%=============================================================================
% APPENDIX AP: CORPUS-CLOSURE THEOREMS (alpha^57 HIERARCHY, MOND SCALE, A+ RIGOR)
% Standalone theorem package adapted from
%   "Density Field Dynamics: Corpus-Closure Theorems for the alpha^57 Hierarchy,
%    the MOND Scale, and A+ Rigor Discipline" (G. Alcock, June 2026).
% Macros expanded to plain LaTeX; sections demoted; internal labels namespaced ap:*;
% status table converted to parbox cells (revtex twocolumn); bibliography dropped
% (no \cite in body). Master compile via DFD_Main.tex.
%=============================================================================

\section{Corpus-Closure Theorems: the $\alpha^{57}$ Hierarchy, the MOND Scale, and A+ Rigor Discipline}
\label{app:alpha57-closure}

\noindent\emph{This appendix is the consolidated corpus-closure theorem package for the
gravitational/MOND sector. It collects, with full proofs, the $\alpha^{57}$ cosmological
clock hierarchy, the action-derived MOND prefactor $a_0 = 2\sqrt\alpha\,cH_0$, and the
$H_0$-free A12 acceleration invariant $a_0^2 G\hbar/c^7 = 4\alpha^{58}$, and it fixes the
corpus-level rigor discipline (branch separation, conservative status taxonomy, retirement
of the minimal-branch black-hole temperature conflict, the A23 rotation-curve repair via the
nonlinear kinetic mechanism, and 2PN lensing branch labelling). The minimal real-kernel
$J=0$ statement below is consistent with, and cross-references, the CP-sector treatment of
Appendix~\ref{app:ckm-update}.}

This paper presents a standalone, integration-ready theorem package for the core closure chain of Density Field Dynamics (DFD).  The primary closed result is the acceleration invariant
\begin{equation*}
        {a_0^2G\hbar\over c^7}=4\alpha^{58},
        \qquad
        a_0=2\alpha^{29}\sqrt{c^7\over G\hbar},
\end{equation*}
which predicts the galactic MOND/RAR acceleration scale without a free continuous parameter.  The paper gives the full proof chain: a Spin$^c$/index mode count $60-3=57$, a primed Gaussian determinant producing $\alpha^{57}$, the finite spectral-action clock dictionary $G\hbar H_0^2/c^5=\alpha^{57}$, a gauge-emergence proof of $k_a=3/(8\alpha)$, an $S^3$ stationarity theorem selecting $\Xi_*=3/2$, and the action-level derivation $a_0=2\sqrt\alpha\,cH_0$.  The paper also imposes corpus-level rigor rules: contradictory branches are separated, unsupported claims are demoted, the black-hole temperature conflict is retired in the minimal branch, the A23 rotation-curve red flag is resolved by the nonlinear kinetic mechanism, and 2PN lensing claims are branch-labeled.  The result is a polished theorem-and-discipline package: the A12 acceleration invariant is proved as a closed theorem package, while the surrounding corpus is organized into explicit theorem, branch theorem, proposition, and program tiers so that no claim hides calibration or contradiction.





%=============================================================================
\subsection{Purpose and standard of proof}
%=============================================================================

The goal of this paper is to turn the current strongest DFD closure results into a standalone theorem package suitable for direct integration into the unified manuscript, with no hidden calibration, missing normalization, or contradictory branch mixing.

The central target is the A12 acceleration invariant
\begin{equation}\label{ap:eq:target-a12}
        \boxed{\Pi_2\equiv {a_0^2G\hbar\over c^7}=4\alpha^{58}}.
\end{equation}
This is equivalent to
\begin{equation}\label{ap:eq:target-a0}
        \boxed{a_0=2\alpha^{29}\sqrt{c^7\over G\hbar}}.
\end{equation}
The old risk was that Eq.~\eqref{ap:eq:target-a12} could be read as numerology: the exponent $58$ might appear to have been inferred from the observed acceleration scale.  The hardened proof below eliminates that by deriving the exponent as
\begin{equation}
        58=(60-3)+1.
\end{equation}
The $60-3=57$ comes from a finite microsector determinant; the extra $+1$ comes from the squared action-derived prefactor $2\sqrt\alpha$ in the scalar acceleration crossover.

\begin{definition}[A+ theorem-grade status]
A result is called \emph{A+ theorem grade} in this paper only if:
\begin{enumerate}
    \item every coefficient has a non-adjustable origin;
    \item every dimensionful identification is stated as either a theorem or a declared dictionary axiom;
    \item no observational value is used to choose an exponent or integer;
    \item branch-dependent calculations are not mixed;
    \item contradictory results are either resolved, retired, or demoted to archived consistency-check notes.
\end{enumerate}
\end{definition}

\begin{remark}[Hard scoping rule]
A corpus can be A+ without every speculative sector being a theorem.  What makes it A+ is that theorem-grade claims are proved and non-theorem claims are not mislabeled as theorems.
\end{remark}

%=============================================================================
\subsection{Notation and DFD primitives}
%=============================================================================

We use
\begin{align}
        c&=\text{reference two-way light speed},\\
        G&=\text{Newton constant},\\
        \hbar&=\text{reduced Planck constant},\\
        \alpha&=\text{fine-structure constant},\\
        H_0&=\text{cosmological clock scale},\\
        a_0&=\text{galactic acceleration crossover}.
\end{align}
The Planck time, length, and acceleration are
\begin{equation}\label{ap:eq:planck-defs}
        \fiteq{t_P=\sqrt{G\hbar\over c^5},\qquad
        \ell_P=\sqrt{G\hbar\over c^3},\qquad
        a_P=\sqrt{c^7\over G\hbar}={c\over t_P}={c^2\over \ell_P}.}
\end{equation}

The DFD scalar density field is denoted by $\psi$.  In the weak/static optical sector,
\begin{equation}\label{ap:eq:n-psi}
        n(x)=e^{\psi(x)},
\end{equation}
and the physical acceleration field is normalized as
\begin{equation}\label{ap:eq:g-psi}
        \mathbf g={c^2\over 2}\nabla\psi.
\end{equation}
This factor $1/2$ is not optional in the closure theorem: it is the conversion between scalar-gradient scale and physical acceleration.

%=============================================================================
\subsection{Corpus-level rigor axioms}
%=============================================================================

\begin{axiom}[Branch discipline]\label{ap:ax:branch}
A calculation performed in one DFD branch cannot be quoted as a theorem in another branch unless an embedding theorem identifies the two branches.  In particular, the minimal optical-exponential exterior, the PPN-closed physical metric, a nonminimal horizon-closure model, and a quantum-corrected vacuum action are distinct until proved equivalent.
\end{axiom}

\begin{axiom}[Status discipline]\label{ap:ax:no-overclaim}
The corpus assigns every claim exactly one of the following statuses:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Status & Meaning\\
\midrule
\parbox[t]{0.22\linewidth}{Theorem} & \parbox[t]{0.68\linewidth}{Derived from declared axioms/dictionary with no fitted continuous parameter.}\\[3pt]
\parbox[t]{0.22\linewidth}{Proposition} & \parbox[t]{0.68\linewidth}{Mathematically derived but dependent on a declared dictionary map.}\\[3pt]
\parbox[t]{0.22\linewidth}{Model prediction} & \parbox[t]{0.68\linewidth}{Computed inside one explicitly named model branch.}\\[3pt]
\parbox[t]{0.22\linewidth}{Phe\-nom\-e\-no\-log\-i\-cal closure} & \parbox[t]{0.68\linewidth}{Data-matching or effective description with stated correction budget.}\\[3pt]
\parbox[t]{0.22\linewidth}{Conjecture} & \parbox[t]{0.68\linewidth}{Structurally motivated but not proved.}\\[3pt]
\parbox[t]{0.22\linewidth}{Program item} & \parbox[t]{0.68\linewidth}{Calculation, simulation, or formal proof not yet completed.}\\[3pt]
\parbox[t]{0.22\linewidth}{Retired} & \parbox[t]{0.68\linewidth}{Contradictory, false, duplicated, or branch-incompatible.}\\
\bottomrule
\end{tabular}
\end{center}
No result may be theorem-grade if one of its coefficients is described as plausible, calibrated, motivated, or empirically selected.
\end{axiom}

\begin{axiom}[Finite spectral-action clock dictionary]\label{ap:ax:clock-dictionary}
The finite DFD spectral action identifies the primed internal Gaussian determinant ratio with the unique dimensionless cosmological clock hierarchy:
\begin{equation}\label{ap:eq:clock-dictionary}
        (H_0t_P)^2={G\hbar H_0^2\over c^5}={Z'_{\alpha}\over Z'_1}.
\end{equation}
This dictionary is the bridge between finite internal topology and the infrared de Sitter clock scale.  If this axiom is not adopted, the $\alpha^{57}$ statement must be downgraded from theorem to proposition.
\end{axiom}

\begin{remark}[What is derived vs.\ what the axiom supplies (2026-07-02)]\label{rem:clock-credit}
The derivation order is clean and non-circular: the exponent $57=60-3$ is pure Hirzebruch--Riemann--Roch arithmetic with no cosmological input, and $H_0$ first appears downstream as a-posteriori validation. Consequently the source law plus the Friedmann equation applied to the derived $E_0=\alpha^{57}M_P^4$ \emph{predicts} the infrared clock to within a factor $\sqrt{8\pi/3}\simeq2.9$ with zero tuning --- the $10^{-122.7}$ hierarchy is forced. What this axiom supplies is only the remaining $O(1)$ coefficient: the identification of $E_0$ with \emph{today's closure density at coefficient exactly one} ($(H_0t_P)^2=\alpha^{57}$), absorbing the $3/8\pi$ and the $\Omega_\Lambda$ reading (the $\sim12\%$ $\rho_\Lambda^{1/4}$ offset is booked separately). Rival conventions are excluded numerically, not by fiat: Friedmann-on-$E_0$ gives $H_0=208.7$, $\Lambda=\alpha^{57}/\ell_P^2$ gives $41.6$, and the $\Omega_\Lambda$-weighted reading gives $86.9$ km/s/Mpc. Bookkeeping note: Lemma~O.hierarchy states the ratio as a squared \emph{scale} ratio while T23 (Extended) states an energy \emph{density}; the two differ by exactly $8\pi/3$ and are reconciled by this axiom's convention --- the coefficient is counted once, here.
\end{remark}

\begin{axiom}[Scalar acceleration action dictionary]\label{ap:ax:scalar-dictionary}
The physical acceleration scale is extracted from the scalar action through Eq.~\eqref{ap:eq:g-psi}; the crossover $a_0$ is the point where the dimensionless acceleration invariant $\Xi$ reaches the stationary value selected by the $S^3$ microsector.  It is not inserted as a phenomenological fit parameter.
\end{axiom}

%=============================================================================
\subsection{The $\alpha^{57}$ hierarchy from topology and determinant scaling}
%=============================================================================

\subsubsection{Index count: $k_{\max}=60$}

\begin{lemma}[Hirzebruch--Riemann--Roch on $\mathbb{C}P^2$]\label{ap:lem:hrr-opn}
Let $H\in H^2(\mathbb{C}P^2,\mathbb Z)$ be the hyperplane class with $\int_{\mathbb{C}P^2}H^2=1$.  For the holomorphic line bundle $\mathcal O(n)$,
\begin{equation}\label{ap:eq:chi-On}
        \chi(\mathbb{C}P^2,\mathcal O(n))={ (n+1)(n+2)\over 2}.
\end{equation}
\end{lemma}

\begin{proof}
Hirzebruch--Riemann--Roch gives
\begin{equation}
        \chi(\mathbb{C}P^2,\mathcal O(n))=\int_{\mathbb{C}P^2}\operatorname{ch}(\mathcal O(n))\operatorname{Td}(T\mathbb{C}P^2).
\end{equation}
For $\mathbb{C}P^2$,
\begin{equation}
        \operatorname{ch}(\mathcal O(n))=e^{nH}=1+nH+{n^2\over2}H^2,
\end{equation}
while
\begin{equation}
        \operatorname{Td}(T\mathbb{C}P^2)=1+{3\over2}H+H^2.
\end{equation}
The integral extracts the coefficient of $H^2$:
\begin{equation}
        {n^2\over2}+{3n\over2}+1={n^2+3n+2\over2}={(n+1)(n+2)\over2}.
\end{equation}
\end{proof}

\begin{theorem}[Finite microsector mode count]\label{ap:thm:mode-count}
For the DFD internal finite sector
\begin{equation}\label{ap:eq:E-bundle}
        E=\mathcal O(9)\oplus\mathcal O^{\oplus5}
\end{equation}
over $\mathbb{C}P^2$, the total mode count is
\begin{equation}\label{ap:eq:kmax-60}
        k_{\max}=60.
\end{equation}
\end{theorem}

\begin{proof}
Using Lemma~\ref{ap:lem:hrr-opn},
\begin{equation}
        \chi(\mathbb{C}P^2,\mathcal O(9))={(10)(11)\over2}=55.
\end{equation}
Also $\chi(\mathbb{C}P^2,\mathcal O)=1$.  Therefore
\begin{equation}
        \chi(\mathbb{C}P^2,E)=55+5\chi(\mathbb{C}P^2,\mathcal O)=55+5=60.
\end{equation}
\end{proof}

\begin{axiom}[Generation kernel]\label{ap:ax:generation-kernel}
The protected generation kernel of the finite DFD microsector has dimension
\begin{equation}
        N_{\rm gen}=3.
\end{equation}
These three zero modes are not included in the primed determinant.
\end{axiom}

\begin{corollary}[Nonzero determinant count]\label{ap:cor:57}
The primed determinant contains exactly
\begin{equation}\label{ap:eq:57-count}
        N_{\rm nz}=k_{\max}-N_{\rm gen}=60-3=57
\end{equation}
nonzero complex modes.
\end{corollary}

\subsubsection{Primed determinant scaling}

\begin{lemma}[One complex Gaussian mode]\label{ap:lem:complex-gaussian}
For $\lambda>0$,
\begin{equation}
        \int_{\mathbb{C}}d^2z\,\exp\left[-{\lambda\over\alpha}|z|^2\right]={\pi\alpha\over\lambda}.
\end{equation}
Consequently, relative to the $\alpha=1$ normalization, one nonzero complex mode contributes a factor $\alpha$.
\end{lemma}

\begin{proof}
Writing $z=x+iy$ gives
\begin{equation}
        \int_{\mathbb{C}}d^2z\,e^{-(\lambda/\alpha)|z|^2}
        =\left(\int_{-\infty}^{\infty}e^{-(\lambda/\alpha)x^2}\,dx\right)^2
        ={\pi\alpha\over\lambda}.
\end{equation}
At $\alpha=1$ the same integral is $\pi/\lambda$, so the ratio is $\alpha$.
\end{proof}

\begin{theorem}[Primed determinant hierarchy]\label{ap:thm:det-alpha57}
Let the nonzero internal kinetic operator have eigenvalues $\lambda_i>0$ for $i=1,\ldots,57$, and let the gauge-normalized quadratic action be
\begin{equation}
        S_{\rm micro}=\sum_{i=1}^{57}{\lambda_i\over\alpha}|z_i|^2.
\end{equation}
Then
\begin{equation}\label{ap:eq:z-alpha57}
        {Z'_{\alpha}\over Z'_1}=\alpha^{57}.
\end{equation}
Equivalently,
\begin{equation}\label{ap:eq:det-alpha57}
        {\det'\mathcal K\over \det'(\mathcal K/\alpha)}=\alpha^{57}.
\end{equation}
\end{theorem}

\begin{proof}
By Lemma~\ref{ap:lem:complex-gaussian}, each nonzero complex mode contributes $\alpha$ to the normalized partition ratio.  There are $57$ such modes by Corollary~\ref{ap:cor:57}.  Hence
\begin{equation}
        {Z'_{\alpha}\over Z'_1}=\prod_{i=1}^{57}\alpha=\alpha^{57}.
\end{equation}
The determinant statement follows because
\begin{equation}
        \det'(\mathcal K/\alpha)=\prod_{i=1}^{57}{\lambda_i\over\alpha}=\alpha^{-57}\det'\mathcal K.
\end{equation}
Thus $\det'\mathcal K/\det'(\mathcal K/\alpha)=\alpha^{57}$.
\end{proof}

\subsubsection{The cosmological invariant}

\begin{lemma}[Uniqueness of the $H_0$ dimensionless group]\label{ap:lem:H0-unique}
The unique primitive dimensionless monomial formed from $G,\hbar,c,H_0$ is
\begin{equation}\label{ap:eq:IH}
        I_H={G\hbar H_0^2\over c^5}=(H_0t_P)^2.
\end{equation}
\end{lemma}

\begin{proof}
Let $G^a\hbar^b c^d H_0^e$ be dimensionless.  With dimensions
\begin{equation}
        \fiteq{[G]=L^3M^{-1}T^{-2},\quad [\hbar]=ML^2T^{-1},\quad [c]=LT^{-1},\quad [H_0]=T^{-1},}
\end{equation}
the mass exponent gives $-a+b=0$, so $b=a$.  The length exponent gives $3a+2b+d=0$, hence $5a+d=0$.  The time exponent gives $-2a-b-d-e=0$, hence $-3a-d-e=0$.  With $d=-5a$, this gives $e=2a$.  Taking the primitive choice $a=1$ gives
\begin{equation}
        G\hbar H_0^2c^{-5}.
\end{equation}
\end{proof}

\begin{theorem}[$\alpha^{57}$ master invariant]\label{ap:thm:alpha57-master}
Under the finite spectral-action clock dictionary, the DFD cosmological hierarchy is
\begin{equation}\label{ap:eq:alpha57-master}
        \boxed{{G\hbar H_0^2\over c^5}=\alpha^{57}}.
\end{equation}
\end{theorem}

\begin{proof}
By Axiom~\ref{ap:ax:clock-dictionary},
\begin{equation}
        {G\hbar H_0^2\over c^5}={Z'_{\alpha}\over Z'_1}.
\end{equation}
By Theorem~\ref{ap:thm:det-alpha57}, the right side is $\alpha^{57}$.  Therefore Eq.~\eqref{ap:eq:alpha57-master} follows.
\end{proof}

\begin{remark}[What is theorem-grade and what is dictionary]
The determinant scaling $Z'_{\alpha}/Z'_1=\alpha^{57}$ is a theorem once $E=\mathcal O(9)\oplus\mathcal O^{\oplus5}$ and $N_{\rm gen}=3$ are declared.  The identification of this determinant ratio with $(H_0t_P)^2$ is a DFD spectral-action dictionary axiom.  The unified paper must state this explicitly.  Hiding this map would demote the result.
\end{remark}

%=============================================================================
\subsection{The action-derived MOND prefactor}
%=============================================================================

The second parent result needed for A12 is
\begin{equation}\label{ap:eq:mond-prefactor-target}
        a_0=2\sqrt\alpha\,cH_0.
\end{equation}
This section derives it from the scalar action and internal gauge normalization.

\subsubsection{Gauge-emergence proof of $k_a=3/(8\alpha)$}

\begin{lemma}[Magnetic dual coupling]\label{ap:lem:magnetic-dual}
In rationalized units with
\begin{equation}
        \alpha={e^2\over4\pi},
\end{equation}
and minimal Dirac quantization
\begin{equation}
        eg=2\pi,
\end{equation}
the magnetic fine-structure constant is
\begin{equation}\label{ap:eq:alphaM}
        \alpha_M={g^2\over4\pi}={1\over4\alpha}.
\end{equation}
\end{lemma}

\begin{proof}
From $eg=2\pi$,
\begin{equation}
        g={2\pi\over e}.
\end{equation}
Therefore
\begin{equation}
        \alpha_M={1\over4\pi}\left({2\pi\over e}\right)^2={\pi\over e^2}.
\end{equation}
Since $e^2=4\pi\alpha$,
\begin{equation}
        \alpha_M={\pi\over4\pi\alpha}={1\over4\alpha}.
\end{equation}
\end{proof}

\begin{axiom}[Internal gauge partition]\label{ap:ax:gauge-partition}
The DFD internal frame partition relevant to the scalar acceleration channel has backbone and doorway dimensions
\begin{equation}
        n_3=3,
        \qquad
        n_2=2.
\end{equation}
The frame-stiffness ratio entering the scalar backreaction channel is $n_3/n_2$.
\end{axiom}

\begin{theorem}[Acceleration-channel self-coupling]\label{ap:thm:ka}
The DFD scalar acceleration-channel coupling is
\begin{equation}\label{ap:eq:ka-value}
        \boxed{k_a={3\over8\alpha}}.
\end{equation}
\end{theorem}

\begin{proof}
By Axiom~\ref{ap:ax:gauge-partition}, the frame factor is
\begin{equation}
        {n_3\over n_2}={3\over2}.
\end{equation}
By Lemma~\ref{ap:lem:magnetic-dual}, the magnetic dual coupling is $1/(4\alpha)$.  The acceleration-channel coupling is the product
\begin{equation}
        k_a={n_3\over n_2}\alpha_M={3\over2}\cdot{1\over4\alpha}={3\over8\alpha}.
\end{equation}
\end{proof}

\begin{corollary}[Generation-normalized loop coefficient]\label{ap:cor:loop-eighth}
If the same coupling is written in the generation-normalized form
\begin{equation}
        k_a=N_{\rm gen}C_{\rm loop}\alpha^{-1},
\end{equation}
with $N_{\rm gen}=3$, then
\begin{equation}
        \boxed{C_{\rm loop}={1\over8}}.
\end{equation}
\end{corollary}

\begin{proof}
Set $3C_{\rm loop}\alpha^{-1}=3/(8\alpha)$.  Cancel $3\alpha^{-1}$.
\end{proof}

\begin{remark}[Why this replaces the old heat-kernel handwave]
The coefficient $1/8$ is not used as an unexplained heat-kernel guess.  It is the generation-normalized form of the independent product
\begin{equation}
        {n_3\over n_2}\cdot{1\over4}={3\over2}\cdot{1\over4}={3\over8}.
\end{equation}
This is stronger because every factor has a named origin.
\end{remark}

\subsubsection{The $S^3$ stationarity theorem}

\begin{lemma}[$S^3$ scaling charge]\label{ap:lem:S3-charge}
For the $SU(2)$ Chern--Simons partition function on $S^3$,
\begin{equation}
        Z_{S^3}(k)=\sqrt{2\over k+2}\,\sin\left({\pi\over k+2}\right),
\end{equation}
the large-level scaling charge is
\begin{equation}\label{ap:eq:S3-charge}
        q_{S^3}= -{\partial\log Z_{S^3}\over \partial\log(k+2)}={3\over2}+\mathcal{O}(k^{-2}).
\end{equation}
The topological scaling exponent is therefore $3/2$.
\end{lemma}

\begin{proof}
For large $k$,
\begin{equation}
        \sin\left({\pi\over k+2}\right)={\pi\over k+2}+\mathcal{O}((k+2)^{-3}).
\end{equation}
Hence
\begin{equation}
        \fiteq{\begin{aligned}
        Z_{S^3}(k)&=\sqrt{2\over k+2}\left({\pi\over k+2}+\mathcal{O}((k+2)^{-3})\right)\\
        &=\sqrt2\pi\,(k+2)^{-3/2}\left(1+\mathcal{O}((k+2)^{-2})\right).
        \end{aligned}}
\end{equation}
Thus
\begin{equation}
        \log Z_{S^3}=\log(\sqrt2\pi)-{3\over2}\log(k+2)+\mathcal{O}((k+2)^{-2}),
\end{equation}
so $-\partial\log Z_{S^3}/\partial\log(k+2)=3/2+\mathcal{O}(k^{-2})$.
\end{proof}

\begin{definition}[Dimensionless acceleration invariant]\label{ap:def:Xi}
The scalar action uses the dimensionless invariant
\begin{equation}\label{ap:eq:Xi-def}
        \Xi=k_a\left({|a|\over cH_0}\right)^2.
\end{equation}
\end{definition}

\begin{theorem}[$S^3$ stationary crossover]\label{ap:thm:Xi-stationary}
The local topological potential
\begin{equation}\label{ap:eq:U-Xi}
        U(\Xi)=\Xi-{3\over2}\log\Xi
\end{equation}
selects the crossover value
\begin{equation}\label{ap:eq:Xi-star}
        \boxed{\Xi_*={3\over2}}.
\end{equation}
\end{theorem}

\begin{proof}
Stationarity gives
\begin{equation}
        {dU\over d\Xi}=1-{3\over2\Xi}=0.
\end{equation}
Solving gives $\Xi=3/2$.
\end{proof}

\subsubsection{Action-level derivation of the MOND prefactor}

\begin{theorem}[MOND prefactor from scalar action]\label{ap:thm:a0-prefactor}
With $k_a=3/(8\alpha)$ and $\Xi_*=3/2$, the scalar acceleration crossover is
\begin{equation}\label{ap:eq:a0-action-derived}
        \boxed{a_0=2\sqrt\alpha\,cH_0}.
\end{equation}
\end{theorem}

\begin{proof}
At the crossover $|a|=a_0$, so Definition~\ref{ap:def:Xi} and Theorem~\ref{ap:thm:Xi-stationary} give
\begin{equation}
        k_a\left({a_0\over cH_0}\right)^2={3\over2}.
\end{equation}
Using Theorem~\ref{ap:thm:ka},
\begin{equation}
        {3\over8\alpha}\left({a_0\over cH_0}\right)^2={3\over2}.
\end{equation}
Cancel $3$ and solve:
\begin{equation}
        \left({a_0\over cH_0}\right)^2=4\alpha.
\end{equation}
Taking the positive square root gives Eq.~\eqref{ap:eq:a0-action-derived}.
\end{proof}

%=============================================================================
\subsection{The nonlinear kinetic action and galaxy law}
%=============================================================================

The correct galaxy mechanism is not a direct source term $k_a a^2/c^2$.  The acceleration scale $a_0$ controls the nonlinear kinetic branch of the scalar action.

\begin{definition}[Static scalar kinetic energy]\label{ap:def:scalar-energy}
Let
\begin{equation}\label{ap:eq:Y-def}
        Y={|\nabla\psi|^2\over \kappa_\alpha^2},
        \qquad
        \kappa_\alpha={2a_0\over c^2}.
\end{equation}
Define
\begin{equation}\label{ap:eq:scalar-energy}
        E_\psi[\psi]
        ={c^4\over32\pi G}\int \kappa_\alpha^2\mathcal F(Y)\,d^3x
        +{c^2\over2}\int \rho\psi\,d^3x.
\end{equation}
The interpolation function obeys
\begin{equation}\label{ap:eq:F-limits}
        \mathcal F(Y)\sim Y\quad(Y\gg1),
        \qquad
        \mathcal F(Y)\sim {2\over3}Y^{3/2}\quad(Y\ll1).
\end{equation}
\end{definition}

\begin{theorem}[Euler--Lagrange equation]\label{ap:thm:EL-aqual}
The scalar energy \eqref{ap:eq:scalar-energy} yields
\begin{equation}\label{ap:eq:psi-aqual}
        \nabla\cdot\left[\mathcal F'(Y)\nabla\psi\right]={8\pi G\over c^2}\rho.
\end{equation}
Equivalently, in terms of $\mathbf g=(c^2/2)\nabla\psi$,
\begin{equation}\label{ap:eq:g-aqual}
        \boxed{\nabla\cdot\left[\mu\left({|\mathbf g|\over a_0}\right)\mathbf g\right]=4\pi G\rho},
\end{equation}
where
\begin{equation}
        \mu(s)=\mathcal F'(s^2),\qquad s={|\mathbf g|\over a_0}.
\end{equation}
\end{theorem}

\begin{proof}
Vary Eq.~\eqref{ap:eq:scalar-energy}.  Since
\begin{equation}
        \delta Y={2\nabla\psi\cdot\nabla\delta\psi\over\kappa_\alpha^2},
\end{equation}
we have
\begin{align}
        \delta E_\psi
        &= {c^4\over32\pi G}\int \kappa_\alpha^2\mathcal F'(Y){2\nabla\psi\cdot\nabla\delta\psi\over\kappa_\alpha^2}\,d^3x
        +{c^2\over2}\int \rho\delta\psi\,d^3x \\
        &= {c^4\over16\pi G}\int \mathcal F'(Y)\nabla\psi\cdot\nabla\delta\psi\,d^3x
        +{c^2\over2}\int \rho\delta\psi\,d^3x.
\end{align}
Integrating by parts and discarding boundary variations,
\begin{equation}
        \delta E_\psi
        =\int\left[-{c^4\over16\pi G}\nabla\cdot(\mathcal F'(Y)\nabla\psi)+{c^2\over2}\rho\right]\delta\psi\,d^3x.
\end{equation}
Stationarity for arbitrary $\delta\psi$ gives Eq.~\eqref{ap:eq:psi-aqual}.  Multiplying by $c^2/2$ inside the divergence converts $\nabla\psi$ to $\mathbf g$ and gives Eq.~\eqref{ap:eq:g-aqual}.
\end{proof}

\begin{corollary}[Newtonian and deep branches]\label{ap:cor:branches}
The limits \eqref{ap:eq:F-limits} imply
\begin{equation}
        \mu(s)\to1\quad(s\gg1),
        \qquad
        \mu(s)\to s\quad(s\ll1).
\end{equation}
\end{corollary}

\begin{proof}
If $Y=s^2\gg1$, $\mathcal F(Y)\sim Y$, so $\mathcal F'(Y)\to1$.  If $Y=s^2\ll1$, $\mathcal F(Y)\sim(2/3)Y^{3/2}$, so $\mathcal F'(Y)\sim Y^{1/2}=s$.
\end{proof}

\begin{theorem}[Baryonic Tully--Fisher relation from the kinetic branch]\label{ap:thm:btf}
For a spherical source in the deep branch, Eq.~\eqref{ap:eq:g-aqual} implies
\begin{equation}\label{ap:eq:btf}
        \boxed{v^4=GMa_0}.
\end{equation}
\end{theorem}

\begin{proof}
In spherical symmetry outside the source, Eq.~\eqref{ap:eq:g-aqual} integrates to
\begin{equation}
        \mu(g/a_0)g=g_N={GM\over r^2}.
\end{equation}
In the deep branch $\mu(g/a_0)=g/a_0$, hence
\begin{equation}
        {g^2\over a_0}=g_N={GM\over r^2}.
\end{equation}
For circular motion $g=v^2/r$.  Therefore
\begin{equation}
        {v^4\over r^2}=a_0{GM\over r^2},
\end{equation}
so $v^4=GMa_0$.
\end{proof}

\begin{corollary}[Resolution of the A23 red flag]\label{ap:cor:A23-fix}
If a term $k_a a^2/c^2$ is too small to act as a direct galactic mass source, that does not invalidate the MOND mechanism.  In the A+ corpus, $k_a(a/cH_0)^2$ is the crossover-control invariant $\Xi$ in the scalar action, while galaxy rotation curves arise from the nonlinear kinetic equation \eqref{ap:eq:g-aqual}.
\end{corollary}

%=============================================================================
\subsection{Buckingham--$\pi$ closure and the $\alpha^{58}$ acceleration invariant}
%=============================================================================

\begin{lemma}[Two independent groups]\label{ap:lem:buckingham}
For the dimensional set $\{a_0,G,\hbar,c,H_0\}$, there are exactly two independent dimensionless groups.  A convenient basis is
\begin{equation}\label{ap:eq:Pi12}
        \Pi_1={G\hbar H_0^2\over c^5},
        \qquad
        \Pi_2={a_0^2G\hbar\over c^7}.
\end{equation}
\end{lemma}

\begin{proof}
The dimension matrix in the $(M,L,T)$ basis is
\begin{equation}
D=\begin{array}{c|ccccc}
 & a_0 & G & \hbar & c & H_0\\\hline
M & 0 & -1 & 1 & 0 & 0\\
L & 1 & 3 & 2 & 1 & 0\\
T & -2& -2& -1& -1& -1
\end{array}.
\end{equation}
The rank is $3$.  Since there are five dimensional variables, Buckingham's theorem gives $5-3=2$ independent dimensionless groups.  Direct substitution verifies both groups in Eq.~\eqref{ap:eq:Pi12} are dimensionless.  Independence follows because $\Pi_1$ contains $H_0$ but not $a_0$, while $\Pi_2$ contains $a_0$ but not $H_0$.
\end{proof}

\begin{theorem}[A12 acceleration invariant]\label{ap:thm:A12}
The DFD acceleration invariant is
\begin{equation}\label{ap:eq:A12-final}
        \boxed{{a_0^2G\hbar\over c^7}=4\alpha^{58}},
\end{equation}
and hence
\begin{equation}\label{ap:eq:a0-final}
        \boxed{a_0=2\alpha^{29}\sqrt{c^7\over G\hbar}}.
\end{equation}
\end{theorem}

\begin{proof}
By Theorem~\ref{ap:thm:alpha57-master},
\begin{equation}
        {G\hbar H_0^2\over c^5}=\alpha^{57}.
\end{equation}
By Theorem~\ref{ap:thm:a0-prefactor},
\begin{equation}
        a_0=2\sqrt\alpha\,cH_0.
\end{equation}
Squaring,
\begin{equation}
        a_0^2=4\alpha c^2H_0^2.
\end{equation}
Then
\begin{equation}
        {a_0^2G\hbar\over c^7}
        ={4\alpha c^2H_0^2G\hbar\over c^7}
        =4\alpha{G\hbar H_0^2\over c^5}
        =4\alpha\alpha^{57}=4\alpha^{58}.
\end{equation}
Solving for $a_0$ gives Eq.~\eqref{ap:eq:a0-final}.
\end{proof}

\begin{remark}[Why the exponent is not fitted]
The exponent is
\begin{equation}
        58=57+1=(60-3)+1.
\end{equation}
The $57$ is the primed determinant count.  The additional $+1$ is the single power of $\alpha$ from squaring $2\sqrt\alpha$.  No galaxy acceleration value enters this count.
\end{remark}

\subsubsection{Numerical prediction and falsifiability}

Using CODATA constants, Eq.~\eqref{ap:eq:a0-final} evaluates to
\begin{equation}\label{ap:eq:a0-number}
        a_0=1.19662\times10^{-10}\,\mathrm{m\,s^{-2}}
\end{equation}
up to the uncertainty dominated by $G$.  The theorem is falsifiable because $a_0$ is fixed by laboratory constants and $\alpha$, not by fitting galaxy rotation curves.  If future RAR determinations converge outside the predicted value beyond their systematic error budget, at least one of the parent inputs--the spectral clock dictionary, the scalar prefactor theorem, or the DFD interpretation of $a_0$--fails.

%=============================================================================
\subsection{Black-hole sector: retirement of the triple-temperature contradiction}
%=============================================================================

The current corpus contained mutually incompatible base-temperature claims.  A+ status requires one branch rule.

\begin{definition}[Minimal optical-exponential exterior]\label{ap:def:minimal-exterior}
The minimal DFD exterior around a spherical mass $M$ is the optical branch
\begin{equation}\label{ap:eq:n-exterior}
        n(r)=\exp\left({2GM\over c^2r}\right),
        \qquad
        \psi(r)={2GM\over c^2r}.
\end{equation}
The local coordinate phase speed is $c_{\rm ph}(r)=c/n(r)$.
\end{definition}

\begin{theorem}[No finite-radius horizon in the minimal branch]\label{ap:thm:no-minimal-horizon}
In the minimal optical-exponential exterior, there is no finite-radius optical horizon and therefore no tree-level finite-radius Hawking temperature.
\end{theorem}

\begin{proof}
A finite-radius optical horizon would require $c_{\rm ph}(r_H)=0$, equivalently $n(r_H)=\infty$ or $\psi(r_H)=+\infty$, at some $r_H>0$.  But Eq.~\eqref{ap:eq:n-exterior} is finite for every $r>0$.  It diverges only as $r\to0$.  Hence there is no finite $r_H$ in the minimal optical branch.  Without a finite horizon and associated Euclidean regularity condition, no finite-radius tree-level Hawking temperature is defined.
\end{proof}

\begin{corollary}[Temperature-claim triage]\label{ap:cor:BH-triage}
Claims such as
\begin{equation}
        \fiteq{T_H=e^{-1/2}T_H^{\rm GR},\qquad
        T_H=T_H^{\rm GR}(1+\alpha/2),\qquad
        T_H=T_H^{\rm GR}(1+\delta_{\rm Casimir})}
\end{equation}
may not coexist as theorem-grade base temperatures.  In the minimal branch they are retired as base-temperature theorems.  They may reappear only as conditional statements inside a named nonminimal horizon-closure model.
\end{corollary}

\begin{remark}[Nonminimal horizon closure]
If a nonminimal DFD branch introduces a physical metric or compact-object boundary condition with a true horizon $\mathcal H$, its base temperature must be computed from the same branch's surface gravity:
\begin{equation}
        T_H^{\mathcal H}={\hbar\kappa_{\mathcal H}\over2\pi k_B}.
\end{equation}
Local Unruh or Casimir corrections must be written as corrections to that branch value, not as separate base temperatures.
\end{remark}

%=============================================================================
\subsection{Lensing and PPN branch discipline}
%=============================================================================

\begin{definition}[Lensing branches]\label{ap:def:lensing-branches}
The corpus distinguishes:
\begin{enumerate}
    \item the \emph{PPN-closed physical metric branch}, used for solar-system parameter comparisons;
    \item the \emph{pure optical-exponential ray branch}, using Fermat propagation through $n=e^\psi$ without all PPN closure terms.
\end{enumerate}
\end{definition}

\begin{theorem}[Allowed lensing theorem statement]\label{ap:thm:lensing-statement}
The A+ corpus may state as a theorem that DFD reproduces the leading 1PN light-deflection term only if that result is computed in the declared physical branch.  Any 2PN coefficient is theorem-grade only after being computed from the same branch action/metric used for the PPN table.
\end{theorem}

\begin{proof}
By Axiom~\ref{ap:ax:branch}, coefficients computed in different branches cannot be co-theorems of one corpus.  A pure optical-exponential 2PN coefficient and a PPN-closed 2PN coefficient answer different mathematical questions unless an embedding theorem identifies the rays and affine structures order by order.  Therefore a 2PN anomaly can remain as a model prediction in the pure optical branch, while the PPN-closed branch may retain exact 2PN agreement only if it is derived from the same declared metric.  Keeping both as theorem-grade violates Axiom~\ref{ap:ax:branch}.
\end{proof}

\begin{corollary}[Required wording]
Replace ``DFD matches GR lensing exactly'' with: DFD reproduces the leading GR deflection in the declared physical branch.  At 2PN and beyond, the coefficient is branch-dependent until computed from the full PPN-closed action.
\end{corollary}

%=============================================================================
\subsection{Flavor/CKM sector: exact theorem boundary}
%=============================================================================

\begin{proposition}[Minimal real-kernel CP null]\label{ap:prop:CKM-null}
If the minimal DFD flavor sector produces only real Yukawa kernels and no additional complex orientation/discrete shape-duality datum, then the CKM matrix can be chosen real and the Jarlskog invariant vanishes:
\begin{equation}
        J=0.
\end{equation}
\end{proposition}

\begin{proof}
If the up- and down-sector mass matrices are real, their diagonalizing transformations may be chosen orthogonal in the absence of an intrinsic complex phase.  Then
\begin{equation}
        V_{\rm CKM}=O_u^TO_d
\end{equation}
is real orthogonal.  The Jarlskog invariant is
\begin{equation}
        J=\operatorname{Im}(V_{ij}V_{kl}V_{il}^*V_{kj}^*)
\end{equation}
for any distinct $i,k$ and $j,l$.  If all entries of $V$ are real, the product is real and its imaginary part is zero.
\end{proof}

\begin{theorem}[Allowed CKM claim]\label{ap:thm:CKM-allowed}
The minimal DFD corpus may claim theorem-grade magnitude relations only to the extent that their projection/operator dictionary is fixed without fitting.  CP violation and nonzero $J$ are theorem-grade only in an explicitly extended DFD--SD branch containing the required complex/discrete orientation datum.
\end{theorem}

\begin{proof}
The null result in Proposition~\ref{ap:prop:CKM-null} shows that minimal real kernels cannot generate CKM CP violation.  Therefore any nonzero $J$ requires an additional datum or branch.  If that datum is declared as DFD--SD, the CP result belongs to that branch.  Quoting it as a minimal DFD theorem would contradict Proposition~\ref{ap:prop:CKM-null}.
\end{proof}

%=============================================================================
\subsection{Inflation and early-universe sector}
%=============================================================================

\begin{theorem}[No mixed inflation claim]\label{ap:thm:no-mixed-inflation}
The scalar $\psi$ sector and any separate topological/phase inflation branch must be kept distinct.  If $\psi$-inflation produces excluded or disfavored $(n_s,r)$, then the corpus may not simultaneously claim that $\psi$ is the successful inflaton and that inflation is unnecessary.
\end{theorem}

\begin{proof}
The statements answer different model questions.  A no-go theorem for $\psi$ as a slow-roll inflaton constrains the scalar density field.  A separate topological phase-inflation mechanism, if present, is a different branch with a different action.  By Axiom~\ref{ap:ax:branch}, success or failure in one cannot be transferred to the other without an embedding theorem.
\end{proof}

\begin{corollary}[Required wording]
The A+ corpus statement is: the DFD scalar $\psi$ is not the conventional slow-roll inflaton in the disfavored branch.  Any viable early-universe smoothing mechanism must be stated as a separate branch with its own predictions.
\end{corollary}

%=============================================================================
\subsection{PDE/well-posedness sector}
%=============================================================================

This paper does not reproduce the full PDE proof from Wave A11.  It records the theorem boundary needed for integration.

\begin{definition}[Critical Sobolev threshold]
For a first-order scalar evolution whose principal nonlinear structure contains one spatial derivative of $\psi$ in three spatial dimensions, the scaling-critical Sobolev index takes the form
\begin{equation}
        s_c=1+{3\over2}={5\over2}.
\end{equation}
\end{definition}

\begin{proposition}[Allowed PDE status]\label{ap:prop:PDE-status}
A statement that the DFD $\psi$ equation is locally well-posed above $s>5/2$ and controlled by a Beale--Kato--Majda type gradient criterion may be theorem-grade only if the unified paper includes the exact PDE, energy estimates, commutator bounds, and continuation proof.  Otherwise it is a proposition classified as a program item unless the complete A11 estimates are included.
\end{proposition}

\begin{proof}
The number $5/2$ follows by scaling, but local well-posedness requires more than scaling: a closed energy inequality, estimates for the nonlinear operator, and a continuation criterion.  Without the exact PDE and estimates, the statement is not reproducible from the corpus text alone.  Therefore the PDE claim is classified here as a program item unless the complete A11 proof is included in the same manuscript.
\end{proof}

%=============================================================================
\subsection{Rigor ledger for the closure results}
%=============================================================================

The table below gives integration status under the A+ discipline.  It is not a scientific verdict on truth; it is an editorial theorem-status ledger.


\begin{description}[leftmargin=1.2cm,style=nextline]
    \item[A12: $a_0^2G\hbar/c^7=4\alpha^{58}$.] \textbf{Status: theorem package.}  Use Sections 4--7 of this paper as the theorem-grade closure block.
    \item[A02/A26: $\alpha$-lock geometry and formal proof.] \textbf{Status: theorem/proposition tier.}  The unified manuscript must include the full heat-kernel and index proof wherever it cites this sector as theorem-grade; normalizations must be explicit.
    \item[A11: $\psi$ PDE well-posedness.] \textbf{Status: program/proposition tier in this paper.}  The exact PDE, commutator estimates, energy inequality, and continuation proof are required in any manuscript that promotes this claim to theorem-grade.
    \item[A16: 2PN lensing anomaly.] \textbf{Status: branch model prediction.}  Keep only with an explicit pure-optical branch label unless embedded into the PPN-closed branch.
    \item[A23: rotation-curve mechanism.] \textbf{Status: repaired theorem.}  Replace direct-source language with the nonlinear kinetic-branch mechanism.
    \item[A07/A06/AH3a: black-hole temperature.] \textbf{Status: retired in the minimal branch.}  No base Hawking temperature is assigned without a named nonminimal horizon closure.
    \item[A08: $\psi$-inflation.] \textbf{Status: no-go/model constraint.}  State that $\psi$ is not a successful conventional slow-roll inflaton in the disfavored branch.
    \item[CKM flavor/CP closure.] \textbf{Status: conditional theorem in DFD--SD.}  Minimal DFD gives $J=0$ unless the shape-duality datum is added.
    \item[A14: PTA variable-$c$ signal.] \textbf{Status: constraint/model prediction.}  Requires a screening mechanism before promotion.
    \item[A28: Sakharov induced $G$.] \textbf{Status: archived consistency check.}  Do not use as the main $G$ derivation if the overshoot remains.
    \item[A03/A19/A21: tabletop tests.] \textbf{Status: model predictions.}  Include only with experimental protocols and branch assumptions.
    \item[A05/A09/A15/A24/A30: GW/photon/CMB/running effects.] \textbf{Status: model predictions.}  Keep as falsifiers with the branch and action stated.
    \item[A13: hidden Noether/Tully--Fisher.] \textbf{Status: theorem candidate.}  Ensure it reduces to the kinetic proof and does not duplicate A12.
\end{description}


%=============================================================================
\subsection{Standalone theorem summary for unification}
%=============================================================================

For direct incorporation into the unified paper, the minimal statement is:

\begin{quote}
The finite DFD microsector has Spin$^c$ index $60$ and a protected three-dimensional generation kernel.  The primed determinant therefore contains $57$ nonzero complex modes.  Gauge-normalized Gaussian integration gives one factor of $\alpha$ per nonzero mode, so the finite spectral-action clock dictionary gives
\begin{equation*}
        {G\hbar H_0^2\over c^5}=\alpha^{57}.
\end{equation*}
The scalar acceleration-channel coupling is fixed by the internal frame ratio and magnetic dual coupling,
\begin{equation*}
        k_a={n_3\over n_2}\alpha_M={3\over2}{1\over4\alpha}={3\over8\alpha}.
\end{equation*}
The $S^3$ scaling potential $U(\Xi)=\Xi-(3/2)\log\Xi$ selects $\Xi_*=3/2$ for
\begin{equation*}
        \Xi=k_a(a/cH_0)^2.
\end{equation*}
Thus
\begin{equation*}
        a_0=2\sqrt\alpha\,cH_0.
\end{equation*}
Combining the two gives the $H_0$-free acceleration invariant
\begin{equation*}
        {a_0^2G\hbar\over c^7}=4\alpha^{58},
        \qquad
        a_0=2\alpha^{29}\sqrt{c^7\over G\hbar}.
\end{equation*}
The galaxy law follows from the nonlinear kinetic equation, not from a direct $k_aa^2/c^2$ source term.
\end{quote}

%=============================================================================

\subsection{Dimension checks}
%=============================================================================

\begin{proposition}[Dimensionlessness of $\Pi_2$]
\begin{equation}
        \Pi_2={a_0^2G\hbar\over c^7}
\end{equation}
is dimensionless.
\end{proposition}

\begin{proof}
Using $[a_0]=LT^{-2}$, $[G]=L^3M^{-1}T^{-2}$, $[\hbar]=ML^2T^{-1}$, and $[c]=LT^{-1}$,
\begin{equation}
        [\Pi_2]=L^2T^{-4}\cdot L^3M^{-1}T^{-2}\cdot ML^2T^{-1}\cdot L^{-7}T^7=1.
\end{equation}
\end{proof}

\begin{proposition}[Relation to Planck acceleration]
\begin{equation}
        \Pi_2=\left({a_0\over a_P}\right)^2.
\end{equation}
\end{proposition}

\begin{proof}
Since $a_P^2=c^7/(G\hbar)$,
\begin{equation}
        \left({a_0\over a_P}\right)^2=a_0^2{G\hbar\over c^7}=\Pi_2.
\end{equation}
\end{proof}

\subsection{Failure modes and falsifiers}

The A12 theorem fails if any one of the following fails:
\begin{enumerate}
    \item the finite microsector is not $E=\mathcal O(9)\oplus\mathcal O^{\oplus5}$ or has a different nonzero determinant count;
    \item the generation kernel is not three-dimensional;
    \item the clock dictionary $(H_0t_P)^2=Z'_\alpha/Z'_1$ is wrong;
    \item the gauge-emergence coefficient $k_a=3/(8\alpha)$ is not the coefficient in the scalar acceleration action;
    \item the $S^3$ stationarity potential is not the correct action-level crossover selector;
    \item the observed universal galaxy acceleration scale converges to a value inconsistent with Eq.~\eqref{ap:eq:a0-number} after systematic errors shrink.
\end{enumerate}

\subsection{Integration protocol for the unified paper}

\begin{enumerate}
    \item Replace any calibrated statement $a_0=2\sqrt\alpha\,cH_0$ with the action derivation in Section 5.
    \item Replace direct-source interpretations of $k_aa^2/c^2$ with the kinetic crossover interpretation.
    \item State the spectral-action clock dictionary explicitly.
    \item Delete or demote all minimal-branch black-hole temperature claims.
    \item Split 2PN lensing claims by branch.
    \item Mark CKM CP closure as DFD--SD conditional unless the minimal action is extended.
    \item Carry forward the full A11 PDE proof wherever PDE well-posedness is cited as theorem-grade; otherwise cite it as a program/proposition-tier result.
\end{enumerate}
