD OBSERVABLE -- CLOSURE
%
% Self-contained section resolving the long-standing OPEN/FITTED entry
% "DFD scalaron mass" (prior status: A_s-route ~2.85e13 GeV vs curvature/a_4
% route ~Planckian; disagreement ~1e11; no heat-kernel number carries alpha^-5).
%
% RESOLUTION (theorem-grade, consistent with a threefold independent verification, tally
%   scalaron-exists 0/3, reframe-valid 3/3, manufactured-flaws 0):
%   The question is MIS-FRAMED. DFD contains no native Starobinsky R+R^2
%   scalaron (Thm. no-native-inflaton, App. AH2b), so there is no scalar field
%   whose mass could be the "scalaron mass." The two routes disagree precisely
%   because NEITHER is a real DFD field: they are two ways of fitting a foreign
%   field onto a theory that does not contain it. DFD inflation is rung-locked:
%   H_star = 8 pi f_chi, with n_s = 1 - 2/60 and r = 4 alpha/pi, and NO
%   scalaron potential.
%
% This appendix manufactures NEITHER a forced scalaron mass NOR an obstruction.
% It DISSOLVES the question; it does not claim to "solve" a scalaron mass, and
% it leaves the genuine residual (the fitted A_s coefficient / eps_W) standing.
% Arithmetic reproduced in dfd_class/scalaron_mass_analysis.py (figure
% scripts/scalaron_mass_analysis.py (shipped generator)).
%=============================================================================

\section{The ``Scalaron Mass'' is Not a DFD Observable}
\label{app:scalaron-mass}

\subsection{Statement of the (mis-framed) problem}
\label{app:sm-statement}

An earlier ledger carried an open entry labelled \emph{DFD scalaron mass}, graded
\textsc{open/fitted}. Two estimates were on record and disagreed by eleven orders
of magnitude:
\begin{itemize}
\item an \emph{$A_s$-route}, which fits a Starobinsky scalaron mass to the
measured scalar amplitude through the $R+R^2$ amplitude relation
$A_s=\tfrac{1}{24\pi^2}(M/\bar M_P)^2 N^2$, giving
$M_{A_s}\simeq2.85\times10^{13}\,$GeV; and
\item a \emph{curvature / $a_4$-route}, which reads a Starobinsky mass from a
dimensionless $R^2$ heat-kernel coefficient through $M^2=M_P^2/(6\,c_{R^2})$,
which with any natural $c_{R^2}=\mathcal O(1)/16\pi^2$ returns $M\sim M_P$
(Planckian).
\end{itemize}
The discrepancy $(M_P/M_{A_s})^2\approx1.8\times10^{11}$ was treated as an
unresolved tension between two estimates of a single quantity. We show it is no
such thing. The two numbers measure two \emph{different} objects, and the
quantity they were supposed to bracket --- a Starobinsky scalaron mass --- does
not exist in DFD.

\subsection{No native scalaron exists}
\label{app:sm-no-scalaron}

\begin{proposition}[No-native-scalaron $\Rightarrow$ scalaron mass undefined]
\label{prop:sm-no-scalaron}
The DFD gravitational action contains no curvature-scalar ($R^2$) or
higher-derivative term, and therefore propagates no Stelle-type scalar degree of
freedom. Consequently there is no field in DFD whose mass can be identified with
a Starobinsky scalaron mass, and ``the DFD scalaron mass'' is
\emph{undefined-because-absent}, not merely undetermined.
\end{proposition}

\begin{proof}
Three independent facts close this.

\emph{(a) No $R^2$ term in the action.} The DFD gravitational sector is the
two-derivative Einstein--Hilbert term arising from the $a_2$ Seeley--DeWitt
coefficient of the spectral action; the propagation analysis records explicitly
that DFD carries \emph{no higher-derivative terms and no curvature-scalar
couplings} (Sec.~\ref{sec:gw}, with $\alpha_T=0$ identically).
A Starobinsky scalaron is the additional propagating scalar polarisation of an
$R+R^2$ theory; with no $R^2$ term there is no such polarisation.

\emph{(b) The $a_4$ coefficient is the gauge sector, not $R^2$.} In the Connes
spectral action the $a_4$ heat-kernel coefficient yields the Yang--Mills/Maxwell
kinetic term together with the chiral anomaly
(Ext.\ Deriv., App.~AH3a; App.~\ref{app:O_alpha57_clean}),
not an Einstein-frame $R^2$ term. The historical
``curvature/$a_4$-route gives a Planckian scalaron'' read the gauge coefficient
as a gravitational $R^2$ coefficient; that identification is incorrect, and with
it the Planckian estimate loses any claim to be ``the scalaron mass.''

\emph{(c) No scalar mode is slow-roll-compatible.} By
Theorem~\ref{thm:no-native-inflaton} (proof in Ext.\ Deriv., App.~AH series~IV--VI; six
independent failure modes), no DFD scalar admits an inflaton role with potential
mass $m^2\lesssim10^{-11}M_P^2$: $\psi$ is FRW-frozen and otherwise
Planck-massive; the squashing modulus $\tau$ is fixed Planck-massive
($m_\tau^2\approx2.94\,M_P^2$); the $57$ Kaluza--Klein modes sit at
$\sqrt\alpha\,M_P$; complex-structure and harmonic-one-form deformations vanish
($h^{2,0}=0$, $b_1=0$); and $k_{\max}$, the Chern--Simons levels are
integer-quantised. There is no continuous, light scalar to carry a scalaron mass.

Since a scalaron mass is by definition the mass of the $R+R^2$ scalar, and no
such field exists in DFD by (a)--(c), the symbol $M_{\rm scalaron}$ has no DFD
referent. $\square$
\end{proof}

\subsection{Why the two routes disagree by $\sim10^{11}$}
\label{app:sm-disagreement}

The disagreement is not numerical noise; it is the exact signature of fitting an
absent field. Both numbers are reproduced to printed precision in
\texttt{scalaron\_mass\_analysis.py}.

\paragraph{The $A_s$-route is the inflationary Hubble in disguise.}
DFD inflation is fixed by a \emph{rung coincidence}, not a potential. With
$f_\chi=\bar M_P\alpha^3$ and the Majorana rung $M_R=M_P\alpha^3$ the same
$\alpha^3$ rung, and $M_P=\sqrt{8\pi}\,\bar M_P$, the inflationary Hubble is
\begin{equation}
\fiteq{H_\star \;=\; 8\pi f_\chi \;=\; \sqrt{8\pi}\,M_R \;=\; 2.38\times10^{13}\,\mathrm{GeV}
\qquad(\text{Thm.~\ref{thm:chi-postinflation}, App.~\ref{app:chi-matter}}),
\label{eq:sm-hstar}}
\end{equation}
verified as an identity ($8\pi f_\chi/\!\sqrt{8\pi}M_R=1$ to ten digits). Through
the de~Sitter dictionary $A_s=H_\star^2/(8\pi^2\varepsilon_W\bar M_P^2)$ with
$\varepsilon_W=\alpha/(4\pi)$, the amplitude collapses to $A_s=32\pi\alpha^5$
(App.~\ref{app:new-results}); equivalently $\sqrt{A_s}\propto H_\star$. Feeding
this $A_s$ back through the Starobinsky amplitude relation therefore returns a
number $M_{A_s}=1.20\,H_\star$, i.e. $\mathcal O(1)$ times the rung-locked Hubble
\eqref{eq:sm-hstar}. The ``$A_s$-route scalaron mass'' is simply $H_\star$
re-read --- an inflationary \emph{Hubble} scale, a genuine DFD quantity, but not a
particle mass.

\paragraph{The $a_4$-route defaults to Planckian.}
A genuine $R+R^2$ scalaron obeys $M^2=M_P^2/(6\,c_{R^2})$ with $c_{R^2}$ the
dimensionless $R^2$ coefficient. To force $M_{a_4}\to M_{A_s}$ would require
$c_{R^2}=M_P^2/(6M_{A_s}^2)\approx3.1\times10^{10}$, larger than the natural
heat-kernel value $1/16\pi^2$ by a factor $\sim4.8\times10^{12}$. The single
DFD power closest to bridging the gap is $\alpha^{-5}=4.8\times10^{10}$ --- and it
still misses, \emph{and} there is no $R^2$ term to host any $c_{R^2}$ in the first
place. No DFD heat-kernel number carries $\alpha^{-5}$; the route defaults to
$M\sim M_P$.

\paragraph{The gap is the missing $\alpha^5$.}
The two routes differ by
\begin{equation}
\fiteq{\left(\frac{M_P}{M_{A_s}}\right)^2 \;=\; 1.84\times10^{11}
\;=\; 3.8\,\alpha^{-5},
\qquad \log_\alpha\!\left[(M_P/M_{A_s})^2\right]=-5.27 .
\label{eq:sm-gap}}
\end{equation}
The $A_s$-route carries the factor $A_s=32\pi\alpha^5$; the $a_4$-route carries
$\alpha^0$. The eleven-order gap \eqref{eq:sm-gap} is exactly that $\alpha^5$
power. It is not a tension between two measurements of one mass; it is the
distance between two different objects --- the inflationary Hubble rung and the
Planck scale --- artificially compared through a parameter (the scalaron mass)
that neither realises.

\subsection{The actual DFD inflationary sector (no scalaron)}
\label{app:sm-reframe}

DFD specifies inflation with topological integers and the $\alpha$-tower, never a
scalaron mass. The dependency tree is \emph{inverted} relative to Starobinsky: in
$R+R^2$ the scalaron mass is the fundamental input and $(n_s,r,A_s)$ are derived
from it; in DFD the inputs are $k_{\max}=60$, $N_{\rm gen}=3$, and the locked
rungs, and the observables come out directly:
\begin{equation}
\fiteq{\begin{aligned}
&H_\star=8\pi f_\chi=2.38\times10^{13}\,\mathrm{GeV},
\quad
n_s=1-\frac{2}{60}=0.9667,
\\
&r=16\varepsilon_W=\frac{4\alpha}{\pi}=9.29\times10^{-3},
\quad
A_s=32\pi\alpha^5=2.08\times10^{-9}.
\end{aligned}
\label{eq:sm-observables}}
\end{equation}
Here $H_\star$ is the rung-locked \emph{primordial} (WdW) Hubble scale, not a slow-roll
inflaton Hubble --- a distinction the next theorem makes precise.

\begin{theorem}[Primordial optical decoupling]
\label{thm:optical-decoupling}
In native DFD the slow-roll energy read-off $H_{\rm inf}=\pi\bar M_P\sqrt{A_s r/2}$ is
\emph{undefined}. It presupposes four objects DFD provably lacks: a slow-roll inflaton
potential $V$ (so that $\varepsilon=\tfrac12\bar M_P^2(V'/V)^2$ exists), a sustained
quasi-de~Sitter clock, a Bunch--Davies vacuum on that clock, and the GR
curvature-tensor normalization $P_t=(2/\pi^2)(H/\bar M_P)^2$
(Theorem~\ref{thm:no-native-inflaton}). With no $V$ there is no $\varepsilon$ and no
consistency relation $r=16\varepsilon$. Therefore $A_s$ and $r$ are not inflaton
observables; they are optical-response amplitudes of the $n=e^{\psi}$ medium,
\begin{equation}
A_s=P_\psi^{\rm opt},\qquad r=P^{\rm opt}_{\rm shear}/P_\psi^{\rm opt},
\label{eq:sm-optical-response}
\end{equation}
the scalar optical-density power and its transverse spin-$2$ (shear) ratio, the TT
sector being an independent native mode ($\Box\,h^{\rm TT}=$source, $c_T=c$;
App.~\ref{app:master-equation}). The map $(A_s,r)\to H_{\rm inf}$ is a category error,
and the de~Sitter dictionary $A_s=H_\star^2/(8\pi^2\varepsilon_W\bar M_P^2)$ is a
\emph{removable bridge}, not the origin of $A_s$. In particular there is no inflaton
epoch to randomize the $\chi$ flux state, which is what protects the dark relic
(Theorem~\ref{thm:chi-stateprep}, App.~\ref{app:chi-matter}).
\end{theorem}

The decoupling is reinforced quantitatively: \emph{even granting} the slow-roll
dictionary, the hybrid is self-excluding.

\begin{theorem}[Inflation-dictionary exclusion]
\label{thm:inflation-exclusion}
Suppose, counterfactually, that DFD's amplitudes $A_s=32\pi\alpha^5$, $r=4\alpha/\pi$
were read as standard slow-roll observables. Then the slow-roll energy relation
$H_{\rm inf}=\pi\bar M_P\sqrt{A_s r/2}$ yields, \emph{exactly},
\begin{equation}
\fiteq{A_s r=32\pi\alpha^5\cdot\frac{4\alpha}{\pi}=128\,\alpha^6
\;\Longrightarrow\;
H_{\rm inf}=\pi\bar M_P\sqrt{64\alpha^6}=8\pi\bar M_P\alpha^3=8\pi f_\chi ,
\label{eq:excl-hinf}}
\end{equation}
since $f_\chi=\bar M_P\alpha^3$. The compact-angle stochastic kick per $e$-fold is then
$\delta\theta=H_{\rm inf}/(2\pi f_\chi)=4$ radians, which saturates the compact circle
in a single $e$-fold, driving $\langle\theta^2\rangle\to\pi^2/3$ and hence
$\Omega_\chi h^2=1.62\,(\pi^2/3)=5.33$ --- a $45\times$ overproduction. Therefore the
slow-roll interpretation of $(A_s,r)$ and the finite-CS $\chi$ relic
($\Omega_\chi h^2=0.1182$, Theorem~\ref{thm:chi-stateprep}) \emph{cannot both be held}.
Retaining the relic forces rejecting the dictionary --- which
Theorem~\ref{thm:optical-decoupling} shows is in any case undefined.
\end{theorem}

\begin{proof}
Eliminating $\varepsilon$ from $r=16\varepsilon$ and $A_s=H^2/(8\pi^2\varepsilon\bar M_P^2)$
gives $H^2=\tfrac12\pi^2\bar M_P^2 A_s r$, i.e.\ $H_{\rm inf}=\pi\bar M_P\sqrt{A_s r/2}$;
\eqref{eq:excl-hinf} is then arithmetic, verified to machine precision
($H_{\rm inf}/8\pi f_\chi=1$ to sixteen digits). The over-horizon variance of a light
spectator on the compact circle accumulates as $\langle\theta^2\rangle\simeq
N(H_{\rm inf}/2\pi f_\chi)^2$ until it saturates at the uniform-circle value $\pi^2/3$;
with per-$e$-fold step $\delta\theta=4>2\pi/\!\sqrt{3}$, saturation is immediate.
The numerical identity $H_{\rm inf}=8\pi f_\chi$ also exposes why the apparent
``$A_s$-route $=H_\star$'' coincidence (\S\ref{app:sm-reframe}) is \emph{circular}: the
dictionary was run forward with this same $H_\star=8\pi f_\chi$ and
$\varepsilon_W=\alpha/4\pi$ to produce $A_s,r$, so inverting it returns its own input,
not an independent measurement of an inflaton scale.
\end{proof}

\begin{corollary}[True tensor ratio; reclassification of $4\alpha/\pi$]
\label{cor:true-tensor}
If the finite-CS $\chi$ relic is retained, the primordial expansion is the optical engine
(App.~\ref{app:optical-engine}), not de~Sitter inflation, and its expansion Hubble during
$\chi$-relevant freeze-out is bounded by survival,
$H_{\rm opt}<2.1\times10^{11}$~GeV (Eq.~\eqref{eq:hopt-bound}). Writing the scalar amplitude
in the conservative slow-roll-form proxy $A_s\sim H_{\rm opt}^2/(8\pi^2\varepsilon_{\rm opt}
\bar M_P^2)$ and saturating the bound gives
\begin{equation}
\fiteq{\varepsilon_{\rm opt}\lesssim4.4\times10^{-8}
\qquad\Longrightarrow\qquad
\boxed{\,r_T\lesssim7\times10^{-7}\,}\quad(r_T=16\varepsilon_{\rm opt}).}
\label{eq:true-tensor}
\end{equation}
Substituting $f_\chi=\bar M_P\alpha^3$ and the survival bound \eqref{eq:hopt-bound} into
$r_T=A_t/A_s$ with $A_t=(2/\pi^2)(H_{\rm opt}/\bar M_P)^2$, and writing $A_s=32\pi\alpha^5$,
collapses to the clean closed form
\begin{equation}
r_T<\frac{\alpha\,Q_{\rm CS}}{4\pi N}=7.1\times10^{-7}\qquad(N\simeq60).
\label{eq:true-tensor-closed}
\end{equation}
(The numerical bound itself uses only the \emph{observed} $A_s=2.08\times10^{-9}$ and is
independent of whether the coefficient $32\pi$ is derived; the algebraic form merely makes the
$\alpha$-scaling explicit.) Hence the number $4\alpha/\pi=9.3\times10^{-3}$ is \emph{not} the
observable primordial
tensor-to-scalar (B-mode) ratio. It is the optical \emph{shear/scalar response} ratio
$P^{\rm opt}_{\rm shear}/P^{\rm opt}_\psi$ \eqref{eq:sm-optical-response} --- a property of
the $n=e^{\psi}$ medium --- whereas the genuine gravitational-wave background generated by the
engine is $r_T\lesssim10^{-6}$, far below any planned experiment.
\end{corollary}

\begin{falsifier}[Tensor B-modes]\label{fals:tensor}
The finite-CS branch predicts $r_T\lesssim10^{-6}$ (Cor.~\ref{cor:true-tensor}). A robust
detection of primordial B-modes at $r\simeq4\alpha/\pi\approx9\times10^{-3}$ ---
or anything $\gtrsim10^{-3}$ --- falsifies the finite-CS $\chi$ relic (it would require the
de~Sitter-dictionary $H_\star=8\pi f_\chi$, which erases $\chi$ by
Thm.~\ref{thm:inflation-exclusion}). A null result / $r\ll10^{-3}$ favors the optical branch.
\end{falsifier}

\begin{remark}[Anatomy of $32\pi$: the $2^3$ is derived, the $4\pi/\alpha$ is the gap]
\label{rem:32pi-anatomy}
A first-principles quantization of the canonical optical scalar settles exactly which part of
$32\pi$ is forced. The high-gradient branch of $S_\psi$ reduces to a canonical massless scalar
with $1/(8\pi G)=\bar M_P^2$ (exact), whose frozen power with the mode measure fixed \emph{a
priori} is
\begin{equation}
P_\psi^{\rm bare}=\frac{H_\star^2}{8\pi^2\bar M_P^2}=8\,\alpha^6=1.21\times10^{-12},
\label{eq:psi-bare}
\end{equation}
a genuinely derived amplitude --- the numerator $8=2^3$ is real, with \emph{no}
$\varepsilon_W$ imported. The entire residual to $A_s=32\pi\alpha^5$ is then \emph{exactly}
$32\pi\alpha^5/(8\alpha^6)=4\pi/\alpha=1/\varepsilon_W=1722$: one inverse power of the
\emph{coupling} $\alpha$, times $4\pi$. No solid-angle / horizon-cell integral can produce
$1/\alpha$ (geometric measures yield $O(1)$ transcendentals, never a coupling), so the reading
$32\pi=(4\pi)(2^3)$ is numerology --- the $2^3$ is the derived Bunch--Davies numerator but the
$4\pi$ enters only through the one-loop measure $\varepsilon_W=\alpha/4\pi$. The forced content
is thus $A_s=8\alpha^6\,\varepsilon_W^{-1}$ with the $8$ derived; the single missing primitive
is a native mechanism supplying $\varepsilon_W^{-1}=4\pi/\alpha$ (one power of the coupling),
equivalently $H_\star/M_R=\sqrt{8\pi}$. This sharpens --- and does not beat --- the
architecture-only grade. (A parallel bookkeeping note: the internally reduced-mass tensor
$A_t=64\alpha^6$ gives $r=2\alpha/\pi$, while the corpus's non-reduced $A_t=128\alpha^6$ gives
$r=4\alpha/\pi$; since $r$ is reclassified as the optical response ratio
(Cor.~\ref{cor:true-tensor}), this factor-$2$ convention does not affect the observable
$r_T\lesssim10^{-6}$.)
\end{remark}

\noindent Here $H_\star$ is pinned by the rung-lock (zero knobs), $n_s$ by the mode count
(zero knobs), $r$ is rigid modulo $\varepsilon_W$, and $A_s$ has a forced power
$\alpha^5$ with a fitted coefficient $32\pi$ (App.~\ref{app:new-results};
App.~\ref{app:fitted-knobs}). A Starobinsky scalaron mass would be a redundant
fifth quantity over-determined by these four, and it is precisely that redundancy
that lets the ``two routes'' return inconsistent values. The reframe therefore
\emph{dissolves} the open entry: there is no fifth number to compute.

\subsection{Status}
\label{app:sm-status}

\begin{status}\label{stat:scalaron-mass}
The entry ``DFD scalaron mass,'' previously graded \textsc{open/fitted}, is
re-graded \textsc{non-question / undefined-because-absent}. DFD has no native
Starobinsky $R+R^2$ scalaron (Prop.~\ref{prop:sm-no-scalaron},
Thm.~\ref{thm:no-native-inflaton}); the symbol $M_{\rm scalaron}$ has no DFD
referent. The historical $\sim10^{11}$ ``disagreement'' is explained
\eqref{eq:sm-gap} as the missing $\alpha^5$ between two unrelated scales (the
rung-locked inflationary Hubble $H_\star$ and $M_P$), not as a tension between two
estimates of one mass.

We make no positive claim of a \emph{forced} scalaron mass (none exists to force),
and we manufacture no obstruction: every massive DFD scalar was checked against
the slow-roll ceiling $m\le3.86\times10^{13}\,$GeV with explicit arithmetic
($\psi,\tau,\text{KK}\sim M_P$--$\sqrt\alpha M_P$; $\chi=5.09\,$eV cold relic), and
none qualifies, in agreement with Thm.~\ref{thm:no-native-inflaton}.

\emph{Residual, unchanged and stated openly.} This reframe resolves only the
scalaron-mass mislabel. It does \emph{not} resolve the genuine fitted item: the
$A_s$ coefficient $32\pi$ --- equivalently $\varepsilon_W=\alpha/(4\pi)$,
equivalently the unfixed ratio $H_\star/M_R=\sqrt{8\pi}$ --- remains
asserted, not derived (App.~\ref{app:new-results}, App.~\ref{app:fitted-knobs}).
With no inflaton potential there is no $V'/V$ from which to compute
$\varepsilon_W$, so $r$ and $A_s$ rest on the same single un-derived
normalization; that WdW-closure freedom \textbf{stands}. What the optical-response
reframe (Theorem~\ref{thm:optical-decoupling}) \emph{does} settle is the
\emph{origin}: $A_s=P_\psi^{\rm opt}$ and $r=P_{\rm shear}^{\rm opt}/P_\psi^{\rm opt}$
are optical-medium amplitudes, so the de~Sitter dictionary is retired and the
``inflation randomizes $\chi$'' objection to the dark relic does not apply
(Theorem~\ref{thm:chi-stateprep}); the coefficients $32\pi$ and $4\alpha/\pi$ remain
architecture-only (forced power, asserted coefficient). The present appendix
gains and loses no DFD win: it converts a mislabelled \textsc{open/fitted}
scalaron-mass tension into a correctly labelled non-question.
\end{status}

\begin{falsifier}\label{fals:scalaron-mass}
This appendix is falsified, and a scalaron mass becomes a real DFD observable, if
any of the following is established:
\begin{enumerate}[label=(\roman*)]
\item a curvature-scalar / $R^2$ (or otherwise higher-derivative) term is shown
to be \emph{required} by the DFD spectral action --- i.e. a non-zero $a_4$
gravitational coefficient distinct from the gauge/anomaly sector --- giving a
propagating Stelle scalar with a computable mass; or
\item a DFD scalar mode is exhibited with potential mass
$m^2\lesssim10^{-11}M_P^2$ and a flat ($V''/V\ll1$) direction, contradicting one
of the six failure modes of Thm.~\ref{thm:no-native-inflaton}; or
\item a first-principles derivation fixes $H_\star/M_R=\sqrt{8\pi}$ (equivalently
the $32\pi$ in $A_s$) \emph{and} that same construction supplies an independent
$R^2$ scalaron mass that lands on $H_\star$ --- which would make the two routes
coincide for a real field rather than by the present $\mathcal O(1)$ coincidence.
\end{enumerate}
Absent (i)--(iii), the scalaron mass remains undefined-because-absent.
\end{falsifier}
