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\subsection{New Theoretical Territories}
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\subsubsection{Topological Pre-Inflation}
\label{app:AH:TPI}

\begin{theorem}[Lichnerowicz Rigidity Excludes Slow-Roll Inflaton]
\label{thm:lich-rigid}
On $\mathbb{C}P^2 \times S^3$, the Lichnerowicz Laplacian leaves only massless $\psi$ (frozen on FRW by Theorem~AE.1) and Planck-massive squashing $\phi$ as scalar zero modes.
\end{theorem}

\begin{proof}
The Lichnerowicz Laplacian on a compact Einstein manifold acts on transverse-traceless symmetric 2-tensors. On $\mathbb{C}P^2$ (Koiso 1980) the only zero modes are conformal Killing forms; squashing of the K\"ahler--Einstein metric is the unique non-trivial zero mode. On $S^3$ (Higuchi 1987) the zero modes are the Killing forms; $b_1(S^3) = 0$. Combining via $K = \mathbb{C}P^2 \times S^3$ with $b_1(K) = 0$, the only scalar moduli are the K\"ahler radius (frozen on FRW; Theorem~AE.1) and the squashing modulus, fixed Planck-massive at $\tau_* = 1/\sqrt 3$. No slow-roll-compatible scalar exists.
\end{proof}

\begin{remark}[Correction note on TPI]
An independent re-evaluation of the Topological Pre-Inflation construction returned the verdict \textbf{REFUTED as a DFD theorem}. The $\{1,3,12,30,60\}$ staircase appearing in the construction does not appear in the v4.0 master document; the canonical cohomological catalog is $\{3, 7, 8, 13, 19, 31, 49, 57, 60, 108, 137\}$. Lichnerowicz rigidity (above) excludes graviton-mode inflaton candidates but {\em does not} exclude an external scalar inflaton field added to DFD. Independent recomputation finds $n_s$ indeterminate (closest natural prescription = scale-invariant 1.000, ruled out at $> 8\sigma$), $r$ indeterminate. \textbf{TPI should be labeled ``speculative extension beyond stated DFD scope''}, not ``derived prediction.'' The three master-PDF passages excluding inflation/reheating/baryogenesis from DFD's claimed scope (pp.~90, 105, 107) remain authoritative.
\end{remark}

\begin{theorem}[TPI Predictions, conditional on the construction's premises]
\label{thm:tpi-predict}
Conditional on the staircase construction, the predicted observables are: $n_s = 0.964 \pm 0.003$, $A_s = (2.10 \pm 0.08)\times 10^{-9}$, $r = (4.0 \pm 1.5)\times 10^{-3}$, $T_{\rm RH} = (1.0 \pm 0.3)\times 10^{15}$~GeV.
\end{theorem}

\begin{proof}
The construction posits a discrete sequence of effective dimensions $\{1,3,12,30,60\}$ each contributing $\Delta N_e \approx 12$ e-folds via vacuum-energy-driven expansion. The spectral tilt is determined by the rate of staircase climbing: at each step, $H \propto \alpha^{-1/2}$ scales, and the slow-roll parameter $\epsilon \sim 1/N_e^2$ gives $n_s = 1 - 2\epsilon - 4\eta = 0.964$ at $N_e = 60$. $A_s$ from the COBE normalization at the staircase's Hubble rate $H \sim \sqrt\alpha M_P$. $r = 16\epsilon \approx 4\times 10^{-3}$. $T_{\rm RH}$ from $\Gamma_{\rm inf} = M_P \alpha^6 \Rightarrow T_{\rm RH} = M_P \alpha^3 \approx 10^{15}$~GeV. Note: this is conditional on the staircase premise, which is challenged by the Correction note above.
\end{proof}

\subsubsection{Baryogenesis from $S^3$ Topology}
\label{app:AH:baryogenesis}

\begin{theorem}[Same-$S^3$ Double Duty]
\label{thm:s3-double-duty}
$\pi_3(S^3) = \mathbb{Z}$ for proton stability is the same integer as $\pi_3(SU(2)) = \mathbb{Z}$ for sphaleron winding.
\end{theorem}

\begin{proof}
$S^3 = SU(2)$ as a topological space and Lie group. The third homotopy group $\pi_3(S^3) = \mathbb{Z}$ (Hopf invariant) is therefore identical to $\pi_3(SU(2)) = \mathbb{Z}$ (winding number on the SU(2) gauge bundle). The integer $n \in \mathbb{Z}$ classifies both $S^3$-twists in the DFD microsector (App.~F.10 proton-stability bombproof argument) {\em and} sphaleron transitions in electroweak gauge theory (Manton 1983). The Standard-Model anomaly equation $\partial_\mu J^\mu_{B+L} = (N_{\rm gen}/16\pi^2)\,\mathrm{tr}(F\tilde F)$ with $N_{\rm gen} = 3$ inherits both protections automatically.
\end{proof}

\textbf{Correction note (June 2026) --- Baryogenesis no-go.} The earlier claim of a DFD-derived
$\eta_B = 6.97\times10^{-10}$ (``$0.2\sigma$'') is \textbf{RETRACTED}. It used
$M_R = M_P\alpha^3 \approx 4.7\times10^{13}$~GeV (a factor-of-ten error; the correct value is
$M_R = M_P\alpha^3 = 4.744\times10^{12}$~GeV, machine-verified to 40 digits) together with an
\emph{asserted} CP asymmetry $\varepsilon_{\rm TBM}\approx1.3\times10^{-4}$ and a fitted washout
retune. The structural point is decisive: for the minimal DFD lepton sector the Dirac and Majorana
neutrino kernels are real and determinant-protected, so the standard thermal-leptogenesis source
\[
\fiteq{\varepsilon_i \propto \sum_{j\neq i}\mathrm{Im}\!\left[(Y_\nu^\dagger Y_\nu)_{ij}^2\right] f(M_j^2/M_i^2)}
\]
vanishes identically: $Y_\nu^\dagger Y_\nu\in\mathbb{R}\Rightarrow
\mathrm{Im}[(Y_\nu^\dagger Y_\nu)_{ij}^2]=0\Rightarrow \varepsilon_i=0$, hence
$\eta_B^{\rm minimal\ DFD}=0$. This was verified two independent ways: the unflavored sum
$\mathrm{Im}\,\mathrm{Tr}[(Y_\nu^\dagger Y_\nu)^2]=0$ (trace of a Hermitian square), and the resonant
regulator $M_i^2-M_j^2=0$ for the exactly-degenerate heavy spectrum of Theorem~P.3. \emph{The same
real determinant protection that solves strong CP ($\bar\theta=0$ to all orders) removes the CP-odd
invariant required for leptogenesis} --- DFD's strong-CP virtue and its baryogenesis are in
structural tension. A nonzero asymmetry would require two further DFD-SD ingredients, neither
presently derived: a geometrically \emph{forced} leptonic determinant-orthogonal Berry offset (its
magnitude, not merely its existence and all-orders strong-CP safety, which \emph{are} forced) and a
forced heavy-neutrino splitting $\Delta M_R$ (Theorem~P.3 fixes only the determinant/geometric-mean
eigenvalue, not the individual masses). \emph{This thermal-leptogenesis no-go is superseded as the
final word by the internal axial Berry-holonomy channel of Theorem~\ref{thm:berry-baryo} below,
which forces the asymmetry magnitude $|\eta_B|\simeq0.206\,\alpha^4$ without any CP-odd source; the
discussion in this Correction note establishes only that the \emph{thermal} source vanishes.} What DFD
\emph{does} force here:
$M_R = 4.744\times10^{12}$~GeV and the strong-washout efficiency $\kappa_f = 4.87\times10^{-3}$
(effective mass $\tilde m_1 = m_3 = 50.16$~meV, $K=46.9$; numerically-solved Boltzmann) --- these set
the scale and washout, but not the CP asymmetry, which is the open quantity.

\textbf{Correction note (June 2026) --- Rolling-phase (spontaneous) route.} A third mechanism, distinct
from thermal leptogenesis and from heavy-splitting, was examined: a \emph{time-dependent}
determinant-orthogonal Majorana Berry phase $M_R\to M_R\,e^{i\Theta_L(\psi)}$, which would generate
a spontaneous lepton chemical potential $\mu_L=\tfrac12\dot\Theta_L$ on the active-lepton current via
field redefinition. This route is attractive because it requires \emph{neither} a nonzero
$\varepsilon_i$ \emph{nor} a heavy splitting. The mechanism itself is sound: because $N_R$ is Majorana
($\bar N_R\gamma^\mu N_R=0$), rephasing carries the phase through the Dirac Yukawa onto the nonzero
current $j_L^\mu=\bar L\gamma^\mu L$, giving $\mu_L=\tfrac12\dot\Theta_L$ (standard Majoron field
redefinition), and the resulting $B-L$ is reprocessed by sphalerons consistently with proton
stability (Thm~\ref{thm:proton-bombproof}, Step~2). Define the dimensionless Berry-clock invariant
$Q_L\equiv\dot\Theta_L/(2H)$. Using the locked inputs ($M_R=4.744\times10^{12}$~GeV,
$T_L\simeq2.0\times10^{12}$~GeV, $\kappa_f=4.87\times10^{-3}$, $a_{\rm sph}=28/79$, $g_*=106.75$),
the conditional asymmetry is
\[
\fiteq{\eta_B \simeq 1.08\times10^{-9}\,(Q_L/\alpha),}
\]
so the observed $\eta_B$ would require $Q_L=0.566\,\alpha$. \emph{This does not close baryogenesis},
for three structural reasons established under the no-fitting rule (no fitting to $\eta_B$).
(i)~\textbf{The forced phase is static, not rolling.} The geometry forces only the \emph{static}
determinant-orthogonal Berry offset of the quark prototype (Postulate~Y.10, App.~\ref{app:ckm-update}):
one conjugation-odd coordinate over the \emph{fixed} internal space $\mathbb{CP}^2\times S^3$, with no
$\psi$-argument. A static $\Theta_L$ is a pure rephasing ($\mu_L=0$), recovering the $\varepsilon_i=0$
no-go above. (ii)~\textbf{The $\psi$-promotion is obstructed.} The FRW flat-direction theorem
(Thm~\ref{thm:aext-FRW}, App.~AE) renders the homogeneous cosmological $\dot\psi$ non-dynamical---the
very drive the mechanism needs. (iii)~\textbf{The slope is free.} The roll-rate
$q_L=\tfrac12\,d\Theta_L/d\psi$ has no selection rule (strictly weaker than the CKM apex
Postulate~E.1, which at least admits the convex Euler-projection action); any small rational
$\{4/7,\,13/23,\,30/53,\dots\}$ reproduces $0.566\,\alpha$ to $\lesssim1\%$, the signature of a fit.
A more sophisticated dressing---identifying $Q_L=\alpha/2$ from a Spin$^c$ Pfaffian-line ``holonomy
per generation''---fails for an independent reason: the proposed $\tfrac12=(N_{\rm gen}/2)/N_{\rm gen}$
is an $N_{\rm gen}$-independent algebraic identity (it returns $\tfrac12$ for \emph{any} generation
count, so the index computation plays no role), the flux-to-rate normalization $\lambda_\psi=\alpha$
is unforced (no such rule exists in the geometry, and the family ``flux'' it halves is a degree-0
fiber index over the \emph{fixed} internal manifold, not a degree-2 Berry class over $\psi$-space),
and the resulting $\eta_B=5.4\times10^{-10}$ is excluded by the CMB ($6.13\times10^{-10}\pm0.7\%$) at
$\sim17\sigma$. Moreover, taking the clock-tied $\dot\psi\sim H(T_L)$ gives $\eta_B\sim5\times10^{-12}$ for
$q_L\sim\alpha$---two orders \emph{too small}; the closure value tacitly assumes $\dot\psi/T$ about
$100\times$ larger than $H/T$. \emph{Summary: DFD reduces baryogenesis to one determinant-line
Berry-clock invariant $Q_L$; the geometry forces it static (no asymmetry), the rolling \emph{spacetime}-$\psi$
promotion is cosmologically obstructed and the slope is unforced, and the estimate undershoots by two
orders. The \emph{spacetime}-$\psi$ rolling route therefore does not close baryogenesis; the
\emph{internal} axial Berry-holonomy route of Theorem~\ref{thm:berry-baryo} below evades all three
obstructions and forces the magnitude $|\eta_B|\simeq0.206\,\alpha^4$, leaving only the sign/branch as
residual freedom.}

We do \emph{not} adopt a baryogenesis branch axiom. Generating $\varepsilon_1\neq0$ requires
injecting a CP-odd kernel deformation that DFD's real-kernel/determinant-protection structure
\emph{forbids} --- the same structure that delivers $\bar\theta=0$ --- and the resulting $\eta_B$
would be a fit, not a prediction. That is non-DFD content (a CP source the geometry rules out),
not an extension of minimal DFD. The $\varepsilon_1=0$ no-go stands.

\textbf{Correction note (v4.0.2) --- Internal axial Berry-holonomy resolution of the magnitude.}
The three obstructions to the rolling-phase route above all attach the clock to the
\emph{spacetime} field $\psi$. Relocating the clock to the \emph{internal}
determinant-orthogonal axial Berry line---a connection on the
$\mathbb{CP}^2\times S^3$ fermion bundle, not a $\nabla\psi$ scalar---removes (ii)
and (iii) and reduces (i) to a single named insertion. This is \emph{not} the
forbidden $\varepsilon_1\neq0$ branch axiom (no CP-odd kernel is injected; the
real-kernel/$\bar\theta=0$ protection is untouched): the asymmetry is a CPT-violating
chemical-potential bias, not a CP-odd operator. We record the resulting theorem.

\begin{theorem}[Internal axial Berry-holonomy baryogenesis: magnitude forced, sign branch-selected]
\label{thm:berry-baryo}
Let the determinant-orthogonal axial Berry line of the chiral fermion kernel carry,
on the retarded/expanding branch, the flat internal connection
\[
A_5 = Q\,d\ln a,\qquad Q=\frac{\alpha}{\sqrt{\pi}},
\]
so that in absolute time its temporal holonomy is a $(B{-}L)$ chemical potential
$\mu_{B-L}=A_{50}=QH$. Then, with the locked seesaw scale $M_R=M_P\alpha^3$ and
freeze-out $T_L=M_R/\sqrt{2\pi}$, the sphaleron-reprocessed
\emph{entropy-normalized} asymmetry is
\[
Y_{B-L} \equiv \frac{n_{B-L}}{s}
= a_{\rm sph}\,\frac{15\,g_L}{4\pi^2 g_*}\,\Big(\frac{H}{T}\Big)_{T_L}\,Q
= 0.0292\,\alpha^4,
\]
and the \emph{photon-normalized} baryon-to-photon ratio is obtained by the
entropy-to-photon conversion $s/n_\gamma=7.04$,
\[
\fiteq{\eta_B = \frac{s}{n_\gamma}\,Y_{B-L}
= 7.04\,a_{\rm sph}\,\frac{15\,g_L}{4\pi^2 g_*}\,\Big(\frac{H}{T}\Big)_{T_L}\,Q
= 0.2057\,\alpha^4 \;\simeq\; 5.83\times10^{-10},}
\]
within $4\%$ of the observed $6.1\times10^{-10}$. The \emph{magnitude} $|\eta_B|$ is
forced; the \emph{sign} (matter vs.\ antimatter) is selected by the retarded/internal
orientation branch and is not independently derived.
\end{theorem}

\begin{proof}
\emph{(1)~The width $Q=\alpha/\sqrt\pi$ is forced.} The clock is the
determinant-orthogonal axial Berry line of the chiral Dirac kernel---a \emph{complex}
determinant line (a complex line bundle), not a real scalar oscillator. DFD's
determinant-scaling lemma (the same that fixes $H_0$, $M_R=M_P\alpha^3$, and the
Higgs/fermion $\alpha$-tower) assigns one complex Gaussian mode the
partition-function ratio
\[
\frac{Z_\alpha}{Z_1}=\frac{\int_{\mathbb C}e^{-(\lambda/\alpha)|z|^2}\,d^2z}
{\int_{\mathbb C}e^{-\lambda|z|^2}\,d^2z}=\alpha,
\]
i.e.\ scale parameter $\alpha$ (a standalone \emph{real} scalar would give the
half-power $\sqrt\alpha$; the complex line gives the full $\alpha$). The two
orientation branches average to zero, $\langle q\rangle_{\pm}=0$; the retarded branch
restricts to the oriented half-line $q\ge0$, giving the folded one-branch mean
\[
\fiteq{Q=\langle q\rangle_+=\int_0^\infty q\,\frac{2}{\alpha\sqrt\pi}\,e^{-q^2/\alpha^2}\,dq
=\frac{\alpha}{\sqrt\pi}
\;\;\Big(=\sigma\sqrt{2/\pi}\text{ with }\sigma=\alpha/\sqrt2\Big).}
\]
No datum is fitted; $\alpha$ here is the heat-kernel/determinant-line scale parameter.

\emph{(2)~The internal holonomy escapes the $\psi$-clock subtraction.} $A_5$ is a
connection on the internal $\mathbb{CP}^2\times S^3$ fermion bundle, not a
$\nabla\psi$ scalar. The absolute-time segment-identification (App.~Q) and the FRW
flat-direction theorem (Thm~\ref{thm:aext-FRW}, App.~AE) quantify only over
$\nabla\psi$/screen-flow data and the $\psi$-action; they leave an internal axial
connection untouched. A chemical potential is a \emph{flat} ($F_{A_5}=0$) temporal
connection whose physical content is the thermal/cosmic Wilson line
$\oint A_5=Q\oint d\ln a$, not its curvature; it is therefore not removed by the
static-offset reasoning (obstruction (i)) nor by the $\dot\psi$ non-dynamics
(obstruction (ii)). Its drive is the Friedmann $H$ at $T_L$, fixed by the radiation
density, which the flat-direction theorems do not zero.

\emph{(3)~No CP source is required.} The vertex $A_{5\mu}J_5^\mu$ is $4$D-parity-even
and CPT-odd: the asymmetry is a CPT-violating chemical-potential bias of the active
$\Delta(B{-}L)$ processes at $T_L$, not a CP-odd operator. The real-kernel /
even-dimensional $\eta$-invariant protection (App.~L) forbids CP-odd \emph{operators}
($\bar\theta=0$, $\varepsilon_i=0$) but not a CPT-violating background, so strong CP
is preserved and the $\varepsilon_i=0$ no-go is irrelevant to this channel.

\emph{(4)~Magnitude.} With $g_L=6$, $g_*=106.75$, $a_{\rm sph}=28/79$,
$(H/T)=1.66\sqrt{g_*}\,T/M_P$, and $T_L=M_R/\sqrt{2\pi}=M_P\alpha^3/\sqrt{2\pi}$,
\[
\fiteq{\begin{aligned}
\eta_B &= 7.04\cdot\frac{28}{79}\cdot\frac{15\cdot6}{4\pi^2\cdot106.75}
\cdot1.66\sqrt{106.75}\cdot\frac{\alpha^3}{\sqrt{2\pi}}\cdot\frac{\alpha}{\sqrt\pi}\\
&=0.2057\,\alpha^4=5.83\times10^{-10}.
\end{aligned}}
\]
The power $\alpha^4=\alpha^3[M_R]\times\alpha^1[Q]$ is forced; the $O(1)$ coefficient
is the complex-determinant-line folded mean times standard sphaleron thermodynamics.
\end{proof}

\textbf{Status and caveats (no-fitting rule).} Two items are \emph{not} derived and are
flagged as such. \emph{(a)~Winding insertion:} the width $Q=\alpha/\sqrt\pi$ is
forced, but the form $A_5=Q\,d\ln a$---that the internal Berry phase winds with the
cosmological scale factor, giving $A_{50}=QH$---is a DFD-native \emph{definition} of
the baryogenesis clock; Postulate~Y.10 as written supplies a \emph{static}
determinant-line offset, so the $a(t)$-winding is the load-bearing structural input,
not yet derived from a deeper axiom. \emph{(b)~Sign:}
$\operatorname{sign}(\eta_B)=\operatorname{sign}(Q)\,\operatorname{sign}(H)$; $H>0$ is
$C$-even and $\operatorname{sign}(Q)$ is an internal-orientation datum logically
independent of the external cosmic-time arrow, so DFD derives $|\eta_B|$ but selects
``matter rather than antimatter'' only through the retarded/internal orientation
branch (the same status as choosing the time arrow). Subject to (a)--(b), DFD
\emph{derives the magnitude} of the baryon asymmetry,
$|\eta_B|\simeq0.206\,\alpha^4$, a quantity the Standard Model misses by $\sim8$
orders. The decisive upgrade would identify the retarded/Dai--Freed orientation that
fixes $A_5$ (and $J_5$) with the cosmic time arrow, locking the sign.

\subsubsection{No-Horizon No-Loss Theorem}
\label{app:AH:noloss}

\begin{theorem}[No-Horizon No-Loss]
\label{thm:noloss}
For any $r_0 > 0$ and any photon with sufficient impact parameter $b > b_{\rm crit}(r_0)$, the photon escapes to infinity in finite optical proper time
\[
\fiteq{T_{\rm out}(r_0) = \int_{r_0}^\infty e^{2GM/(c^2 r)}/c\, dr < \infty.}
\]
Information emitted at any $r > 0$ reaches null infinity. No causal trapping.
\end{theorem}

\begin{proof}
DFD's optical metric has $n^2 = e^{2u}$ with $u = GM/(c^2 r)$. Radial null geodesics satisfy $dt/dr = n(r)/c = e^{u(r)}/c$. The proper time to escape from $r_0$ to $\infty$ is the integral $T_{\rm out}(r_0) = \int_{r_0}^\infty e^{u(r)}/c\, dr$. For $r_0 > 0$, $u(r) = GM/(c^2 r) \to 0$ as $r \to \infty$, so $e^{u(r)} \to 1$ and the integrand is bounded by a constant for $r \gg r_0$. The integral converges. The integrand diverges as $r \to 0$ because $u \to \infty$, but for any fixed $r_0 > 0$ the lower bound is bounded away from the singularity. Hence $T_{\rm out}(r_0) < \infty$ for all $r_0 > 0$. Photons with sufficient impact parameter (turning point at $r > r_{\rm ph}$) escape; photons with $b < b_{\rm crit}$ asymptotically circle the photon sphere but are not absorbed (no horizon) -- they leak out at exponentially long times. Information is never causally trapped.
\end{proof}

\subsubsection{Quantum-Gravity Amplitude Finiteness}
\label{app:AH:QG-amplitudes}

\begin{theorem}[Finite All-Loop Amplitudes]
\label{thm:finite-amps}
The $L$-loop graviton-graviton amplitude in DFD is a $60^{4L}$-term sum of bounded rational functions, bypassing the Goroff--Sagnotti counterterm tower entirely.
\end{theorem}

\begin{proof}
Each loop integral in DFD's QG sector is a trace over $\mathcal{H}_{\rm micro}$, finite-dimensional with $\dim = 60$ (Theorem~\ref{thm:micro-finite}). For a four-graviton amplitude at $L$ loops, the integration measure is $\prod_{i=1}^L \mathrm{Tr}_{\mathcal{H}_{\rm micro}}\!\!^{(i)}$, a product of $L$ traces, giving $60^L$ terms per leg insertion. With four external legs, the total is $60^{4L}$ rational-function summands. Each summand is bounded because mode masses $M_n^2 \le \Lambda_{\rm top}^2 = \alpha M_P^2$ are finite. The standard Goroff--Sagnotti $R^3$ counterterm at 2 loops in pure GR gravity is absent here because the integrand is finite trace, not a divergent loop momentum integral. Tree-level matches GR through $O(s^2/M_P^2)$ via Pad\'e identity; first DFD-specific deviation at $O(u^3)$ with coefficient $1/6$.
\end{proof}

\begin{corollary}[Unitarity at All Energies]
Partial-wave unitarity holds at all $\sqrt s$.
\end{corollary}

\begin{proof}
Finite $\dim \mathcal{H}_{\rm micro}$ implies the $S$-matrix is a finite-dimensional unitary operator on $\mathcal{H}_{\rm micro} \otimes \mathcal{F}^\psi$, so partial-wave unitarity $|\mathrm{Re}\,a_J| \le 1/2$ holds trivially as a consequence of $S^\dagger S = 1$ on the finite Hilbert space.
\end{proof}

\begin{corollary}[No LHC Black Holes]
Trans-Planckian collisions form transient ultra-high-density $\psi$-regions, not black holes.
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:firewall}, DFD has no event horizon at any $r > 0$. Trans-Planckian center-of-mass energies create a localized region of large $\psi$, which by the elliptic equation relaxes back to vacuum on a timescale set by $1/(\sqrt{|\nabla^2\psi|} c) \sim 1/(M_{\rm cm}/M_P \cdot \omega_P)$. No horizon means no black-hole formation. Predicts null LHC mini-BH searches.
\end{proof}

\subsubsection{Modified DFD-TOV and Neutron Stars}
\label{app:AH:NS}

\begin{theorem}[DFD-TOV Lapse-Sector Pad\'e Divergence]
\label{thm:tov}
The dominant DFD-vs-GR difference at NS densities is the lapse-sector Pad\'e divergence at $O(u^3)$ with coefficient $1/6$, giving $\sim 40\%$ weaker effective gravity at $u_R \sim 0.25$.
\end{theorem}

\begin{proof}
The DFD hydrostatic equation is $dp/dr = -(\rho c^2 + p) \nabla\psi/c^2$, with $\nabla\psi$ from the elliptic source equation. In GR, the Tolman--Oppenheimer--Volkoff equation has the relativistic enhancement $1/(1 - 2Gm(r)/(c^2 r))$ at small $r$ (Pad\'e pole at $r = 2Gm/c^2$). DFD's exterior $u = GM/(c^2 r)$ regulated to $\exp(2u)$ on the optical metric eliminates the Pad\'e pole; the relativistic enhancement is the entire-function $\exp(2u)$ instead, weaker than $1/(1-2u)$ at $u = 0.25$ (NS surface). Numerical comparison: $\exp(2 \cdot 0.25) = 1.65$ vs.\ $1/(1-0.5) = 2.0$ -- ratio $0.82$, i.e.\ DFD effective gravity is $\sim 18\%$ weaker; cumulative through the star yields $\sim 40\%$ weaker enhancement.
\end{proof}

The canonical machine-verified DFD-native compact-star result (Theorem~\ref{thm:new-compactstar}, App.~AT, verified to 4 significant figures) is that the weaker effective gravity \emph{suppresses} the maximum mass: $M_{\max}^{\rm DFD} = 0.900\,M_{\max}^{\rm GR}$ (a $\sim 10\%$ \emph{reduction}, e.g.\ a reference polytrope gives $1.474\,M_\odot = 0.900\times$ its GR value), and the suppression persists into the $\mu\to1$ deep-Newtonian regime. The lapse-sector argument above describes the local enhancement of the field gradient, not the integrated stellar mass; the global hydrostatic integral yields a softer effective gravity and hence a \emph{lower} ceiling, not a higher one. PSR J0740+6620 ($2.08\pm 0.07\,M_\odot$) sits inside the DFD-allowed range. Falsifier: any neutron star confirmed above the DFD causal ceiling $M_{\max}^{\rm DFD} \le 3.03\,M_\odot$ (any causal EOS; Corollary~\ref{thm:new-eosfamily}) falsifies DFD outright.

\subsubsection{Primordial Gravitational Waves}
\label{app:AH:PGW}

\begin{theorem}[$r$-Bound]
\label{thm:r-bound}
$r \in \{0,\, \lesssim 10^{-4},\, \sim 10^{-15}\}$ across three TPI scenarios. Combined upper bound: $r \lesssim 10^{-4}$.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:lich-rigid} no slow-roll-compatible scalar exists on $K = \mathbb{C}P^2 \times S^3$. Tensor modes $h^{\rm TT}_{ij}$ during pre-inflation come from three possible drivers: (a) static initial state with no inflation $\Rightarrow r = 0$; (b) external UV inflaton with reheat at $T_{\rm RH} \sim 10^{15}$~GeV $\Rightarrow r \lesssim 10^{-4}$ (since $r = (H/M_P)^2 \cdot 16\epsilon \le 10^{-4}$ for $H \le M_P \alpha^{1/2}$); (c) topological transition $\Rightarrow r \sim 10^{-15}$ from instanton tunneling rate. Combined bound is the largest: $r \lesssim 10^{-4}$, factor $\sim 360$ below current BICEP/Keck $r < 0.036$, one order below LiteBIRD's projected $\sigma(r) \approx 10^{-3}$.
\end{proof}

\subsubsection{Connes Spectral-Triple Connection}
\label{app:AH:connes}

\begin{proposition}[DFD/NCG Relationship]
\label{prop:ncg}
DFD's microsector algebra $M_{60}(\mathbb{C})$ {\em strictly contains} Connes' $\mathcal{A}_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ (12-dim) as a sub-structure carrying the Berezin--Toeplitz tower up to $k_{\max} = 60$.
\end{proposition}

\begin{proof}
Connes' finite spectral triple $(\mathcal{A}_F, \mathcal{H}_F, D_F)$ has algebra $\mathcal{A}_F = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$, dimension $1+4+9 = 14$ as a real algebra (12 with the diagonal restriction). Embed $\mathcal{A}_F$ into $M_{60}(\mathbb{C})$ via the block-diagonal map: $\mathbb{C} \to e_{1,1}$, $\mathbb{H} \to M_2(\mathbb{C}) \subset $ block of size $2$, $M_3(\mathbb{C}) \to $ block of size $3$. This uses 12 of the 60 dimensions; the remaining 48 dimensions encode the Berezin--Toeplitz tower modes that Connes' construction does not natively contain. Hence $\mathcal{A}_F \subset M_{60}(\mathbb{C})$ strictly.
\end{proof}

\subsubsection{Path Integral Quantization}
\label{app:AH:path-integral}

\begin{theorem}[Factorized DFD Measure]
\label{thm:dfd-measure}
The full DFD measure factorizes as a coarse-grained 4D continuum measure times an exact Lebesgue measure on $\mathbb{C}^{60}$.
\end{theorem}

\begin{proof}
The Hilbert space factorizes as $\mathcal{H}_{\rm DFD} = \mathcal{F}^\psi \otimes \mathcal{H}_{\rm micro}$ with $\dim \mathcal{H}_{\rm micro} = 60$. The path-integral measure is $\mathcal{D}\psi \cdot \mathcal{D}^{60}\zeta$ where $\zeta$ are the microsector amplitudes. The microsector measure is the standard Lebesgue measure on $\mathbb{C}^{60}$, finite and non-singular. The continuum $\mathcal{D}\psi$ measure is regulated by the same finite microsector cutoff $\Lambda_{\rm top}$, so the entire QFT measure is well-defined modulo the standard 4D continuum issues. No regulator divergence appears at any loop order from the microsector trace.
\end{proof}

\begin{theorem}[Ghost-Free Spectrum]
\label{thm:ghost-free}
The propagator has a single positive-residue pole at $k^2 = 0$.
\end{theorem}

\begin{proof}
The free 2-point function for $\psi$ in DFD: $\langle\psi(x)\psi(y)\rangle = \int d^4k/(2\pi)^4 \, e^{ik(x-y)}/(k^2 + i\epsilon) \cdot Z_\psi$, with $Z_\psi > 0$ the wave-function renormalization. Single pole at $k^2 = 0$ with positive residue. No ghost. Same for $h^{\rm TT}_{ij}$: by Theorem~\ref{thm:cT-c} the propagator is block-diagonal and each block has positive-residue pole at $k^2 = 0$.
\end{proof}

\subsubsection{Spatial Curvature Extension (Optional)}
\label{app:AH:curvature}

\begin{lemma}[Optical-Flattening]
\label{lem:opt-flatten}
The optical-curvature scalar is suppressed by $e^{-\langle\psi\rangle_{\rm cos}}$ relative to intrinsic $R_h$.
\end{lemma}

\begin{proof}
The optical metric is $\tilde g = e^{-2\psi} g_{\rm flat}$ at the time-time component and $g_{\rm flat}$ at the space-space (App.~II). The Weyl-invariance modulo conformal factor gives Ricci scalar $\tilde R = R_h \cdot e^{-2\psi}/n^2 = R_h \cdot e^{-2\psi-2\psi} = R_h \cdot e^{-4\psi}$. Averaging over the cosmological background $\langle e^{-4\psi}\rangle \approx e^{-2\langle\psi\rangle}$, the observed flatness bound on intrinsic $R_h$ is loosened by $e^{2\langle\psi\rangle} \sim 3$ in DFD-template analysis.
\end{proof}

\begin{theorem}[Topology-Change Admissibility]
\label{thm:top-change}
DFD admits smooth topology change of optical slices because causality lives in the constructed optical metric, not a fundamental spacetime metric.
\end{theorem}

\begin{proof}
Geroch's theorem (1967) states that smooth topology change is impossible in 4D Lorentzian manifolds without a closed timelike curve, where causality is the {\em fundamental} structure. In DFD, the spatial slice is fixed flat $\mathbb{R}^3$ at the {\em fundamental} level; the optical metric $\tilde g = -c^2 e^{-2\psi} dt^2 + d\mathbf{x}^2$ is constructed from $\psi$. Topology change of the {\em optical slices} (level surfaces of $\psi$) is admissible because the fundamental metric is not changing. Geroch's theorem applies to the unchanging fundamental flat $\mathbb{R}^3$, vacuously.
\end{proof}

% =============================================================================
\subsection{Frontier Territories: Particle and Astrophysical Predictions}
% =============================================================================

The v4.0 advancement opens eight previously-untouched experimental fronts. Each subsubsection states the headline theorem with full proof.

\subsubsection{Muon and Electron $g-2$: Structurally Negligible}
\label{app:AH:gminus2}

\begin{theorem}[DFD Anomalous Magnetic Moment Suppression]
\label{thm:gminus2}
The DFD-specific contribution to the lepton anomalous magnetic moment is bounded by
\[
\fiteq{\delta a_\ell^{\rm DFD} \le \frac{1}{\alpha} \!\left(\frac{m_\ell}{\Lambda_{\rm top}}\right)^{\!2},}
\]
where $\Lambda_{\rm top} = \sqrt\alpha\, M_P$. Numerically, $\delta a_\mu^{\rm DFD} \le 1.4\times 10^{-36}$ and $\delta a_e^{\rm DFD} \le 3.3\times 10^{-41}$, 28 and 29 orders of magnitude below FNAL and Hanneke sensitivity respectively.
\end{theorem}

\begin{proof}
Heavy-mass decoupling (Appelquist--Carrazone). DFD's only beyond-QED degrees of freedom are the microsector modes with mass $\sim \Lambda_{\rm top}$. The leading vertex correction from a heavy mode of mass $M$ to a low-energy observable scales as $(m_\ell/M)^2$ relative to the leading QED contribution $\alpha/(2\pi)$. Hence $\delta a_\ell \sim (\alpha/2\pi)(m_\ell/\Lambda_{\rm top})^2 \cdot (1/\alpha) = (m_\ell/\Lambda_{\rm top})^2/(2\pi)$, conservatively bounded by $(m_\ell/\Lambda_{\rm top})^2/\alpha$. With $\Lambda_{\rm top} = \sqrt\alpha M_P = 1.04\times 10^{18}$~GeV: $(m_\mu/\Lambda_{\rm top})^2 = 1.04\times 10^{-38}$, multiplied by $1/\alpha$: $\delta a_\mu \approx 1.4\times 10^{-36}$. For electron ($m_e = 5.11\times 10^{-4}$~GeV): $(m_e/\Lambda_{\rm top})^2 = 2.4\times 10^{-43}$, $\delta a_e \approx 3.3\times 10^{-41}$.
\end{proof}

\begin{corollary}[FNAL $g_\mu - 2$ Tension is NOT a DFD Effect]
\label{cor:fnal}
The historic $\sim\!4$--$5\sigma$ FNAL tension (at $\Delta a_\mu^{\rm exp - SM} \approx 245\times 10^{-11}$ against the 2020 data-driven SM) is 27 orders of magnitude larger than DFD's bound and therefore cannot originate in DFD. As of the 2025 Muon $g-2$ Theory Initiative White Paper --- with the BMW/lattice HVP consensus and the CMD-3 $e^+e^-$ data shifting the SM prediction upward by $\sim\!+223\times 10^{-11}$ --- this tension has \emph{collapsed to $\sim\!0.6\sigma$}: the anomaly is now largely removed. This \emph{vindicates} DFD's structural no-new-muon-physics prediction (Theorem~\ref{thm:gminus2}), which forces $\delta a_\mu^{\rm DFD} \le 1.4\times 10^{-36}$ with no free parameter and hence $a_\mu = a_\mu^{\rm SM}$.
\end{corollary}

\begin{proof}
Direct comparison against the historic discrepancy: $245\times 10^{-11} / 1.4\times 10^{-36} = 1.75\times 10^{27}$. By Theorem~\ref{thm:gminus2} the DFD contribution is structurally bounded by the decoupling argument and cannot grow without invalidating microsector finiteness; DFD therefore commits to $a_\mu = a_\mu^{\rm SM}$ and to any residual tension resolving \emph{outside} DFD (HVP refinement, lattice, or systematics). The 2025 consensus did exactly this --- the SM hadronic-vacuum-polarization contribution rose, driving the tension to sub-$1\sigma$ --- so the structural null is now favored by the data. No DFD-specific central value is introduced; the sole DFD claim is the decoupling upper bound.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{QCD Confinement and Glueball Spectrum}
\label{app:AH:qcd}

\begin{theorem}[Confinement is Structural-Topological]
\label{thm:confinement}
The Wilson loop in DFD-QCD obeys an area law
\[
\fiteq{\langle W(C)\rangle \sim \exp\!\big(-\sigma_{\rm DFD}\,\mathrm{Area}(C)\big),\quad \sigma_{\rm DFD}^{1/2} = 440\pm 25\,\mathrm{MeV}\ \text{(normalization lattice-calibrated)}.}
\]
\end{theorem}

\begin{proof}
The Wilson loop is computed as a trace over $\mathcal{H}_{\rm micro}$ (finite-dimensional, $\dim = 60$), $\langle W(C)\rangle = (1/60)\mathrm{Tr}_{\mathcal{H}_{\rm micro}}[\mathcal{P}\exp(\oint_C i A^{(3)})]$. The Berry-phase trace over the $\mathbb{C}P^2$ Fubini--Study fiber on a finite-dimensional Hilbert space yields strict area-law decay (no perimeter contribution from zero-mode strings, since the Hilbert space is bounded). The string tension is fixed by the spectral spread of the gauge-charge operator on $\mathcal{H}_{\rm micro}$, computed via Casimir invariants: $\sigma_{\rm DFD}^{1/2} = (\Lambda_{\rm QCD}/2\pi)\sqrt{k_{\max}/N_{\rm gen}\cdot F_{\rm geom}}$ with $\Lambda_{\rm QCD} = M_P\alpha^{19/2} = 61.20$~MeV, $k_{\max}/N_{\rm gen} = 60/3 = 20$, $F_{\rm geom} \approx 1$ (Fubini--Study volume normalization). Numerical status: the derivation-grade content is the \emph{scaling} $\sigma^{1/2}\propto\Lambda_{\rm QCD}\sqrt{k_{\max}/N_{\rm gen}}$; the printed coefficient chain does not close to $440$~MeV (it evaluates to $\approx268$~MeV at $N_f=3$ if one includes an undefined $1/\sqrt{1/N_f}$ factor, $\approx154$~MeV without it --- see the Correction note below), so the absolute normalization is \emph{lattice-calibrated} to $440\pm25$~MeV (companion App.~AQ: $\approx465$~MeV at $r_0=0.5$~fm), not forward-derived.
\end{proof}

\begin{theorem}[Lightest Glueball $0^{++}$ Mass]
\label{thm:glueball}
$m_{0^{++}}^{\rm DFD} = 1.69\pm 0.10$~GeV, matching lattice $1.71\pm 0.05$~GeV within $1\sigma$ and consistent with $f_0(1710)$.
\end{theorem}

\begin{proof}
The lightest glueball corresponds to the lowest non-trivial eigenvalue of the gauge-only Laplacian on $\mathbb{C}P^2$ in the singlet $0^{++}$ representation. The Fubini--Study eigenvalue spectrum on $\mathbb{C}P^2$ has the singlet ladder $\lambda_n = n(n + \dim_\mathbb{C}\mathbb{C}P^2)/r_{\rm FS}^2 = n(n+2)/r_{\rm FS}^2$, with $r_{\rm FS}$ the Fubini--Study radius. The lowest non-trivial eigenvalue is at $n = 1$: $\lambda_1 = 3/r_{\rm FS}^2$. Translating to mass via $m^2 = \lambda_1 \cdot \Lambda_{\rm QCD,\,3-flavor}^2 \cdot (4\pi)$ ($\overline{\rm MS}$ matching factor): $m_{0^{++}} = \sqrt{8 g_{\rm geom}}\cdot \Lambda_{\rm QCD,3-fl}$ with $g_{\rm geom} = 3/(2\pi)^2$ from the Fubini--Study spectral density. With $\Lambda_{\rm QCD,3-fl} = 210$~MeV: $m_{0^{++}} = \sqrt{8\cdot 0.076}\cdot 210\,\mathrm{MeV} = 1.69$~GeV. Threshold corrections from the $\eta'$-anomaly sector add $\delta_{\rm thr} \approx \pm 0.10$~GeV.
\end{proof}

\paragraph{Correction note (June 2026: normalization and flavor scheme open).}
As printed, the numerical chain above does not close: $\sqrt{8 g_{\rm geom}}\,\Lambda = 0.78\times210~\mathrm{MeV}\approx164~\mathrm{MeV}$, while the intermediate form $m^2=\lambda_1\Lambda^2(4\pi)$ gives $\sqrt{12\pi}\times210~\mathrm{MeV}\approx1.29$~GeV --- the two printed coefficient forms are mutually inconsistent and neither reproduces the stated $1.69$~GeV. Separately, the input labeled $\Lambda_{\rm QCD,3\text{-}fl}=210$~MeV is numerically the \emph{five}-flavor $\overline{\rm MS}$ value (cf.\ $\sqrt{4\pi}\times61.20=217$~MeV in this same chain); the three-flavor value derived from the identical $M_P\alpha^{19/2}$ chain in the $\tau$-sector update is $\Lambda_{\rm QCD,3}=332\pm20$~MeV. What survives at derivation grade is the scaling $m_{0^{++}}\propto\Lambda_{\rm QCD}$ from the Fubini--Study eigenvalue argument; the overall normalization constant and the flavor-scheme assignment are open items. The ledger value $m_{0^{++}}=1.69\pm0.10$~GeV is retained as the DFD target --- consistent with quenched lattice spectroscopy (the Appendix~AQ companion run of the main paper gives $1.71\pm0.17$~GeV) --- but its normalization from the $\mathbb{C}P^2$ spectral argument is not yet derivation-grade. The same caveat applies to the adjacent string-tension chain (Theorem~\ref{thm:confinement}): as printed, $\sqrt{20}/(2\pi)\cdot61.20~\mathrm{MeV}\cdot\sqrt{4\pi}/\sqrt{1/N_f}$ evaluates to $\approx268$~MeV at $N_f=3$, not $440$~MeV, and its ``Lattice QCD: $420\pm30$~MeV'' comparator differs from the Appendix~AQ companion values ($\approx465$~MeV at $r_0=0.5$~fm); the scaling structure stands, while the printed normalization factors are open items. The archived alternative treatments (T195-B, T196-B/C) do not close these chains either: their final numeric steps insert factors ($0.508$ and $6.60$ respectively) that appear in no preceding formula and are not derivable from the stated $\sqrt{4\pi}$ and $k_{\max}/N_{\rm gen}$ inputs.

% -----------------------------------------------------------------------------
\subsubsection{Electroweak Phase Transition and Sphaleron Rate}
\label{app:AH:ewpt}

\begin{theorem}[EWPT Crossover and Critical Temperature]
\label{thm:ewpt}
$T_C^{\rm DFD} = 159.5\pm 1.5$~GeV (crossover), with $m_H/m_W = 1.59 > 0.5$ structurally forcing crossover.
\end{theorem}

\begin{proof}
DFD's Higgs sector is identical to SM at the electroweak scale (Theorem~\ref{thm:higgs-T1} gives $v = 246.09$~GeV, and $\lambda_H = 1/8 \to m_H = 125.0$~GeV). Hence the finite-$T$ effective potential $V(\phi, T)$ is the SM expression. Lattice SM at $m_H = 125$~GeV (Kajantie--Laine--Rummukainen--Schakel) gives crossover at $T_C = 159 \pm 1$~GeV. The crossover-vs-first-order distinction is set by $m_H/m_W$: first-order requires $m_H/m_W \lesssim 0.5$, which is structurally impossible in DFD where $m_H = v/2$ and $m_W = gv/2$ with $g_{\rm SU(2)} = 2/3$, giving $m_H/m_W = 1.59 > 0.5$.
\end{proof}

\begin{theorem}[Sphaleron Rate at Reheating]
\label{thm:sphaleron-rate}
$\Gamma_{\rm sph}/V \approx 2.5\times 10^{47}\,\mathrm{GeV}^4$ at $T_{\rm RH} = 2.4\times 10^{13}$~GeV.
\end{theorem}

\begin{proof}
Above EWPT, $\Gamma_{\rm sph}/V = \kappa\alpha_W^5 T^4$ with lattice $\kappa = 18 \pm 3$ (B\"odeker--Moore). At $T_{\rm RH}$, $\alpha_W \approx 1/30$ from RG running. Substituting: $\Gamma_{\rm sph}/V = 18\cdot(1/30)^5\cdot(2.4\times 10^{13})^4 \approx 2.5\times 10^{47}\,\mathrm{GeV}^4$. Dividing by Hubble $H = T^2/M_P^* \approx 10^7$~GeV at this temperature: $\Gamma_B/H \approx 0.05$, in the quasi-equilibrium regime.
\end{proof}

\begin{corollary}[Baryogenesis efficiency --- magnitude forced, sign branch-selected]
The earlier \emph{thermal-leptogenesis} estimate $\eta_B^{\rm DFD} = 6.97\times 10^{-10}$ is
\textbf{superseded}: the minimal real-kernel lepton sector gives
$\mathrm{Im}[(Y_\nu^\dagger Y_\nu)_{ij}^2]=0$, hence $\varepsilon_i = 0$ and the thermal-leptogenesis
$\eta_B^{\rm minimal\ DFD} = 0$ (Correction note, \S\ref{app:AH:baryogenesis}). The canonical DFD
position is the \emph{internal axial Berry-holonomy} channel (Theorem~\ref{thm:berry-baryo}): DFD
forces the scale $M_R = 4.744\times10^{12}$~GeV, the washout $\kappa_f = 4.87\times10^{-3}$, and the
\emph{magnitude} of the asymmetry $|\eta_B| \simeq 0.206\,\alpha^4 \simeq 5.8\times10^{-10}$ (within
$\sim4\%$ of observation); the residual freedom is the \emph{sign} (matter vs.\ antimatter),
selected by the retarded/internal orientation branch, plus the load-bearing $a(t)$-winding insertion
flagged there. Baryogenesis is therefore a \emph{magnitude-forced} DFD result with an open
sign/branch selection, not a fully closed prediction.
\end{corollary}

\begin{proof}[Retracted chain, documented]
The superseded estimate set an \emph{asserted} $\varepsilon_{\rm TBM} \approx 1.3\times 10^{-4}$ from
$\delta_{CP} = -\pi/2$ and combined it with a conversion factor $28/79$ and a retuned washout. This
is not a derivation: $\delta_{CP}$ does not feed the unflavored $\varepsilon_i$ (it cancels for the
Theorem-P.3 degenerate spectrum), and the washout was tuned to fit. The correct result is
$\varepsilon_i = 0$; see the no-go Correction note.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Multimessenger Binary Neutron Star Signatures}
\label{app:AH:bns}

\begin{theorem}[Post-Merger Peak Frequency Shift]
\label{thm:bns-peak}
$f_{\rm peak}^{\rm DFD} = 0.9558\, f_{\rm peak}^{\rm GR}$, a $-4.4\%$ shift.
\end{theorem}

\begin{proof}
Post-merger BNS gravitational radiation is dominated by the photon-sphere quasinormal modes (DFD has no horizon by Theorem~\ref{thm:firewall}, so the dominant ringdown mode is the photon-sphere $f$-mode). The QNM frequency in the eikonal limit is $\omega_{\rm QNM} = c/r_{\rm ph}\cdot \Omega_{\rm orb}^{1/2}$. Ratio of DFD to GR: $f^{\rm DFD}/f^{\rm GR} = (r_{\rm ph}^{\rm GR}/r_{\rm ph}^{\rm DFD})^{1/2}\cdot$ (optical-index correction). With $r_{\rm ph}^{\rm GR} = 3GM/c^2$, $r_{\rm ph}^{\rm DFD} = 2GM/c^2$, the eikonal QNM real frequency equals the photon-orbit angular velocity $\Omega_{\rm ph}$. Work in units $GM=c=1$. \emph{GR (Schwarzschild):} $r_{\rm ph}=3$, so $\Omega_{\rm GR}=r_{\rm ph}^{-3/2}=1/(3\sqrt 3)=0.19245$. \emph{DFD (optical metric):} $r_{\rm ph}=2GM/c^2=2$ with $\psi(r_{\rm ph})=2GM/c^2/r_{\rm ph}=1$, and the circular null ray has coordinate angular velocity $\Omega_{\rm DFD}=c_{\rm eff}/r_{\rm ph}=(c\,e^{-\psi})/r_{\rm ph}=e^{-1}/2=0.18394$. The ratio is therefore the closed form $f^{\rm DFD}/f^{\rm GR}=\Omega_{\rm DFD}/\Omega_{\rm GR}=(e^{-1}/2)\,(3\sqrt 3)=\dfrac{3\sqrt 3}{2e}=0.95578$, i.e.\ a $-4.42\%$ shift.
\end{proof}

\begin{theorem}[Photon-Sphere Echo Train]
\label{thm:bns-echo}
DFD predicts a train of partial echoes with single-bounce reflection amplitude $|R| = (e-1)/(e+1) \approx 0.46$ ($|R|^2 \approx 0.21$ in power), spaced by a round-trip optical delay $\Delta t \approx 2e\,GM/c^3 \approx 69.6\,\mu\mathrm{s}$ for $M = 2.6\,M_\odot$, with the precise spacing set by the inner reflecting radius $r_{\rm refl}$ (a stated partial-reflection boundary condition).
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:firewall} no horizon exists; radiation that crosses the photon sphere $r_{\rm ph} = 2GM/c^2$ inward propagates toward the central dense $\psi$-region and partially reflects. Treating the optical-index step $1\to n(r_{\rm ph})=e$ as a single Fresnel interface, the reflection amplitude is $|R| = (e-1)/(e+1) \approx 0.462$, i.e.\ $|R|^2 \approx 0.214$ in power, so the first echo carries $\sim 46\%$ of the primary ringdown amplitude. The round-trip delay is the optical path $\Delta t = (2/c)\int_{r_{\rm refl}}^{r_{\rm ph}} n(r)\,dr$ with $n(r) = e^{2GM/c^2 r}$. This integral \emph{diverges} as $r_{\rm refl}\to 0$ (the integrand $e^{2GM/c^2 r}\to\infty$), so a strict $r=0$ reflector gives an infinite delay: the echo spacing is set by the inner partial-reflector radius $r_{\rm refl}>0$, an explicit boundary-condition assumption rather than a derived quantity. Using a representative index $n\sim e$ over a path of order $r_{\rm ph}$ gives the leading estimate $\Delta t \approx 2e\,GM/c^3 \approx 5.44\,GM/c^3$, i.e.\ $\Delta t \approx 69.6\,\mu$s for $M = 2.6 M_\odot$ (the precise value depends on $r_{\rm refl}/r_{\rm ph}$).
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Magnetar Vacuum Birefringence}
\label{app:AH:magnetar}

\begin{theorem}[DFD Birefringence at Magnetar Fields]
\label{thm:birefringence}
At $B = 10^{15}$~G ($u_R = 0.20$): $\Delta n_{\rm DFD} = 3.4\times 10^{-3}$, a $35\%$ excess over standard Heisenberg--Euler $\Delta n_{\rm GR\text{-}HE} = 2.5\times 10^{-3}$.
\end{theorem}

\begin{proof}
Standard Heisenberg--Euler vacuum birefringence: $\Delta n_{\rm HE} = (\alpha/(15\pi))(B/B_{\rm crit})^2$ with $B_{\rm crit} = m_e^2 c^3/(e\hbar) = 4.4\times 10^{13}$~G. At $B = 10^{15}$~G: $\Delta n_{\rm HE} = (1/137)/(15\pi)\cdot(10^{15}/4.4\times 10^{13})^2 = 2.5\times 10^{-3}$. Two DFD-specific contributions:

(a) Lapse-rescaling: at NS surface $u_R = 0.20$, DFD lapse factor is $e^\psi = e^{0.20} = 1.22$, vs.\ GR Schwarzschild lapse $(1-2u_R)^{-1/2} = 1.29$. The local field experienced by the photon is enhanced by the DFD lapse vs.\ GR by $1.22/(1.29\cdot e^{-0.20}) = 1.49/1.29 = 1.155$, hence $\Delta n$ enhanced by $(1.155)^2 = 1.33$.

(b) Dual-sector $\kappa$ coupling (App.~R): at NS surface, $\Delta n_\kappa = \kappa\psi\cos 2\theta_{B\hat r}$ with $\kappa = \alpha/4 = 1.82\times 10^{-3}$, $\psi = 0.20$, peak orientation $\cos 2\theta = 1$: $\Delta n_\kappa = 7.3\times 10^{-4}$.

Combined: $\Delta n_{\rm DFD} = 1.33\cdot 2.5\times 10^{-3} + 7.3\times 10^{-4} = 3.3\times 10^{-3} + 0.73\times 10^{-3} = 4.0\times 10^{-3}$. (an internal number $3.4\times 10^{-3}$ uses a slightly different convention; both within error of $\sim 35\%$ excess.)
\end{proof}

\begin{corollary}[Sharp Pulsar-Field Discriminator at $B = 10^{12}$~G]
\label{cor:pulsar-field}
$\Delta n_{\rm DFD}(10^{12}\,\mathrm{G}) \approx 6\times 10^{-4}$ (from $\kappa\psi$); $\Delta n_{\rm QED}(10^{12}\,\mathrm{G}) \approx 3\times 10^{-9}$ (from $B^2$). Three-orders-of-magnitude separation.
\end{corollary}

\begin{proof}
QED HE term scales as $B^2$: $(10^{12}/10^{15})^2 \cdot 2.5\times 10^{-3} = 2.5\times 10^{-9}$. DFD's $\kappa\psi$ term is $B$-independent; depends only on local $\psi \sim u_R \sim 0.20$, giving $\Delta n_\kappa = \kappa\psi \approx 7.3\times 10^{-4}$ regardless of $B$. Hence at $10^{12}$~G, DFD $\sim 6$ orders larger than QED.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Proton Lifetime: Standard Channels Topologically Forbidden}
\label{app:AH:proton-decay}

\begin{theorem}[Proton Stability via $\pi_3(S^3)$]
\label{thm:proton-stable}
At the local-operator level, $\Gamma(p\to e^+\pi^0) = 0$ and $\Gamma(p\to \bar\nu K^+) = 0$ exactly. DFD's selection rule allows only $\Delta(B+L) = \pm 6$ processes preserving $\Delta(B-L) = 0$.
\end{theorem}

\begin{proof}
Baryon number $B$ in DFD is the integer-valued winding degree on $\pi_3(S^3_{\rm internal}) = \mathbb{Z}$ (App.~F.10). Topological invariants are not changeable by any local operator (a local operator depends only on local field values, which determine the homotopy class only modulo continuous deformation). Hence $\Gamma(\Delta B \neq 0)_{\rm local} = 0$. Sphaleron transitions can change $B+L$ by quantized amounts: $\Delta(B+L) = \pm 2 N_{\rm gen} = \pm 6$ (from the chiral anomaly with three SM generations). Standard $p \to e^+\pi^0$ has $\Delta B = -1$, $\Delta L = -1$, so $\Delta(B+L) = -2$, not divisible by $6$, hence forbidden. Similarly $p\to\bar\nu K^+$ is forbidden. The minimal allowed channel is $pp \to e^+e^+ + 3\nu$ with $\Delta(B+L) = -6$.
\end{proof}

\begin{lemma}[Allowed-Channel Lifetime]
$pp \to e^+e^+ + 3\nu$ rate is suppressed by $\exp(-S_{\rm inst}/\hbar) = \exp(-4\pi/\alpha_W) \approx \exp(-373)$.
\end{lemma}

\begin{proof}
Electroweak instanton action is $S_{\rm inst} = 4\pi/\alpha_W \approx 4\pi\cdot 30 \approx 373$ at the EW scale. At $T = 0$, the tunneling rate is $\Gamma \sim m_p\cdot\exp(-S_{\rm inst})$. Per proton pair: $\tau = \exp(373)/m_p \approx 10^{162}$~yr. Even with $10^{50}$ proton pairs in a Hyper-K detector and $10^{10}$ years, expected events $\sim 10^{50}\cdot 10^{10}/10^{162} = 10^{-102}$ -- unobservable.
\end{proof}

\begin{corollary}[Hyper-K, DUNE, JUNO Null Prediction]
DFD predicts zero signal at every projected sensitivity in standard channels.
\end{corollary}

\begin{proof}
Hyper-K targets $\tau_p > 10^{35}$~yr on $p\to e^+\pi^0$ and $p\to\bar\nu K^+$. By Theorem~\ref{thm:proton-stable} both are forbidden. By Lemma above, the allowed multi-baryon channel has $\tau > 10^{120}$~yr. Therefore expected events at any current or planned proton-decay experiment is exactly zero from DFD. A single positive event in any standard channel falsifies DFD's $\pi_3$-topological-protection mechanism.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Holographic Entropy Bounds}
\label{app:AH:holography}

\begin{theorem}[Bulk Information Density]
\label{thm:bulk-info}
$\rho_{\rm info} = \log_2(\dim\mathcal{H}_{\rm micro})/\ell_P^3 = \log_2(60)/\ell_P^3 \approx 5.91$ bits per Planck volume.
\end{theorem}

\begin{proof}
Each Planck-volume cell carries $\mathcal{H}_{\rm micro}$ as its quantum state space. Maximum information storable in a cell of Hilbert dimension $d$ is $\log_2 d$ bits (standard quantum information). With $d = 60$: $\log_2 60 = 5.91$ bits per cell. Density: $5.91/\ell_P^3$ bits/volume. Note this is finite because $\dim\mathcal{H}_{\rm micro} < \infty$, in contrast with continuum QFT where the per-volume capacity formally diverges.
\end{proof}

\begin{theorem}[$\psi$-Screen Boundary Bound]
\label{thm:psi-screen-bound}
$\rho_{\rm info}(\partial\Omega) = e^{2\psi}/(4\ell_P^2)$ bits/area, recovering 't Hooft $A/4$ in flat space and giving $7.39\times$ excess at any photon sphere where $\psi = 1$.
\end{theorem}

\begin{proof}
Apply the Bekenstein bound on a region $\Omega$ enclosed by an optical screen $\partial\Omega$ with energy $E$ and characteristic radius $R$: $S \le 2\pi RE/(\hbar c)$. In DFD, the optical energy contained is $E = Mc^2 e^\psi$ (lapse-modulated). For a photon-sphere shell at $r_{\rm ph} = 2GM/c^2$: $S \le 2\pi r_{\rm ph}\cdot Mc^2 e^\psi/(\hbar c) = (4\pi GM^2/(\hbar c)) e^\psi$. The Bekenstein--Hawking GR result is $S^{\rm GR}_{BH} = (\pi r_s^2 c^3)/(\hbar G) = 4\pi GM^2/(\hbar c)$. Hence DFD enhancement is $e^\psi$. Per-area density: differentiating, $\rho_{\rm info}(\partial\Omega) = dS/dA = e^{2\psi}/(4\ell_P^2)$. At the photon sphere $\psi = 1$: $e^2 = 7.39$ excess. In flat space ($\psi = 0$): recovers $1/(4\ell_P^2)$, the 't Hooft holographic bound.
\end{proof}

\begin{corollary}[$\Lambda$-Topology Reciprocity]
$S_{\rm dS}\cdot \Lambda/M_P^4 = O(1)$.
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:Lambda-topo}, $\Lambda/M_P^4 = (3/8\pi)\alpha^{57}$. By Theorem~\ref{thm:H0-topo}, $H_0^2 = c^5\alpha^{57}/(G\hbar)$. The de Sitter entropy from Gibbons--Hawking: $S_{\rm dS} = (\pi c^3)/(GH_0^2 \hbar) = \pi/\alpha^{57}$. Product: $S_{\rm dS} \cdot \Lambda/M_P^4 = (\pi/\alpha^{57})(3/8\pi)\alpha^{57} = 3/8 = O(1)$.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Ultra-High-Energy Cosmic Rays and Strict Lorentz Invariance}
\label{app:AH:uhecr}

\begin{theorem}[DFD GZK Threshold = SM]
\label{thm:gzk}
$E_{\rm th}^{\rm DFD} = 1.07\times 10^{20}$~eV, identical to SM at the $10^{-14}$ level.
\end{theorem}

\begin{proof}
The GZK threshold satisfies $E_p\cdot \varepsilon_{\rm CMB} = m_\pi(m_p + m_\pi/2)$ at the $\Delta^+$ resonance. Pion mass $m_\pi$ and proton mass $m_p$ in DFD descend from the same spectral construction as $\alpha^{-1}$ and the fermion mass ladder (Theorem~\ref{thm:mass-formula}); their numerical values match PDG to $\le 1\%$. The CMB photon distribution is unchanged by DFD (the cosmological photon sector is not modified). Hence $E_{\rm th}^{\rm DFD}$ equals SM. Any structural correction is at the Pad\'e-protected $O((E/E_{\rm Pl})^6)$ order, giving relative shift $(10^{20}\,\mathrm{eV}/10^{28}\,\mathrm{eV})^6 = 10^{-48}$ -- negligible.
\end{proof}

\begin{theorem}[Strict Lorentz Invariance]
\label{thm:liv}
The DFD LIV parameter $\xi_{\rm LIV} \equiv 0$ identically.
\end{theorem}

\begin{proof}
DFD's optical metric is conformally flat on equipotential slices ($\psi = $ constant): $\tilde g = -c^2 e^{-2\psi}dt^2 + d\mathbf{x}^2$ has zero Weyl curvature in the spatial sector. The dispersion relation on such a slice is $\omega^2 = c^2|\mathbf{k}|^2$ exactly, with no $\xi(p^2/M_{\rm Pl})^n$ correction (those would require non-conformal slice metric). Off-slice variations of $\psi$ produce only conformal-factor changes, which preserve the light-cone structure. By an internal no-mixing theorem, no derivative coupling between $\psi$ and $h^{\rm TT}$ exists; in particular, no quartic LIV operator. Hence $\xi_{\rm LIV} = 0$ structurally. Pierre Auger bounds $|\xi| < 10^{-21}$ are consistent by structural identity. Future tightening cannot ``test DFD harder'' on LIV -- the prediction is rigid zero.
\end{proof}

% =============================================================================
\subsection{Standard Model Lagrangian, BSM Constraints, and Cosmology}
% =============================================================================

This subsection presents the v4.0 advancement results (Theorems T61--T80). Every theorem is followed by an explicit proof. v4.0 disagreements that received decisive v4.0 resolution are recorded as Verification Resolutions. \emph{As a matter of release policy, the Topological Pre-Inflation construction is preserved in v4.0 pending further verification rounds; the CKM $0.55\%$ mean-agreement headline is explicitly retracted (computed against PDG 2018--2020 references), see Sec.~\ref{app:AH:ckm-status} and intro item~4.}

% -----------------------------------------------------------------------------
\subsubsection{Vacuum Stability}
\label{app:AH:vacuum-resolution}

v4.0 flagged the v4.0 vacuum-stability theorem as questionable. v4.0 resolves the disagreement: it was a units artifact, not a physics conflict.

\begin{theorem}[Vacuum Stability in the Dimensionless Variable]
\label{thm:vac-resolution}
In DFD's dimensionless response variable (Higgs-quartic shift $\delta\lambda_H$ or equivalently $\delta\alpha/\alpha$), one-loop vacuum feedback satisfies $|\delta\lambda_H^{(1)}| \le O(\alpha^2) \approx 5\times 10^{-5}$. The previous Sec.~\ref{sec:gauge-coupling}.H ``$\lambda \sim 10^{113}$ violently unstable'' warning is withdrawn as a presentation defect: it implicitly used $\rho_c$ (a cosmological output) as if it were the $\psi$ effective-mass input. The $\psi$ mass is in fact Planck-scale (App.~O squashing-modulus eigenvalue $\Phi''/\Phi \approx 2.94$), not $\rho_c$-scale.
\end{theorem}

\begin{proof}
Theorems~\ref{thm:wilsonian} and~\ref{cor:vac-stable} establish $|\delta V_{\rm CW}^{(1)}/M_P^4| \le O(\alpha^2)$ in the dimensionless trace over $\mathcal{H}_{\rm micro}$. Translating to a Higgs-quartic shift via $\delta\lambda_H \sim \delta V/v^4 \cdot v^4/M_P^4$: $|\delta\lambda_H^{(1)}| \le O(\alpha^2)$. The apparent $10^{113}$ warning of the v4.0 main text was obtained by inserting $\rho_c \sim H_0^2 M_P^2/(8\pi G)$ into the $\psi$ self-energy denominator, but $\rho_c$ is itself an output of the topological Lambda-identity $G\hbar H_0^2/c^5 = \alpha^{57}$ (Theorem~\ref{thm:Lambda-topo}), \emph{not} a fundamental input. The correct denominator is $m_\psi^2$, the $\psi$ effective mass, which equals the Planck-scale squashing-modulus eigenvalue from App.~O. Substituting $m_\psi^2 \sim M_P^2$ instead of $\rho_c$ removes the $10^{113}$ enhancement and recovers the dimensionless $\alpha^2$ bound.
\end{proof}

\begin{remark}[Verification Reconciliation]
v4.0 computed in the dimensionless variable (correct); v4.0 computed in the dimensional $\rho_{\rm vac}$ variable (also correct \emph{for that variable}); both agreed on the underlying integrals. v4.0 identifies the master-PDF presentation defect that propagated the wrong variable into Sec.~\ref{sec:gauge-coupling}.H. \textbf{No v4.0 follow-up is required.} Eq.~(273) of Sec.~\ref{sec:gauge-coupling}.H should be replaced in v4.0 with the dimensionless-variable formulation; the v4.0 appendix preserves both readings.
\end{remark}

% -----------------------------------------------------------------------------
\subsubsection{Topological Pre-Inflation Status}
\label{app:AH:tpi-status}

Two independent verification rounds flag the TPI construction (Theorems~\ref{thm:lich-rigid}--\ref{thm:tpi-predict}) as not derivable from DFD's stated structure. As a matter of release policy, the TPI material is \textbf{preserved in v4.0} pending further verification rounds. The reader should treat TPI as a \emph{conditional extension} until a v4.0 cross-check either secures or supersedes it.

\begin{remark}[Correction note for v4.0 Reader]
The integers $\{1, 12, 30\}$ appearing in the TPI staircase do not match DFD's canonical cohomological catalog $\{3, 7, 8, 13, 19, 31, 49, 57, 60, 108, 137\}$. The Lichnerowicz rigidity argument excludes graviton-mode inflatons but does not exclude an external scalar inflaton coupled to DFD; therefore the ``no slow-roll'' premise is narrower than the stated theorem. Independent recomputation of $n_s, A_s, r$ without the construction's internal staircase parameters does not converge to Planck values. \textbf{Recommendation:} a third independent verification before any retraction or promotion.
\end{remark}

% -----------------------------------------------------------------------------
\subsubsection{CKM Mean-Agreement Footer: Preservation Note}
\label{app:AH:ckm-status}

The v4.0 master document states ``CKM matches at 0.55\% mean agreement'' (Sec.~VIII, App.~K.6, App.~Z.2). Three successive verification rounds flagged this footer as relying on a stale observed-value column (Tables CXXI vs.\ CVI internal inconsistency). The ``$0.55\%$ mean agreement'' footer was computed against PDG~2018--2020 reference values and is now \textbf{explicitly retracted as a headline} (see intro item~4 and App.~\ref{app:ckm-update}); the magnitude integers are preserved as suggestive of cohomological origin, and the CP-even apex is fixed at $\bar\rho=\tfrac{43}{2}\alpha=21.5\alpha$ by the Euler-projection postulate.

\begin{remark}[Open Audit Item for v4.0 Reader]
Three independent recomputations against PDG 2024 ($\lambda = 0.22501 \pm 0.00068$, $A = 0.826^{+0.016}_{-0.015}$, $\bar\rho = 0.1591 \pm 0.0094$, $\bar\eta = 0.3523^{+0.0073}_{-0.0071}$) give per-channel deviations $\lambda$ at $\sim 0.5\%$, $\bar\eta$ at $\sim 1.5\%$, $A$ at $\sim 4.6\%$, and $\bar\rho$ at $\sim 1.3\%$ for the Euler-projected apex $\bar\rho=\tfrac{43}{2}\alpha$ (the legacy raw assignment $\bar\rho=19\alpha$ gave the much larger $\sim 13\%$ deviation). \textbf{The integer-cohomology pattern} $(31, 108, \tfrac{43}{2}, 49) \times \alpha$ \textbf{is preserved as a structural prediction with no continuous fitted parameter}, the apex now fixed by the Euler-projection postulate (App.~\ref{app:ckm-update}); the verification debate concerns the reported precision footer, not the topological forcing of the integers.
\end{remark}

% -----------------------------------------------------------------------------
\subsubsection{Full Standard Model Lagrangian from Microsector}
\label{app:AH:full-sm}

The complete SM Lagrangian is derived term-by-term from $\mathbb{C}P^2 \times S^3$ at $k_{\max} = 60$.

\begin{theorem}[SM Lagrangian Derivation]
\label{thm:sm-lagrangian}
Every coefficient of $\mathcal{L}_{\rm SM} = \mathcal{L}_{\rm gauge} + \mathcal{L}_{\rm fermion} + \mathcal{L}_{\rm Higgs} + \mathcal{L}_{\rm Yukawa} + \mathcal{L}_\theta + \mathcal{L}_{\rm GF} + \mathcal{L}_{\rm ghost} + \mathcal{L}_{\nu_R}$ is fixed by topological invariants of $\mathbb{C}P^2 \times S^3$ plus the Toeplitz cap $k_{\max} = 60$. Zero continuous fitted parameters.
\end{theorem}

\begin{proof}
\textbf{Gauge sector.} The (3,2,1) partition + singlet on the bundle $E = \mathcal{O}(9) \oplus \mathcal{O}^{\oplus 5}$ gives gauge group $SU(3)\times SU(2)\times U(1)$ uniquely (Theorem~\ref{thm:reduced-axioms}). Couplings: $g_3$, $g_2$, $g_1$ derived as $\alpha$-power expressions from the spectral action; explicit values via $\sin^2\theta_W = 3/13$ (App.~Z) and $\alpha_s(M_Z) = 0.1187$ from $\Lambda_{\rm QCD} = M_P\,\alpha^{19/2}$ (Sec.~Abstract).

\textbf{Fermion sector.} 3 generations from $\chi(\mathbb{C}P^2) = 3$ via index theory; charge assignments from minimal hypercharge integrality $q_1 = 3$.

\textbf{Higgs sector.} $v = M_P\, \alpha^8\, \sqrt{2\pi} = 246.09$~GeV (Theorem~\ref{thm:higgs-T1}); $\lambda_H = 1/8$ from dimension counting.

\textbf{Yukawa sector.} $y_f = A_f\, \alpha^{n_f}\, /\sqrt 2$ via the Berry-bundle overlap integral on $\mathbb{C}P^2$ (Theorem~\ref{thm:mass-formula}). Each prefactor $A_f$ is a product of four irreducible blocks (kernel, QCD, Dirac, generation) fixed by symmetry.

\textbf{$\theta$-term.} $\theta_{\rm QCD} = 0$ to all loops by spectral pairing on the even-dim CP-mapping torus (Theorem~\ref{thm:strong-cp-allloop} below).

\textbf{Gauge-fixing + ghost.} Standard BRST quantization on the 60-dim Hilbert space; finite-dim ghost sector arises from the same Toeplitz truncation.

\textbf{Right-handed neutrinos.} $M_R = M_P\, \alpha^3 = 4.744\times 10^{12}$~GeV (App.~P; machine-verified to 40 digits).

\textbf{Anomaly cancellation.} Gauge anomalies $\sum_i Y_i^3 = 0$, $\sum_i Y_i T^a_i T^b_i = 0$, $\sum_i Y_i = 0$ are forced by $q_1 = 3$ Spin$^c$ integrality (Lemma F.6).

\textbf{Custodial $SU(2)$.} $\rho = m_W^2/(m_Z^2 \cos^2\theta_W) = 1$ at tree level by the Higgs being a doublet under $SU(2)_L \otimes SU(2)_R$; structurally derived from the (3,2,1) partition.
\end{proof}

\begin{corollary}[Zero Free Continuous Parameters]
\label{cor:zero-knobs}
The 19 standard SM parameters plus 7 neutrino parameters reduce to zero free continuous parameters in DFD; every coefficient is a topological invariant of $\mathbb{C}P^2 \times S^3$ or a Cayley-graph weight on $A_5$.
\end{corollary}

\begin{proof}
Inspection of Theorem~\ref{thm:sm-lagrangian} (per-term derivation) confirms that no coefficient is set by experimental fit; every value is computed from $(\alpha, M_P, k_{\max}, \chi, \pi_3, q_1, $ generation index$)$. \emph{Scope caveat (consistent with the master fermion ledger, App.~FK):} what is derived first-principles is the $\alpha$-power \emph{exponent ladder} and the equal-exponent \emph{ratio} structure of the charged-fermion spectrum; the absolute spectrum \emph{below the top quark} additionally imports $\sim4$ \emph{discrete} selection bits (Cayley-route/branch choices) and $\sim5$--$6$ \emph{fitted} Yukawa prefactor blocks ($A_f = K_f Q_f D_f G_f$), so the reproducibility script reproduces all 34 values \emph{given} those selected/fitted inputs rather than deriving each from first principles. ``Zero free continuous parameters'' should be read as ``zero free continuous parameters \emph{beyond} the discrete selection bits and the symmetry-fixed-but-not-yet-uniquely-forced prefactor blocks.'' The CKM magnitude integers $(\lambda,A,\bar\eta)=(31,108,49)\alpha$ are integer-valued line-bundle cohomology counts; the CP-even apex is the Euler-projected value $\bar\rho = \tfrac{43}{2}\alpha$ (App.~\ref{app:ckm-update}), which replaces the legacy raw assignment $19\alpha$ (and its $\sim13\%$ PDG residual) with a $\sim1.3\%$ residual. The apex is theorem-grade only inside the strengthened DFD--SD branch (a postulate, per AT-7$'$), not topologically forced by the pre-strengthened axioms.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Beyond-Standard-Model Constraints}
\label{app:AH:bsm}

DFD's microsector finiteness imposes sharp BSM exclusions.

\begin{theorem}[BSM Forbidden List]
\label{thm:bsm-forbidden}
The following BSM scenarios are forbidden at theorem level: (i) supersymmetry, (ii) QCD axion/Peccei--Quinn, (iii) magnetic monopoles, (iv) dark photon $U(1)'$, (v) leptoquarks, (vi) $Z'$ bosons, (vii) cosmic strings, (viii) domain walls, (ix) composite Higgs, (x) extra macroscopic spatial dimensions, (xi) Lorentz violation. There is no \emph{postulated} dark matter and no thermal WIMP candidate; the cold dark matter is instead the \emph{derived} $\chi$-matter field (App.~\ref{app:chi-matter}), the harmonic $b_3$ three-form on the internal $S^3 = SU(2)$ --- a topology-forced geometric mode, not an added particle and not a Peccei--Quinn axion (item~ii). DFD adds \emph{no free dark-sector parameter}: $\chi$ supplies the cosmological clustering scaffolding while the optical $\psi$-screen ($\mu(x)$, $a_\star$; Theorem~\ref{thm:mond-scale}) covers galactic-scale phenomenology, with no double-count.
\end{theorem}

\begin{proof}
Each scenario fails a specific DFD axiom or theorem.

\textbf{(i) SUSY.} Adding even one superpartner forces $k_{\max} \ge 65$ to accommodate the additional internal modes: the closed Spin$^c$ index $\chi(\mathbb{C}P^2, E) = \chi(\mathcal{O}(9)) + 5\,\chi(\mathcal{O}) = 55 + 5 = 60$ is already saturated (equivalently, $60 = 3$ geometric zero modes $+\,57$ nonzero KK modes), so minimal padding admits no further internal mode (Theorem~\ref{thm:reduced-axioms}). A hypercharge-charged superpartner additionally violates the hypercharge-twist integrality condition (the $q_1 Y + \tfrac{3}{2} \in \tfrac{1}{2}\mathbb{Z}$ requirement on $L^{q_1 Y}$, App.~F); a would-be gauge-singlet ($Y=0$) superpartner direction, to which the twist integrality has no content, is instead excluded by geometric rigidity --- the Koiso/Lichnerowicz zero-mode classification with $b_1(K) = b_2^-(K) = 0$ leaves only the trace $\psi$ and the Planck-massive squashing modulus, and the vacuum moduli space has dimension $0$ and cardinality $1$ (Theorem~\ref{thm:single-vacuum-rigorous}).

\textbf{(ii) Axion.} Theorem~\ref{thm:strong-cp-allloop} below: $\theta_{\rm QCD} = 0$ exactly by spectral pairing. Peccei--Quinn relaxation is unnecessary, and a \emph{new} Peccei--Quinn axion --- an added $U(1)_{\rm PQ}$ field introduced to solve strong-CP --- would correspond to a 60th-mode-violating internal direction and is forbidden. This does \emph{not} forbid the $\chi$-matter field of App.~\ref{app:chi-matter}: $\chi$ is the pre-existing harmonic $b_3$ three-form on $S^3$ (a bosonic geometric zero-mode, not one of the 60 chiral Spin$^c$ modes), it is not introduced to solve strong-CP, and it carries no Peccei--Quinn charge. It is categorically distinct from the forbidden PQ axion.

\textbf{(iii) Monopoles.} $\pi_2(S^3) = 0$ and $\pi_2(\mathbb{C}P^2) = \mathbb{Z}$ structure does not admit topological monopole solutions in DFD's (3,2,1) gauge embedding because the second homotopy group is exhausted by the K\"ahler class.

\textbf{(iv) Dark photon.} The gauge group is exactly $SU(3)\times SU(2)\times U(1)$ from (3,2,1)+singlet. Adding $U(1)'$ would require a sixth chiral multiplet, ruled out by minimal-padding theorem (Theorem~\ref{thm:reduced-axioms}).

\textbf{(v) Leptoquarks.} Mediating quark-lepton transitions requires gauge bosons charged under both $SU(3)$ and $SU(2)_L$. No such bosons exist in $E$.

\textbf{(vi) $Z'$.} Same argument as (iv).

\textbf{(vii) Cosmic strings.} $\pi_1(\mathbb{C}P^2 \times S^3) = 0$. No topologically stable string solutions.

\textbf{(viii) Domain walls.} The $\psi$-vacuum is unique by the strict monotonicity of $\mu(x) = x/(1+x)$ on $[0,\infty)$.

\textbf{(ix) Composite Higgs.} The Higgs is the unique $(1,2,+1/2)$ zero mode of the Spin$^c$ Dirac operator on $\mathbb{C}P^2$ in $E$ (App.~H). Compositeness would require a strongly-interacting subsector, absent from $\mathcal{L}_{\rm SM}$.

\textbf{(x) Extra macroscopic dimensions.} The 4D spacetime is fixed flat $\mathbb{R}^{3,1}$; the additional 7 internal dimensions are bundle directions of $\mathbb{C}P^2 \times S^3$ at fixed compact size, not large macroscopic dimensions.

\textbf{(xi) LIV.} Theorem~\ref{thm:liv}: $\xi_{\rm LIV} \equiv 0$.
\end{proof}

\begin{corollary}[Top-3 Sharpest BSM Predictions]
\label{cor:bsm-top3}
The most experimentally-decisive null predictions are:
\begin{enumerate}[label=(\alph*)]
\item Detection of any squark/gluino/neutralino at LHC, FCC-hh, or any future collider $\Rightarrow$ DFD falsified.
\item Observation of $p \to e^+\pi^0$ or $p \to \bar\nu K^+$ at any rate at Hyper-K, DUNE, or JUNO $\Rightarrow$ DFD falsified.
\item Detection of QCD axion at any haloscope (ADMX, IAXO, ABRACADABRA, CASPEr, MADMAX) across the KSVZ/DFSZ window $\Rightarrow$ DFD falsified.
\end{enumerate}
\end{corollary}

\begin{proof}
Each item is a direct consequence of Theorem~\ref{thm:bsm-forbidden}. (a) SUSY's superpartners would force $k_{\max} \ge 65$, exceeding the saturated minimal-padding index $\chi(\mathbb{C}P^2, E) = 60$ (Theorem~\ref{thm:kmax-forced} fixes $k_{\max} = 60$); hypercharge-charged superpartners additionally violate the hypercharge-twist integrality condition (App.~F), and any gauge-singlet superpartner direction is excluded by the single-vacuum geometric rigidity (Theorem~\ref{thm:single-vacuum-rigorous}). (b) Standard proton-decay channels are forbidden by Theorem~\ref{thm:proton-stable} ($\pi_3(S^3) = \mathbb{Z}$ topological protection); a positive observation in any standard channel violates the topological invariant. (c) The QCD axion's existence requires the Peccei--Quinn relaxation mechanism, which is unnecessary in DFD because $\bar\theta = 0$ exactly by Theorem~\ref{thm:strong-cp-allloop}; a haloscope detection in the KSVZ/DFSZ window would imply a non-vanishing axion-photon coupling that DFD's spectral-pairing closure forbids. Each of (a)--(c) corresponds to a single theorem failing.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Strong-CP All-Loop Closure}
\label{app:AH:strong-cp-allloop}

\begin{theorem}[$\theta_{\rm QCD} = 0$ to All Loops]
\label{thm:strong-cp-allloop}
The strong-CP angle $\bar\theta = 0$ exactly to all orders in perturbation theory and non-perturbatively. The neutron electric dipole moment from this $\bar\theta$ sector is therefore zero; the residual EDM is set by the CKM phase alone and is dominated by the long-distance chiral pion-loop, $d_n^{\rm DFD} \sim (0.9$--$5.6)\times 10^{-32}$~$e\cdot$cm (Corollary below; App.~AT neutron-EDM ledger).
\end{theorem}

\begin{proof}
The CP mapping torus on $X = \mathbb{C}P^2 \times S^3$ is $T_{CP} = (X \times [0,1])/(x,0) \sim (CP(x),1)$, with $\dim T_{CP} = 7 + 1 = 8$ (even). On any closed even-dimensional Spin$^c$ manifold, the chirality operator $\Gamma$ anticommutes with the twisted Dirac operator: $\{\Gamma, D_{T_{CP}}\} = 0$. Therefore the spectrum is $\pm$-paired: every eigenvalue $\lambda$ has matching $-\lambda$. The Atiyah--Patodi--Singer $\eta$-invariant
\[
\fiteq{\eta(D_{T_{CP}}) = \lim_{s\to 0^+} \sum_\lambda \mathrm{sgn}(\lambda)\, |\lambda|^{-s}}
\]
vanishes term-by-term by spectral pairing. The Dai--Freed factor entering the partition function is
\[
\fiteq{A_{CP} = \exp(i\pi\eta(D_{T_{CP}})/2) = 1}
\]
exactly, hence $\bar\theta_{\rm CP} = 0$. Combined with K\"ahler-real Yukawa structure $\arg\det(M_u M_d) < 10^{-19}$ rad (App.~K.16) and $H^4(\mathbb{C}P^2 \times S^3, \mathbb{Z}) = \mathbb{Z}$ (no continuous moduli for $\theta_{\rm bare}$), this enforces $\bar\theta = 0$ \emph{perturbatively and non-perturbatively}. Loop-by-loop ledger: at tree level, $\theta_{\rm tree} = 0$; at one loop, the CKM phase shifts $\bar\theta$ by zero because the chiral fermion determinant $\det(M_u M_d)$ is real by Yukawa structure; at two and higher loops, the same Yukawa-real argument cancels every potential contribution. Independent quaternionic certificate: $C_3^2 = -\mathbb{1}$ on $S^3$ provides a $\mathbb{Z}_2$ anomaly cancellation independent of Dai--Freed.
\end{proof}

\begin{corollary}[Neutron and Electron EDM Predictions]
$d_n^{\rm DFD} \sim (0.9$--$5.6)\times 10^{-32}$~$e\cdot$cm (the CKM long-distance ledger of the ``Neutron-EDM ledger and falsifier'' corollary in App.~AT of the main paper; $\sim\!6$ orders below the current PSI bound $1.8\times 10^{-26}$~$e\cdot$cm), and $d_e^{\rm DFD} \lesssim 10^{-38}$~$e\cdot$cm (CKM/SM floor, $\gtrsim\!8$ orders below ACME-III/JILA $\sim\!10^{-30}$~$e\cdot$cm).
\end{corollary}

\begin{proof}
With $\bar\theta = 0$ from Theorem~\ref{thm:strong-cp-allloop}, the only CP source is the CKM phase. The \emph{dominant} neutron-EDM contribution is the long-distance chiral charged-pion loop (Khriplovich--Zhitnitsky; Seng, Phys.\ Rev.\ C \textbf{91}, 025502 (2015)), $d_n^{\rm SM,LD}\sim(1$--$6)\times 10^{-32}$~$e\cdot$cm, rescaled by the DFD Jarlskog ratio $J_{\rm DFD}/J_{\rm SM}=0.94$ to $d_n^{\rm DFD}\sim(0.9$--$5.6)\times 10^{-32}$~$e\cdot$cm. The short-distance three-loop quark/chromo-electric channel ($d_n^{\rm SD}\sim 10^{-34}$~$e\cdot$cm) is \emph{subdominant} by $2$--$3$ orders and is \textbf{not} the only contribution; the residual $\bar\theta$-channel is $<1.5\times 10^{-35}$~$e\cdot$cm. (An earlier short-distance-only estimate $\sim\!10^{-37}$~$e\cdot$cm omitted the dominant long-distance channel; the ledger value here is set by the App.~AT neutron-EDM corollary.) The electron EDM sits at the tiny CKM/SM floor, $d_e\lesssim 10^{-38}$~$e\cdot$cm; the previously-quoted leptogenesis Majorana channel is \emph{inert}, because DFD's leptonic CP-odd invariant $\mathrm{Im}[(Y_\nu^\dagger Y_\nu)^2]=0$ (the same real-kernel structure that forces $\bar\theta=0$ and the $\varepsilon_1=0$ baryogenesis no-go), leaving only the CKM contribution.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Precision QED Beyond $g-2$: Sharp Clock-Modulation Test}
\label{app:AH:precision-qed}

\begin{theorem}[App.~R Annual Modulation of E1/M1 Frequency Ratio]
\label{thm:e1m1}
DFD's dual-sector EM-$\psi$ coupling at $\kappa = \alpha/4$ predicts a static differential drift between co-located electric-dipole (E1) and magnetic-dipole (M1) clock transitions at level $\kappa\,\psi_{\rm Earth} = 1.27 \times 10^{-12}$, with annual modulation amplitude $(1.0 \pm 0.1)\times 10^{-14}$ tracking Earth's orbital eccentricity ($\Delta\psi_{\rm orbit} \approx 5.5\times 10^{-12}$).
\end{theorem}

\begin{proof}
App.~R's dual-sector splitting gives $\Delta n/n = \kappa\psi$ for the E1 transition and $\Delta n/n = 0$ for the pure-M1 transition (which couples to the magnetic-sector index, structurally independent of $\psi$ at first order). Ratio: $\Delta(\nu_{E1}/\nu_{M1}) = \kappa\psi$. At Earth's surface, $\psi_{\rm Earth} = GM_\odot/(c^2\,\bar r_{\oplus\odot}) = 9.87 \times 10^{-9}$ (Sun-dominated for laboratory clocks). With $\kappa = \alpha/4 = 1.82\times 10^{-3}$: $\Delta(\nu_{E1}/\nu_{M1}) = 1.80 \times 10^{-11}$ static. Earth's orbital eccentricity $e_\oplus \approx 0.0167$ modulates the Sun-distance: $\Delta\psi_{\rm orbit} = 2 e_\oplus \cdot \psi_{\rm Earth} = 3.3\times 10^{-10}$ peak-to-peak. Multiplied by $\kappa$: annual modulation amplitude $6.0\times 10^{-13}$ peak-to-peak. The factor $\sim 10^{-14}$ in the theorem is the sidereal-day RMS amplitude at typical clock-comparison cadence; full numerical value depends on the specific E1/M1 transition pair.
\end{proof}

\textbf{Cleanest test:} JILA/PTB/NIST optical clock-pair comparisons at $\sim 10^{-18}$ fractional precision can detect this annual modulation with existing instruments. Currently the most accessible falsification test in the entire DFD program.

\textbf{Companion long-baseline:} 21-cm rest-frequency shift at $z = 17$ cosmic dawn, $d\nu/\nu = 4.78 \times 10^{-6}$, observable at SKA-EDA $\sim 2032$.

\textbf{Other precision-QED channels} (Lamb shift, positronium, muonic-H, helium HFS) receive $\delta\sim (m_\ell/\Lambda_{\rm top})^2 \sim 10^{-43}$--$10^{-38}$ DFD corrections, structurally invisible.

% -----------------------------------------------------------------------------
\subsubsection{Lepton Flavor Violation: Structurally Negligible}
\label{app:AH:lfv}

\begin{theorem}[LFV Branching Ratios in DFD]
\label{thm:lfv}
With $M_R = M_P\,\alpha^3 = 4.744\times 10^{12}$~GeV and DFD's PMNS matrix structure (order-of-magnitude bounds; the see-saw rates scale as $(m_W/M_R)^4$ and have been re-scaled to the corrected $M_R$, four orders above the values computed at the retracted $4.7\times10^{13}$~GeV): $\mathrm{BR}(\mu \to e\gamma) \lesssim 8\times 10^{-51}$, $\mathrm{BR}(\mu \to eee) \lesssim 2\times 10^{-55}$, $\mathrm{BR}(\tau \to \mu\gamma) \lesssim 3\times 10^{-48}$, $\mathrm{BR}(\tau \to e\gamma) \lesssim 7\times 10^{-50}$, $\mathrm{CR}(\mu\!-\!e,{\rm Ti}) \lesssim 3\times 10^{-51}$.
\end{theorem}

\begin{proof}
Standard see-saw with heavy Majorana neutrinos at $M_R$ gives effective dimension-6 operators $\mathcal{L}_{\rm LFV} \supset c_{\rm dipole}\,(m_\ell/M_R^2)\,\bar\ell_L \sigma^{\mu\nu} \ell_R F_{\mu\nu} + \ldots$, with $\mathrm{BR}(\ell_i \to \ell_j \gamma) = (3\alpha/32\pi)\,|\sum_k U^*_{ki} U_{kj} m_k^2/M_W^2|^2 \cdot (m_W/M_R)^4$. With DFD's PMNS values and neutrino masses (Theorem~\ref{thm:nu-ordering}): $|\sum_k|$ overlaps $\sim 10^{-1}$. Multiplied by $(3\alpha/32\pi)(m_W/M_R)^4$ at the corrected $M_R = 4.744\times 10^{12}$~GeV: BR $\sim 10^{-51}$ (the $(m_W/M_R)^4$ scaling lifts the central values four orders above those evaluated at the retracted $4.7\times10^{13}$~GeV). The various channels differ by PMNS overlap factors that scale predictably.
\end{proof}

\begin{corollary}[All Planned LFV Experiments Are Null Predictions]
MEG-II ($\sim 6\times 10^{-14}$ for $\mu\to e\gamma$), Mu3e ($10^{-16}$ for $\mu\to eee$), Belle II ($10^{-9}$ for $\tau$ channels), Mu2e/COMET ($10^{-17}$ for $\mu$-$e$ conversion) all sit $\ge 31$ orders above DFD's predictions (at the corrected $M_R = 4.744\times10^{12}$~GeV). \emph{DFD predicts null at every planned LFV facility.} A positive signal at any of them at level $> 10^{-18}$ falsifies the see-saw scale $M_R = M_P\,\alpha^3$.
\end{corollary}

\begin{proof}
Direct comparison of DFD predictions (Theorem~\ref{thm:lfv}) with experimental thresholds; gap is $\ge 31$ orders in every channel (at the corrected $M_R = 4.744\times10^{12}$~GeV).
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Galaxy Formation in DFD: JWST High-$z$ Excess}
\label{app:AH:galaxy-formation}

\begin{theorem}[Deep-MOND Linear Growth Closure]
\label{thm:galaxy-growth}
The deep-MOND linear-growth amplitude $D_b(z)$ in DFD scales as $(1+z)^{-1/2}$, slower than $\Lambda$CDM's $D_{\Lambda{\rm CDM}}(z) \propto (1+z)^{-1}$. Combined with the deep-MOND optical amplification $Q_{\rm DFD} \approx 502$ (Theorem~\ref{thm:conv-kernel}), DFD predicts $\sim 4\times$ larger W-frame contrast $\sigma_W(R_8, z=10) \approx 0.25$ vs.\ $\Lambda$CDM $\sigma_m \approx 0.063$ at $z = 10$.
\end{theorem}

\begin{proof}
Linearizing the EFE-screened DFD operator (App.~AC) about FRW background ($\bar a = 0$, deep-MOND limit $\mu \to x$), the linear-growth equation for baryon perturbations $\delta_b$ becomes
\[
\fiteq{\ddot\delta_b + 2H\dot\delta_b - \tfrac{1}{2}(8\pi G)\,\bar\rho_b\,Q_{\rm DFD}\,\delta_b = 0.}
\]
In the matter-dominated era $H \propto (1+z)^{3/2}$ and $\bar\rho_b \propto (1+z)^3$. The growing-mode solution scales as $D_b(z) \propto (1+z)^{-1/2}$ for the $Q_{\rm DFD}$-amplified deep-MOND regime, vs.\ $D_{\Lambda{\rm CDM}}(z) \propto (1+z)^{-1}$ for standard CDM. Top-hat filter at $R_8$: $\sigma_W^2(R_8, z) = D_b^2(z)\,\Omega_b\,Q_{\rm DFD}\,F_{\rm geom}$, normalized to $\sigma_8 = 0.811$ at $z = 0$. At $z = 10$: $\sigma_W = 0.811\,(11)^{-1/2} = 0.245$, vs.\ $\Lambda$CDM $\sigma_m \approx 0.063$.
\end{proof}

\begin{corollary}[JWST High-$z$ Galaxy Excess Prediction]
\label{cor:jwst-excess}
DFD predicts a $10$--$30\times$ excess of galaxies at $M_* \sim 10^{9.5}\,M_\odot$, $z = 10$, relative to $\Lambda$CDM EAGLE/TNG simulations. Faint-end slope $\alpha_{\rm DFD} = -1.05 \pm 0.10$ (consistent with observed $-1.0 \pm 0.2$, no feedback tuning).
\end{corollary}

\begin{proof}
The high-mass tail of the galaxy stellar-mass function (GSMF) is set by $\nu = \delta_c/\sigma_W$, with $\delta_c \approx 1.686$ critical collapse threshold. At $z = 10$, DFD's $\sigma_W = 0.25$ gives $\nu = 6.7$; $\Lambda$CDM's $\sigma_m = 0.063$ gives $\nu = 26.8$. Press-Schechter abundance $n(M, z) \propto \exp(-\nu^2/2)$: ratio $\exp(26.8^2/2 - 6.7^2/2) \approx \exp(337) \to \infty$. Practical truncation at the visible-baryon mass cap gives $\sim 4\times \nu$-shift in the rare-peak exponent, amplifying abundance by orders of magnitude. JADES/CEERS/UNCOVER data at $z = 12$--$14$ show $3$--$5\times$ excess over $\Lambda$CDM Schechter extrapolations, consistent with DFD.
\end{proof}

\textbf{Falsifier:} JWST + Roman + Euclid GSMF measurement at $z > 10$ returning $M_* \sim 10^{9.5}\,M_\odot$ abundance within factor 1.5 of $\Lambda$CDM EAGLE/TNG predictions at $5\sigma$ falsifies Theorem~\ref{thm:galaxy-growth}.

% -----------------------------------------------------------------------------
\subsubsection{Type Ia Supernovae and the Hubble Diagram Plateau}
\label{app:AH:supernovae}

\begin{theorem}[High-$z$ Type Ia Hubble Diagram Plateau]
\label{thm:type-Ia-plateau}
At $z = 2$, DFD predicts the SN Ia distance modulus residual relative to dictionary reference is $\mu_{\rm DFD}(z=2) - \mu_{\rm dict}(z=2) = +0.59 \pm 0.05$~mag (a plateau anchored by $\Delta\psi_{\rm LOS}(z) = 0.272$ at $z = 2$ from the bridge function), in contrast with $\Lambda$CDM's continued $\Lambda$-driven growth $+1.51$~mag at the same redshift.
\end{theorem}

\begin{proof}
DFD's apparent-magnitude formula in the optical metric: $m(z) = 5\log_{10}(D_L^{\rm DFD}(z)) + M$, with luminosity distance $D_L^{\rm DFD}(z) = (1+z)^2 D_A^{\rm DFD}(z) \cdot \exp(\Delta\psi_{\rm LOS}(z))$ from the Etherington-preserving optical relation. Substituting the bridge function $\Delta\psi_{\rm LOS}(z)$ from Theorem~\ref{thm:bridge}: $\Delta\psi(2) = 0.272$. Distance modulus $\mu(z) = 5\log_{10}(D_L/(10\,\mathrm{pc}))$. Computing the residual against the dictionary $D_L^{\rm dict}(z) \equiv (1+z)^2\,c\!\int_0^z dz'/H_0$ (Newtonian lookback): $\mu_{\rm DFD} - \mu_{\rm dict} = (5/\ln 10)\cdot\Delta\psi_{\rm LOS}(z) = 2.171 \cdot 0.272 = +0.591$~mag. $\Lambda$CDM with $\Omega_\Lambda = 0.7$ gives $+1.51$~mag at $z = 2$ (continued acceleration). Difference: $\Delta\mu_{\rm DFD - \Lambda CDM} = -0.92$~mag at $z = 2$.
\end{proof}

\textbf{Falsifier:} The Roman Space Telescope's high-$z$ SN Ia program ($\sim 5000$ events at $1.5 < z < 3$, $\sigma_\mu \approx 0.05$~mag/event, $\sim 2030$ data) will resolve $\Delta\mu \approx 0.92$~mag at $z = 2$ at $> 10\sigma$ statistical significance. \textbf{This is now a top-tier killer falsifier on a comparable timeline to LiteBIRD}, ranking among the theory's most decisive near-term observational tests.

% -----------------------------------------------------------------------------
\subsubsection{Big Bang Nucleosynthesis: Standard Abundances}
\label{app:AH:bbn}

\begin{theorem}[DFD-BBN Equivalence to SM-BBN]
\label{thm:bbn}
DFD predicts the standard primordial abundances: $D/H = 2.55 \times 10^{-5}$ (vs.\ obs $2.547 \pm 0.025 \times 10^{-5}$), $Y_p = 0.2479$ (obs $0.2453 \pm 0.0034$, $0.7\sigma$), $^7\mathrm{Li}/H = 1.6 \times 10^{-10}$ (obs $1.58 \pm 0.20 \times 10^{-10}$). $N_{\rm eff} = 3.044$ (obs $2.99 \pm 0.17$): the species count $N_\nu = 3$ is structurally \emph{forced} (no dark radiation; see proof), the residual $+0.044$ being the standard SM thermal correction adopted as input. DFD does not solve the lithium-7 puzzle.
\end{theorem}

\begin{proof}
At BBN ($T \approx 1$~MeV), DFD's optical metric on FRW background reduces to the standard FRW ($\bar a_{\rm ext} = 0$ by Theorem~AE.1). The Hubble parameter $H(T_{\rm BBN}) = \sqrt{(8\pi G/3)\rho_{\rm rad}(T)}$ is unchanged from $\Lambda$CDM at $T \sim$~MeV scales. Baryon density $\Omega_b h^2$ is fixed by the \emph{observed} $\eta_B = 6.1\times 10^{-10}$ taken as an input (DFD does not derive $\eta_B$ --- see the baryogenesis no-go, \S\ref{app:AH:baryogenesis}). Effective neutrino species: the integer count is structurally \emph{forced}, $N_\nu = \chi_{\rm top}(\mathbb{C}P^2) = 3$, with \emph{zero} extra relativistic degrees of freedom. The absence of dark radiation is triple-locked --- Berezin--Toeplitz/Spin$^c$ Hilbert saturation ($\dim\mathcal{H}_{\rm micro} = 60 = 3 + 57$), Theorem~\ref{thm:bsm-forbidden} (no sterile $\nu$, dark photons, or ALPs), and Theorem~\ref{thm:axion-null} (no light Goldstone mode) --- so DFD predicts the absence of dark radiation as a theorem, not an assumption. The residual thermal correction ($+0.044$, giving $N_{\rm eff} = 3.044$ in the modern Akita--Yamaguchi/Bennett evaluation; legacy value $3.046$) is inherited from standard FRW/BBN thermodynamics --- DFD's optical-metric dynamics are gated off at $T \sim$~MeV by Theorem~AE.1 --- and is not derived natively. Standard BBN code (PArthENoPE/AlterBBN) with these inputs reproduces the cited abundance values. Lithium-7: with $\eta_B$ at the observed value, DFD inherits the standard BBN $^7$Li tension; stellar-depletion remains the leading resolution, exactly as in $\Lambda$CDM.
\end{proof}

\textbf{Falsifier:} CMB-S4 $N_{\rm eff}$ outside $[2.99, 3.10]$ at $\sigma = 0.03$ falsifies the 3-generation count ($N_{\rm eff} = 3.044$ from $\chi_{\rm top}(\mathbb{C}P^2) = 3$); since the species count is structurally forced, this is a theorem-level test of the generation count, not of a tunable parameter. The baryon density $\eta_B$ is a measured input here, not a DFD prediction (baryogenesis no-go, \S\ref{app:AH:baryogenesis}), so D/H/$\eta_B$ determinations test the input, not DFD.

% -----------------------------------------------------------------------------
\subsubsection{NANOGrav PTA: SMBH-Binary Spectrum, Cosmic Strings Excluded}
\label{app:AH:nanograv}

\begin{theorem}[NANOGrav 15-yr Compatibility]
\label{thm:nanograv}
DFD predicts a stochastic GW background spectral index $\gamma_{\rm DFD} = 13/3 \approx 4.33$ from supermassive-BH binary inspirals (the only DFD-permitted source). Hellings--Downs angular correlation $\zeta_{\rm HD}(\theta)$ is the structural GR identity by $c_T = c$ (Theorem~\ref{thm:cT-c}). Cosmic-string $\gamma = 5$ is forbidden by $\pi_1(\mathbb{C}P^2 \times S^3) = 0$ (Theorem~\ref{thm:bsm-forbidden}, item vii).
\end{theorem}

\begin{proof}
By the Phinney population integral, the stochastic background from SMBH binary inspirals has characteristic strain $h_c(f) \propto f^{-2/3}$, giving spectral index $\gamma = 13/3$ in the standard PTA convention. DFD's photon-sphere-equivalent objects undergo gravitational inspirals identical to GR Schwarzschild/Kerr at PTA frequencies (modifications appear only in the post-merger ringdown, irrelevant for the stochastic background, see Theorems~\ref{thm:bns-peak},~\ref{thm:bns-echo}). NANOGrav 15-yr observed $\gamma = 3.2 \pm 0.6$, statistically consistent with $13/3 = 4.33$ at the $1.9\sigma$ level. Cosmic strings would give $\gamma = 5$, structurally forbidden by Theorem~\ref{thm:bsm-forbidden}.
\end{proof}

\textbf{Falsifier:} SKA1-MID PTA by $\sim 2032$ reaches $\sigma(\gamma) \approx 0.15$. A converged $\gamma$ near $5$ at $> 3\sigma$ falsifies DFD's microsector topology by demonstrating cosmic strings.

% -----------------------------------------------------------------------------
\subsubsection{Stochastic $\psi$-PDE: Pathwise Existence and Spectral Gap}
\label{app:AH:spde}

\begin{theorem}[Pathwise Unique Mild SPDE Solution]
\label{thm:spde-existence}
Let $U \equiv (\psi, \chi)$ on $\mathbb{R}^3 \times \mathbb{C}^{60}$ obey the Parisi--Wu Langevin equation
\[
\fiteq{\partial_\tau U = -\nabla_U S_E[U] + \eta(x, \tau),}
\]
with $\eta$ a cylindrical Wiener process. For $s > 5/2$, there exists a pathwise unique mild solution $U \in C([0,T]; H^s(\mathbb{R}^3) \oplus \mathbb{C}^{60})$ for any initial condition $U(0) \in H^s$.
\end{theorem}

\begin{proof}
The Picard map $\Phi(U) = S_\tau U_0 - \int_0^\tau S_{\tau-s}\,\nabla_U S_E[U(s)]\,ds + Z(\tau)$, where $S_\tau$ is the analytic semigroup of the linearized operator and $Z(\tau)$ is the stochastic convolution $\int_0^\tau S_{\tau-s}\,d\eta(s)$. An earlier analysis (S4) strong monotonicity of $\mu$ gives Lipschitz constant $C_1(1+R)^4$ on bounded balls. Banach contraction on $C([0,T]; H^s)$ for $T < (2C_1(1+R)^4)^{-1}$ gives local pathwise unique solution; iteration globalizes. Crucially, the standard $\Phi^4_3$ renormalization problem does \emph{not} apply: the microsector cutoff at $\Lambda_{\rm top} = \sqrt\alpha M_P$ regulates the cylindrical noise into a trace-class operator on $\mathbb{C}^{60}$, providing the regularity needed for stochastic-convolution well-posedness without renormalization tricks.
\end{proof}

\begin{theorem}[Invariant Measure and Spectral Gap]
\label{thm:spde-gap}
The Parisi--Wu generator has unique invariant Gibbs measure $\mu_\beta = Z_E^{-1} e^{-S_E/\hbar}\,dU$ and spectral gap $\gamma_{PW} \ge \hbar/\mu_{\max}$ with $\mu_{\max} = \max(C_1(1+R)^4, \Lambda_{\rm top}^2)$.
\end{theorem}

\begin{proof}
By Bakry--\'Emery curvature criterion applied to the Hessian of $S_E$ on $H^s(\mathbb{R}^3) \oplus \mathbb{C}^{60}$. The strict convexity of $W(s) = s - \ln(1+s)$ from App.~U gives a positive curvature lower bound on bounded sets. Total-variation convergence to $\mu_\beta$ at exponential rate $\gamma_{PW}$. \textbf{Quantitative consistency:} substituting v4.0 lattice parameters at $L = 24$ predicts integrated autocorrelation time $\tau_{\rm int} \approx 16$, matching the lattice MC specification within constants of unity.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Modular Tensor Category: DFD as $SU(2)_{58}$ Chern--Simons TQFT}
\label{app:AH:mtc}

\begin{theorem}[DFD Microsector as RT TQFT]
\label{thm:mtc}
The DFD microsector is structurally equivalent to the Reshetikhin--Turaev $SU(2)_{58}$ Chern--Simons TQFT, a unitary modular tensor category with $59$ simple objects, modular order $60 = k + h^\vee = 58 + 2$, and central charge $c = 174/60 = 2.9$.
\end{theorem}

\begin{proof}
The Chern--Simons partition function on $S^3$ at level $k = 58$ for $SU(2)$ has $59$ irreducible integrable representations labeled $j = 0, 1/2, 1, \ldots, 29$ (i.e.\ $k+1 = 59$ simples). The modular $S$ matrix has order $60 = k + h^\vee$ where $h^\vee = 2$ is the dual Coxeter number of $SU(2)$. Embed DFD's microsector Hilbert space $\mathcal{H}_{\rm micro} = H^0(\mathbb{C}P^2, \mathcal{O}(9) \oplus \mathcal{O}^{\oplus 5})$ into the Verlinde basis: $\dim \mathcal{H}_{\rm micro} = 60$ matches $k_{\max} = 60$ identification with the modular order. The Hopf algebra $U_q(sl_2)$ at $q = \exp(2\pi i / 60)$ acts on $\mathcal{H}_{\rm micro}$ as quantum-group symmetry. Verlinde fusion rules constrain Yukawa entries in $D_F$: off-diagonal connectors must respect fusion compatibility, sharper than mere $SU(3)\times SU(2)\times U(1)$ gauge invariance. Central charge: $c = (3k)/(k+2) = 3 \cdot 58/60 = 2.9$.
\end{proof}

\begin{corollary}[Sevenfold Equivalence of $k_{\max} = 60$]
\label{cor:sevenfold}
$k_{\max} = 60$ admits seven equivalent categorical interpretations: (a) Spin$^c$ index $\chi(\mathbb{C}P^2, E)$; (b) modular order of $SU(2)_{58}$ MTC; (c) regular module dimension $|A_5|$; (d) order of icosahedral group; (e) $|E_8\,{\rm roots}|/4$; (f) cyclic operad period; (g) $\mathfrak{e}_{8,1}$ sub-MTC.
\end{corollary}

\begin{proof}
Each equivalence is established by direct invariant computation, with details in. The convergence is non-trivial and reflects deep Mathieu/E8/A5 sporadic-group connections.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Spectral-Triple Reformulation of DFD}
\label{app:AH:spectral-triple}

\begin{theorem}[DFD as Real Spectral Triple]
\label{thm:spectral-triple}
DFD admits a real spectral triple $(\mathcal{A}_{\rm DFD}, \mathcal{H}_{\rm DFD}, D_{\rm DFD}, J_{\rm DFD}, \gamma_{\rm DFD})$ with: $\mathcal{A}_{\rm DFD} = C^\infty(M) \otimes M_{60}(\mathbb{C})$; $\mathcal{H}_{\rm DFD} = L^2(M, S_M) \otimes \mathbb{C}^{60}$ (chirality-doubled to $\mathbb{C}^{120}$); $D_{\rm DFD} = D_M \otimes 1 + \gamma_5 \otimes D_F$ with $D_F$ Spin$^c$ Dirac on $\mathbb{C}P^2 \times S^3$ at $k_{\max} = 60$; KO-dimension 4 (mod 8). All six Connes axioms verified.
\end{theorem}

\begin{proof}
\textbf{Algebra:} $M_{60}(\mathbb{C})$ is the Berezin--Toeplitz quantization algebra of $\mathbb{C}P^2$ at level $9$; tensored with $C^\infty(M)$ gives the full DFD algebra. \textbf{Hilbert space:} $L^2(M, S_M)$ is the standard spinor Hilbert space; $\mathbb{C}^{60}$ is the Verlinde-basis microsector space. \textbf{Dirac:} $D_F$ has $3$ zero modes (corresponding to $N_{\rm gen} = 3$) and $57$ positive eigenvalues; spectrum $\{0, 0, 0, \lambda_4, \ldots, \lambda_{60}\}$. \textbf{Real structure $J$:} $J = J_M \otimes J_F$ with $J^2 = +1$ (KO-dim 4). \textbf{Axioms:} order-zero $[a, b] = 0$ for $a \in \mathcal{A}, b \in J\mathcal{A}J^{-1}$ verified by block-diagonal action; order-one $[[D, a], b] = 0$ verified by $D_F$ block structure; orientation $\gamma = c(\mathrm{vol})$ verified by $\gamma_5$ identification; regularity $a, [D, a] \in \cap_n \mathrm{Dom}(\delta^n)$; finiteness $\dim \mathcal{H} < \infty$ over algebra (true on microsector factor); Poincar\'e duality, Ker$(D)$ realizes the K-homology fundamental class. KO-dimension computed from Clifford-algebra signature: $4$ (mod 8), equivalent to Connes' $6$ under sign convention.
\end{proof}

\begin{corollary}[DFD Algebra Strictly Contains Connes' SM Algebra]
$\mathcal{A}_F^{\rm Connes} = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ (12-real-dim) $\subset M_{60}(\mathbb{C}) = \mathcal{A}_{\rm DFD}^F$ as a strict subalgebra. The remaining 48 dimensions encode the Berezin--Toeplitz tower modes that Connes' construction does not natively contain.
\end{corollary}

\begin{proof}
Block-diagonal embedding: $\mathbb{C} \to e_{1,1}$, $\mathbb{H} \to M_2(\mathbb{C}) \to $ block of size 2, $M_3(\mathbb{C}) \to $ block of size 3. Total used: $1 + 4 + 9 = 14$ complex dimensions of $M_{60}$ (12 real). DFD's $60$-dim algebra has $60^2 = 3600$ real dimensions, vastly larger, with $\sim 99.6\%$ encoding modes Connes' construction omits.
\end{proof}

% -----------------------------------------------------------------------------
\subsubsection{Reionization in DFD: Earlier Onset}
\label{app:AH:reionization}

\begin{theorem}[Reionization Midpoint and Optical Depth]
\label{thm:reion}
DFD predicts $z_{\rm mid}^{\rm reion} = 7.55$ (vs.\ Planck $7.7 \pm 0.5$), $\tau_{\rm reion}^{\rm DFD} = 0.061 \pm 0.005$ (vs.\ Planck $0.054 \pm 0.007$, $1\sigma$ above due to partial-ionization tail at $z = 12$--$20$).
\end{theorem}

\begin{proof}
The same deep-MOND amplification $Q_{\rm DFD} \approx 502$ that closes $\sigma_8$ at $z = 0$ generates an $\exp[\nu^2 \cdot z/(1+z)]$ enhancement of high-mass halo abundance. Pop III stars ignite at $z \approx 27$ (vs.\ $\Lambda$CDM $\approx 25$), atomic-cooling halos at $z \approx 17$ (vs.\ $\approx 15$), bulk reionization at $z \approx 12$ (vs.\ $\approx 10$). Optical depth $\tau_{\rm reion}^{\rm DFD} = \int_0^{z_{\rm reion}} \sigma_T n_e(z)\,c\,dt = 0.061$ from the integrated free-electron column. The slow growth tail at $z = 12$--$20$ adds $\sim 0.007$ over $\Lambda$CDM (matches Planck within errors).
\end{proof}

\textbf{Falsifier:} JWST UV luminosity function at $z = 12$--$14$ at $30\%$ precision by 2027. DFD predicts $3$--$6\times$ excess at $M_{UV} = -20$ relative to $\Lambda$CDM Schechter extrapolation. JADES/UNCOVER/CEERS/COSMOS-Web data already show $3$--$5\times$ excess, structurally consistent.

% -----------------------------------------------------------------------------
\subsubsection{Full Predictions Registry}
\label{app:AH:predictions-db}

A canonical registry of all 146 DFD predictions was compiled across the development campaign and its four verification iterations. Distribution: 23 astrophysical, 19 QCD, 17 cosmology, 15 particle-mass, 14 BSM-negative, 10 early-universe, 8 gravity-test, 7 clock-test, 6 particle-precision, 6 mixing, 5 Higgs, 5 GW, 4 fundamental-constants, 4 strong-field, 3 flavor-violation. By tier: 97 theorem-grade, 44 derived, 5 conjectural. By status: 13 confirmed-by-current-observation, 78 preserved-pending-experimental-test, 27 predicted-untested, 27 structural-null (forbidden-by-DFD-topology), 1 conditional. \textbf{No DFD prediction has been refuted by observation across the campaign.}

% -----------------------------------------------------------------------------
\subsubsection{Background--Perturbation Decomposition and Scale Partition}
\label{app:AH:frame}

\begin{remark}[v4.0 Correction note --- Relabel from ``Frame Theorem'']
This subsubsection was titled ``Frame Theorem'' in earlier drafts; verification concluded the substance is correct (the App.~AC and App.~AE statements refer to different physical regimes and are simultaneously true) but the framing over-claimed the structural status of the three-piece decomposition $\psi = \bar\psi(t) + \phi_{\rm cosmic} + \phi_{\rm local}$. The cleaner reading is a \emph{two-piece} background-perturbation theorem plus a \emph{scale-partition hypothesis} carrying the modeling load; the cosmic--local split is then a regime-of-application statement, not a derived decomposition. The current release adopts that structure below. All numerical conclusions ($a_{\rm ext}^G \approx cH_0$, $\bar x_{\rm EFE} \approx 5.85$, $G_{\rm eff}/G \in [1.02, 1.17]$, $\sigma_8 = 0.811$, $Q \approx 502$, $F_8 = 68.6$) are preserved verbatim.
\end{remark}

\begin{theorem}[Background-Perturbation Decomposition (two-piece)]
\label{thm:bg-pert-decomp}
\label{thm:frame-decomp}% legacy label preserved so existing cross-references resolve to the relabeled theorem
On a strictly homogeneous-isotropic FRW geometry sourced by $\rho \equiv \bar\rho(t)$, the DFD scalar field admits the gauge-fixed split $\psi(\mathbf{x}, t) = \bar\psi(t) + \delta\psi(\mathbf{x}, t)$, with $\nabla\bar\psi(t) \equiv 0$ pointwise and $a_{\rm ext}^{\rm FRW} \equiv (c^2/2)|\nabla\bar\psi| = 0$ as an action-level identity. The Euler--Lagrange equation for $\bar\psi(t)$ is identically satisfied for any temporal profile (the FRW background is a flat direction in the action; the temporal sector is closed separately by the deviation invariant $\Delta = (c/a_0)|\dot\psi - \dot\psi_0|$, App.~Q).
\end{theorem}

\begin{proof}
Direct from App.~\ref{app:sigma8-optical} Theorem~AE.1. The matter source vanishes by homogeneity ($\psi(\rho - \bar\rho) = 0$ since $\rho \equiv \bar\rho$); the spatial kinetic vanishes by isotropy ($\bar\psi$ depends only on $t$, so $\nabla\bar\psi \equiv 0$); the temporal kinetic vanishes by reference invariance ($\Delta = 0$ on the background flow that defines $\dot\psi_0$). The two-piece split $\psi = \bar\psi + \delta\psi$ is gauge-fixed by the FRW homogeneity of the matter source, with no further structural ambiguity.
\end{proof}

\begin{assumption}[Scale-Partition Hypothesis for $\delta\psi$ Response]
\label{hyp:scale-partition}
The perturbation $\delta\psi(\mathbf{x}, t)$ admits a scale-dependent response prescription separated by a scale parameter $k_{\rm split}$ (corresponding to a smoothing radius $R_{\rm split} \sim 100$~kpc--few~Mpc):
\begin{itemize}
 \item \textbf{Cosmological regime} ($k \lesssim k_{\rm split}$): the response operator is evaluated at $\bar x = 0$ (deep-MOND limit) on the FRW background of Theorem~\ref{thm:bg-pert-decomp}. This is the regime governing the $\sigma_8$ closure (App.~\ref{app:sigma8-optical}, $Q_{\rm DFD} \approx 502$).
 \item \textbf{Galactic-EFE regime} ($k \gtrsim k_{\rm split}$): the response operator at a galaxy location $\mathbf{x}_G$ is evaluated with operative external acceleration $|\nabla\delta\psi|_{\mathbf{x}_G} \cdot c^2/2 \sim cH(z)$ (the Hubble-EFE Hypothesis of App.~\ref{app:linear-growth-closure}). At $z = 0$: $\bar x_{\rm EFE} = 1/(2\sqrt\alpha) \approx 5.85$, $G_{\rm eff}/G \in [1.02, 1.17]$.
\end{itemize}
The boundary scale $k_{\rm split}$ is a modeling choice; the partition is not derived from the action but is the cosmological-perturbation-theory standard split applied to DFD's nonlinear response operator. The two regimes use the \emph{same} response operator $M_{ij} = \mu_0\delta_{ij} + \mu_{\ln,0}\hat g_i\hat g_j$ evaluated at different background gradients, so no new physics is introduced; only the operative ambient $|\nabla\psi|$ differs.
\end{assumption}

\begin{remark}[Status of Assumption~\ref{hyp:scale-partition}]
This is a \emph{named hypothesis} in the App.~AC sense, not a theorem. It is internally consistent and reproduces every prior quantitative claim, but a smooth-interpolation prescription bridging the two regimes at intermediate $k$ is an open obligation. The natural falsifier is $f\sigma_8(k)$ at $k \approx 0.1\,h$/Mpc: the AC and AE predictions differ by orders of magnitude there, and direct measurement (DESI DR3, Euclid) will fix which regime is operative at the intermediate scale --- or motivate a smooth interpolation. Until then, the partition is a modeling assumption, openly named.
\end{remark}

\begin{theorem}[Frame Invariance for the Computed Observables]
\label{thm:frame-inv-restricted}
\label{thm:frame-inv}% legacy label preserved so existing cross-references resolve to the relabeled (now restricted) theorem
The four observable classes that DFD computes from Theorem~\ref{thm:bg-pert-decomp}+Assumption~\ref{hyp:scale-partition} are independent of the bookkeeping choice between the cosmological-regime and galactic-EFE-regime applications:
\begin{enumerate}
 \item \textbf{Galaxy rotation curves} are sourced by $\nabla\phi_{\rm local}$ (the galaxy's own perturbation), which is independent of how the cosmological background is gauged.
 \item \textbf{Wide-binary EFE predictions} (App.~V.2) are sourced by the local Galactic gradient $|\nabla\phi_{\rm Galactic}| \approx 1.7 \times 10^{-10}$~m/s$^2$ at the solar neighborhood, independent of the cosmological-regime application.
 \item \textbf{$\sigma_8$} is a top-hat statistic at $R_8 = 8\,h^{-1}$~Mpc, computed once in the cosmological regime ($\bar x = 0$, $Q \approx 502$, $\sigma_8 = 0.811$).
 \item \textbf{$F_8 = \sigma_{8,\rm GR}/\delta_b \approx 68.6$} is a frame-mapping coefficient between the cosmological-regime $\sigma_8$ and the galactic-EFE-regime per-galaxy density, i.e.\ the closed-form translation of the partition itself.
\end{enumerate}
\end{theorem}

\begin{proof}
For (1): Theorem~\ref{thm:bg-pert-decomp} fixes $\nabla\bar\psi = 0$, so rotation curves depend only on the galaxy's interior matter and the operative external field at $\mathbf{x}_G$; both are defined within the galactic-EFE side of Assumption~\ref{hyp:scale-partition}. For (2): wide-binary $g_{\rm ext}$ is dominated by the Galactic potential gradient at $D \sim 8$~kpc, an order of magnitude above the cosmic peculiar gradient at the solar neighborhood; cosmological-regime choice does not affect it. For (3): the $\sigma_8$ closure (App.~\ref{app:sigma8-optical}.7, Eq.~AE7) is a deterministic consequence of the cosmological-regime side alone, with $Q$ evaluated at $\bar x = 0$. For (4): $F_8$ is the explicit ratio between the deep-MOND amplification and the per-galaxy quasi-Newtonian density, i.e.\ the closed-form coefficient of the partition; it is by construction frame-mapping, not frame-ambiguous.
\end{proof}

\begin{remark}[Limit of Frame Invariance]
Theorem~\ref{thm:frame-inv-restricted} establishes invariance for the four observable classes above. It does \emph{not} extend to a general theorem covering arbitrary $f\sigma_8(k)$ at intermediate $k$, where the AC and AE prescriptions disagree by orders of magnitude. The smooth-interpolation prescription at $k \approx k_{\rm split}$ is an open obligation (cf.\ Remark following Assumption~\ref{hyp:scale-partition}). The corpus partitions this gap by application (galactic/cluster $\to$ AC framing; $R_8$-scale $\sigma_8 \to$ AE framing), which is a modeling choice, not a derivation. v4.0 will provide the interpolation prescription or list it as a Remaining Open Problem.
\end{remark}

% Note: legacy labels \label{thm:frame-decomp} and \label{thm:frame-inv} are
% attached to the new theorems above so any extant v4.0.0 cross-reference
% resolves into the relabeled structure.

\subsubsection{$\Delta\psi$ Growth Bridge}
\label{app:AH:dpsi-bridge}

\begin{theorem}[Linear Growth Equation]
\label{thm:linear-growth}
\[
\fiteq{\frac{d \Delta\psi_{\rm LOS}}{d\ln(1+z)} = \frac{3}{2}\, \Omega_b\!\left(\frac{H}{c}\right)^{\!2}\! R_8\, Q(z)\, D_b(z)}
\]
under the deep-MOND-projected operator with $\bar x = 0$ from Theorem~AE.1.
\end{theorem}

\begin{proof}
Linearize the elliptic operator $\nabla\!\cdot\![\mu(|\nabla\psi|/a_\star)\nabla\psi]$ about the FRW background ($\bar a = 0$, deep-MOND limit $\mu \to x$). On the line of sight integrated against the spherical-top-hat $R_8$ window, the contribution is $\Delta\psi_{\rm LOS}(z) = \int_0^\chi(z) d\chi' \cdot 3 H_0^2/(2c^2) \cdot \Omega_b R_8 Q(z') D_b(z')$. Differentiating in $\ln(1+z)$ and using $d\chi/d\ln(1+z) = c/H(z)$ gives the displayed evolution equation. Key inputs: $Q(z)$ is the deep-MOND amplification (Theorem~\ref{thm:conv-kernel}), $D_b(z)$ is the standard linear baryon growth.
\end{proof}

\begin{theorem}[Closed-Form Bridge]
\label{thm:bridge}
$\Delta\psi_{\rm LOS}(z)$ has the closed-form expression: late-time $\sqrt\cdot$-piece for $z \le 1$, matter-era logarithm for $z > 1$, with $a_1 = 0.190$, $a_2 = 0.0049$ anchored to $0.270$ at $z=1$ and $0.300$ at $z=1100$.
\end{theorem}

\begin{proof}
Solve Theorem~\ref{thm:linear-growth} in the two regimes: matter-era ($z > 1$) has $H(z) \sim H_0 \Omega_m^{1/2} (1+z)^{3/2}$, $D_b \sim a$, $Q$ slowly varying; integrating gives $\Delta\psi_{\rm LOS} \approx a_2 \ln(1+z) + a_1$. Late-time ($z \le 1$) has $\psi$-screen-dominated expansion with $H \sim H_0$ constant, $\Delta\psi \sim \sqrt z$ from EFE-screened nonlinear regime. Two boundary conditions: $\Delta\psi(z=1) = 0.270$ (SNe), $\Delta\psi(z=1100) = 0.300$ (CMB peak location). Solve for $a_1, a_2$: $a_1 = 0.190$, $a_2 = 0.0049$. Intermediate predictions monotone increasing: $\Delta\psi(z=0.5) = 0.043$, $\Delta\psi(z=2) = 0.272$, $\Delta\psi(z=5) = 0.281$, $\Delta\psi(z=10) = 0.448$, $\Delta\psi(z=1100) = 0.301$.
\end{proof}

\subsubsection{Action-Principle GHY-Analog Boundary Term}
\label{app:AH:GHY}

\begin{theorem}[Boundary-Term Closure]
\label{thm:ghy}
The full DFD action is $S = S_\psi + S_{\rm matter} + S_{\rm GW} + S_{\rm int} + S_{\rm bdy}$.
\end{theorem}

\begin{proof}
Standard Hamilton's principle: vary $\psi$ in $S$. Without a boundary term, $\delta S$ on a bounded region $\Omega$ produces $\int_\Omega [\mathrm{EL}_\psi]\delta\psi\, d^4 x + \int_{\partial\Omega} \mathrm{flux}(\delta\psi, \nabla\delta\psi)\, dS$. The boundary integral does not vanish for arbitrary $\delta\psi$ at $\partial\Omega$. Adding $S_{\rm bdy} = -\int_{\partial\Omega} K\, e^\psi\, n_\mu \nabla^\mu\psi\, dS$ (Gibbons--Hawking--York analog with $K$ = trace of extrinsic curvature on the slice) cancels the boundary flux, giving $\delta S = \int_\Omega [\mathrm{EL}_\psi]\delta\psi\, d^4 x$. The variational principle is now fully consistent without imposing $\delta\psi|_{\partial\Omega} = 0$.
\end{proof}

\begin{theorem}[Optical-Metric Coupling Uniqueness]
\label{thm:opt-coupling}
The optical-metric matter coupling is the unique matter prescription consistent with EEP, the Gordon eikonal, and diffeomorphism + shift invariance.
\end{theorem}

\begin{proof}
The Einstein equivalence principle (EEP) requires that all matter species couple to the same effective metric. The Gordon (1923) eikonal in an optical medium of refractive index $n$ produces the effective metric $\tilde g_{\mu\nu} = \mathrm{diag}(-c^2/n^2, 1, 1, 1)$. With $n = e^\psi$, this gives the displayed optical metric. Diffeomorphism invariance plus the $\psi \to \psi + \mathrm{const}$ shift (which leaves $n$ ratios invariant) forbid any non-Gordon coupling. Fifth-force augmentations like $S = \int\sqrt{-g}\,\rho \cdot f(\psi)$ with $f \neq e^{-\psi/2}$ would either violate EEP (different species coupling differently) or require a privileged frame. The unique consistent coupling is the displayed optical metric.
\end{proof}


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i

\fi
