%=============================================================================
% APPENDIX AT: CROSS-SECTOR EXACT IDENTITIES, STRONG-FIELD THEOREMS, AND
% NEW PARAMETER-FREE PREDICTIONS.
% Results obtained by an in-framework discovery program (June 2026) operating
% strictly inside the DFD axiom system. Each result is graded:
%   (E) EXACT IDENTITY  -- machine-verified algebra among printed DFD quantities
%                          (proof = the derivation; reuses only locked integers).
%   (T) THEOREM         -- exact closed-form consequence of a printed DFD object.
%   (P) PREDICTION      -- parameter-free data confrontation; look-elsewhere stated.
% Exact identities reuse ZERO new integers (the rigidity signature); their
% empirical content lives in the inputs. The predictions in AT.5 are the
% genuinely new empirical contact and carry explicit look-elsewhere estimates.
%=============================================================================

\section{Cross-Sector Exact Identities, Strong-Field Theorems, and New Parameter-Free Predictions}
\label{app:new-results}

This appendix collects results derived inside the DFD axiom system that were not
tabulated in the body. Throughout we use the locked ledger values
$\alpha^{-1}=137.036$, $M_P = 1.220890\times10^{19}$~GeV,
$v = M_P\,\alpha^{8}\sqrt{2\pi} = 246.09$~GeV,
$\Lambda_{\rm DFD}=M_P\,\alpha^{19/2}=61.20$~MeV,
$\Lambda_{\overline{\rm MS}}^{(5)}=\sqrt{4\pi}\,\Lambda_{\rm DFD}=216.95$~MeV,
$M_R=M_P\,\alpha^{3}=4.744\times10^{12}$~GeV, and
$m_3=\tfrac{14}{13}\pi M_P\,\alpha^{14}=50.16$~meV (App.~\ref{app:complete-params},
\ref{app:neutrino}). Every exact identity below was verified symbolically and to
$\geq 20$ digits in arbitrary-precision arithmetic.

%-----------------------------------------------------------------------------
\subsection{Exact cross-sector identities}\label{app:new-identities}
%-----------------------------------------------------------------------------

\begin{proposition}[Seesaw-triangle closure, (E)]\label{thm:new-yD}
The Dirac neutrino Yukawa coupling is forced to the exact value
\begin{equation}
\fiteq{y_D^2 = \frac{14}{13}\,\alpha = \Bigl(1 + \frac{\sin^2\theta_W}{N_{\rm gen}}\Bigr)\alpha,}
\end{equation}
with $\sin^2\theta_W=3/13$ and $N_{\rm gen}=3$. The neutrino absolute-scale exponent
$14$ is therefore not an independent postulate.
\end{proposition}
\begin{proof}
The Type-I seesaw gives $y_D^2 = 2 m_3 M_R/v^2$. Substituting the printed scales,
$y_D^2 = 2\bigl(\tfrac{14}{13}\pi M_P\alpha^{14}\bigr)\bigl(M_P\alpha^{3}\bigr)/\bigl(M_P^2\alpha^{16}\,2\pi\bigr)
= \tfrac{14}{13}\,\alpha^{14+3-16} = \tfrac{14}{13}\,\alpha$, the $\pi$'s cancelling
against the $\sqrt{2\pi}$ of $v$. The second form follows from
$1+\tfrac{1}{N_{\rm gen}}\sin^2\theta_W = 1+\tfrac{1}{3}\cdot\tfrac{3}{13}=\tfrac{14}{13}$,
so the neutrino Yukawa and the weak mixing angle share the denominator
$13 = N_{\rm gen}+\mathrm{Tr}(Y^2) = 3+10$. Equivalently the Dirac mass is the
seesaw geometric mean $m_D=\sqrt{m_3 M_R}=15.43$~GeV $=\sqrt{14\alpha/13}\,v/\sqrt2$.
\end{proof}

\begin{proposition}[Four-sector scale lock, (E)]\label{thm:new-fourlock}
The QCD, gravitational, electroweak, and neutrino scales satisfy
\begin{equation}
\fiteq{2\pi\,M_P\,\Lambda_{\rm DFD}^2 = v^2\,M_R,}
\end{equation}
exactly, with no constant other than the $2\pi$ inherited from $v$.
\end{proposition}
\begin{proof}
$v^2 M_R = (M_P^2\alpha^{16}\,2\pi)(M_P\alpha^{3}) = 2\pi M_P^3\alpha^{19}$, while
$2\pi M_P\Lambda_{\rm DFD}^2 = 2\pi M_P(M_P\alpha^{19/2})^2 = 2\pi M_P^3\alpha^{19}$.
The equality encodes the exponent identity $2\cdot\tfrac{19}{2} = 16+3 = 19$.
\end{proof}

\begin{proposition}[QCD--Majorana cube identity, (E)]\label{thm:new-MRcube}
With the microsector matching scale $\Lambda_{\rm top}=\sqrt{\alpha}\,M_P$,
\begin{equation}
\fiteq{M_R^3 = \frac{\Lambda_{\rm DFD}\,M_P^3}{\Lambda_{\rm top}},}
\end{equation}
exactly, with prefactor unity.
\end{proposition}
\begin{proof}
$\Lambda_{\rm DFD}M_P^3/\Lambda_{\rm top}
= (M_P\alpha^{19/2})M_P^3/(\alpha^{1/2}M_P) = M_P^3\alpha^{9} = (M_P\alpha^3)^3 = M_R^3$,
encoding $3\cdot 3 = \tfrac{19}{2}-\tfrac12 = 9$.
\end{proof}

\begin{proposition}[QCD--Hubble bridge, (E)]\label{thm:new-qcdhubble}
The Hubble energy is the cube of the confinement scale over the Planck mass squared,
\begin{equation}
\fiteq{\hbar H_0 = \frac{\Lambda_{\rm DFD}^3}{M_P^2}\,c^2,}
\end{equation}
a consequence of $57 = 6\cdot\tfrac{19}{2} = 3\cdot 19$ and the printed invariant
$G\hbar H_0^2/c^5 = \alpha^{57}$ (App.~\ref{app:O_alpha57_clean}).
\end{proposition}
\begin{proof}
From $G\hbar H_0^2/c^5=\alpha^{57}$ and $t_P=\sqrt{G\hbar/c^5}$ one has
$H_0 = \alpha^{57/2}/t_P$, hence $\hbar H_0 = (\hbar/t_P)\alpha^{57/2} = M_P c^2\,\alpha^{57/2}$
since $\hbar/t_P = \sqrt{\hbar c^5/G} = M_P c^2$. On the other side
$\Lambda_{\rm DFD}^3/M_P^2 \cdot c^2 = (M_P\alpha^{19/2})^3/M_P^2\cdot c^2 = M_P c^2\,\alpha^{57/2}$,
using $3\cdot\tfrac{19}{2}=\tfrac{57}{2}$. Equivalently the heaviest neutrino is the
$\alpha$-primed geometric mean of the Hubble rate and the Planck mass,
$m_3 = \tfrac{14}{13}\pi\sqrt{\hbar H_0\,M_P}\,\alpha^{-1/4}$
(from $\tfrac{57}{4}-\tfrac14 = 14$).
\end{proof}

\begin{proposition}[Strange-quark mass as a rung fraction, (E)]\label{thm:new-ms}
$m_s = \tfrac{3}{7}\,\Lambda_{\overline{\rm MS}}^{(5)}$ exactly (a pure rational), and
more generally every charged-fermion mass is an exact closed form
$m_f = (\text{rational}\times 2\text{-power})\,\Lambda_{\overline{\rm MS}}^{(5)}\,\alpha^{k}$.
\end{proposition}
\begin{proof}
$m_s=\tfrac{6}{7}\alpha^{3/2}\,v/\sqrt2 = \tfrac{6}{7}\sqrt{\pi}\,M_P\alpha^{19/2}$ while
$\Lambda_{\overline{\rm MS}}^{(5)}=2\sqrt{\pi}\,M_P\alpha^{19/2}$, so the ratio is
$\tfrac67/2=\tfrac37$; $\alpha$ and $\pi$ cancel identically. The same equal-exponent
mechanism gives the pure-number ratios $m_\tau/m_c=\sqrt2$, $m_t/m_b=42$,
$m_e/m_d=1/9$, $m_\mu/m_s=7/6$, with the down-type prefactor integers
$7=b_0$ and $36=N_f^2$ reappearing verbatim from the QCD running.
\end{proof}

%-----------------------------------------------------------------------------
\subsection{Exact strong-field and gravitational theorems}\label{app:new-strongfield}
%-----------------------------------------------------------------------------

These follow from the physical optical metric
$\tilde g=\mathrm{diag}(-c^2 e^{-\psi},\,e^{\psi}\delta_{ij})$ with the exterior
$\psi = 2u$, $u\equiv GM/(c^2 r)$ (Sec.~\ref{sec:ppn}, App.~\ref{app:pade-identity}).

\begin{theorem}[Golden-ratio ISCO, (T)]\label{thm:new-isco}
The innermost stable circular orbit sits at
$r_{\rm ISCO} = (3+\sqrt5)\,GM/c^2 = 2\varphi^2\,GM/c^2$ ($\varphi$ the golden ratio),
with $\psi_{\rm ISCO}=\varphi^{-2}$, specific energy $E_{\rm ISCO}/mc^2 = 0.945213$,
and accretion efficiency $\eta = 5.479\%$ (GR: $6\,GM/c^2$, $5.719\%$).
\end{theorem}
\begin{proof}
For timelike circular geodesics of $\tilde g$ the effective potential gives
circularity and marginal stability as $dV/dr=d^2V/dr^2=0$, which reduce (with
$m\equiv GM/c^2$) to $r^2 - 6 m r + 4 m^2 = 0$; the stable root is
$r=(3+\sqrt5)m$. Then $\psi_{\rm ISCO}=2m/r=(3-\sqrt5)/2=\varphi^{-2}$, and
$E_{\rm ISCO}/mc^2 = \sqrt{(3+\sqrt5)/4}\;e^{-(3-\sqrt5)/4}=0.945213$.
\end{proof}

\begin{corollary}[Epicyclic / QPO law, (T)]\label{thm:new-qpo}
The radial epicyclic-to-orbital frequency ratio is the exact closed form
$(\kappa/\Omega)^2 = 1 - 6u + 4u^2$, with roots $u=(3\pm\sqrt5)/4$ (the inner root is
$\psi_{\rm ISCO}/2$). The leading periastron advance $6\pi u$ matches the 1PN value;
the $+4u^2$ term is the DFD-specific 2PN signature.
\end{corollary}

\begin{theorem}[Photon ring, redshift, and ringdown, (T)]\label{thm:new-photonring}
The photon sphere is at $r_{\rm ph}=2GM/c^2$ with critical impact parameter
$b_{\rm crit}=2e\,GM/c^2$; the shadow diameter exceeds Schwarzschild by the exact
factor $2e/(3\sqrt3)=1.0463$, while the self-similar subring spacing $e^{-\pi}$ is
unchanged. The gravitational redshift is $z(r)=e^{u}-1$ exactly, finite for all
$r>0$. In the eikonal limit $\lambda_{\rm Lyap}/\Omega_c = 1$ (as in GR), so ringdown
frequencies scale by $3\sqrt3/(2e)=0.95578$ relative to Schwarzschild.
\end{theorem}
\begin{proof}
Null circular orbits of the optical metric satisfy $(\ln A)'=2/r$ with
$A=c^2 e^{-2\psi}$, giving $r_{\rm ph}=2m$ and
$b_{\rm crit}=r_{\rm ph}/\sqrt{A(r_{\rm ph})}/c = 2e\,m$; the ratio to the GR value
$3\sqrt3\,m$ is $2e/(3\sqrt3)$. The redshift is
$1+z=1/\sqrt{-\tilde g_{tt}}/c=e^{\psi/2}=e^{u}$. The Lyapunov exponent equals the
coordinate angular frequency at $r_{\rm ph}$ by direct evaluation of the geodesic
deviation, giving the stated frequency ratio.
\end{proof}

\begin{theorem}[No nonlinear gravitational-wave memory, (T)]\label{thm:new-memory}
In the deposited action (Sec.~\ref{sec:formalism}) the transverse-traceless sector
$S_h$ is exactly quadratic in $h^{\rm TT}_{ij}$ and its source is the matter stress
alone (there is no graviton self-energy or scalar-stress source). Consequently DFD
produces only the linear (ordinary) gravitational-wave memory and \emph{zero}
Christodoulou (nonlinear) memory.
\end{theorem}
\begin{proof}
$\partial^3 S/\partial h^3 \equiv 0$ identically because $S_h$ is quadratic, and the
mixed variation $\partial^3 S/(\partial h\,\partial\psi\,\partial\psi)=0$ because
$S_{\rm int}$ contains no term modifying the principal part of the TT operator. Hence
$\Box h^{\rm TT}_{ij} = -(8\pi G/c^4)\,T^{\rm TT}_{ij}$ with $T$ the matter stress;
the nonlinear (radiation-sourced) memory integral that GR generates is absent.
\end{proof}
\begin{remark}[Channel scope]\label{rem:memory-scope}
The zero-memory statement is a theorem for the \emph{tensor} channel: it rests only on
$\partial^3S/\partial h^3\equiv0$ and the vanishing mixed cubic, independent of whether
the scalar $\psi$ radiates. A possible \emph{scalar-channel} memory contribution is a
separate question: for quasi-circular BBH inspirals the $\psi$-sector sourcing is rigorous
and negligible, while the general (e.g.\ head-on/burst) scalar case is open pending the
far-zone non-radiativeness analysis of the $\psi$ sector. Where the corpus elsewhere writes
a $c$-speed d'Alembertian for $\psi$, that form is the numerical hyperbolic-relaxation of
the elliptic AQUAL constraint, not an independent claim that $\psi$ radiates at $c$.
\end{remark}
\begin{falsifier}\label{fals:new-memory}
For an equal-mass nonspinning merger GR's null memory is $27\%$ of the edge-on strain
peak; DFD predicts none. LISA reaches single-event memory SNR $11$--$35$ for
$10^5$--$10^6\,M_\odot$ at $z=1$ -- a decisive yes/no test.
\end{falsifier}

\begin{proposition}[Cole--Hopf lapse family, (E/T)]\label{thm:new-colehopf}
The substitution $\chi=e^{k\psi/2}$ linearises the master acceleration equation
$\nabla\!\cdot\mathbf a + (k/c^2)\mathbf a^2 = -4\pi G\rho$ (with
$\mathbf a=(c^2/2)\nabla\psi$) to $\nabla^2\chi = -(4\pi G k/c^2)\rho\,\chi$. Its
point-mass exterior is the one-parameter family $n_k(u)=(1+k u)^{2/k}$, with
$\beta(k)=1+k/2$, $\gamma(k)=1$, and $k\to0$ recovering DFD's entire-function lapse
$e^{2u}$. The GR isotropic lapse-squared is the exact geometric mean of the
$k=\pm\tfrac12$ members.
\end{proposition}
\begin{proof}
Writing $\mathbf a=(c^2/2)\nabla\psi$ turns the equation into
$\nabla^2\psi + (k/2)|\nabla\psi|^2 = -(8\pi G/c^2)\rho$; with $\chi=e^{k\psi/2}$,
$\nabla^2\chi = (k/2)\chi[\nabla^2\psi+(k/2)|\nabla\psi|^2]$, the stated linear
equation. The vacuum solution $\chi = 1+kGM/(c^2 r)$ gives $n_k=(1+ku)^{2/k}$;
expanding the optical metric yields $\beta=1+k/2$. Lunar laser ranging
($\beta-1=(-4.5\pm5.6)\times10^{-5}$) pins the vacuum member to
$k=(-0.9\pm1.1)\times10^{-4}$ -- gravity sits at $k=0$ to one part in $10^4$.
\end{proof}

%-----------------------------------------------------------------------------
\subsection{Compact-star structure and the maximum-mass falsifier}\label{app:new-compactstar}
%-----------------------------------------------------------------------------

\begin{theorem}[DFD-native compact-star structure, (T)]\label{thm:new-compactstar}
Varying the matter action of a perfect fluid minimally coupled to the optical
metric (Sec.~\ref{sec:formalism}, App.~\ref{app:master-equation}) with respect to
$\psi$ uniquely fixes the relativistic source: in the $\mu\to1$ regime the static
structure obeys
\begin{equation}
\fiteq{\nabla^2\psi = -\frac{8\pi G}{c^4}\,(\varepsilon+3P)\,e^{\psi},\qquad
\frac{dP}{dr} = \tfrac12(\varepsilon+P)\,\frac{d\psi}{dr},}
\end{equation}
with $\varepsilon$ the energy density and $P$ the pressure. The resulting
mass--radius relation is \emph{not} identical to the general-relativistic one: for
the reference $\Gamma=2$, $K=100$ polytrope the DFD maximum mass is
$M_{\max}^{\rm DFD}=1.474\,M_\odot$ at central density
$\rho_c=1.46\times10^{15}\,\mathrm{g\,cm^{-3}}$, exactly $0.900\times$ the
GR value $1.637\,M_\odot$, with circumferential radius $R(M_{\max})=11.6$~km.
\end{theorem}
\begin{proof}
The point-particle Lagrangian $-mc^2 e^{-\psi/2}\sqrt{1-e^{2\psi}v^2/c^2}$ of the
optical metric gives $\partial\mathcal L/\partial\psi=\tfrac12(\varepsilon+3P)e^{\psi}$
on the static fluid (verified symbolically, residual $0$), which is the active
(Tolman) source; the lapse $g_{tt}=-c^2e^{-\psi}$ gives the equilibrium relation by
the standard static-fluid first integral. Integrating the coupled system for the
reference polytrope reproduces the quoted $M$--$R$ curve; the $0.900$ ratio is
stable under grid refinement. The reduction $\nabla^2\psi=-(8\pi G/c^2)\rho$
(Sec.~\ref{sec:ppn}) is recovered in the nonrelativistic limit $P\ll\varepsilon$,
$\psi\ll1$.
\end{proof}

\textbf{Reproducibility Note.} The stellar-structure integrations reported here shoot
the central scalar value $\psi_c$ so that the exterior solution satisfies asymptotic
flatness, $\psi(\infty)=0$ and $n(\infty)=1$. Holding $\psi_c$ fixed instead computes
a non-asymptotically-flat normalization and can produce spurious runaway behavior.
Radii quoted in this subsection are circumferential radii,
\[
\fiteq{R_{\rm circ}=e^{\psi(R)/2}\,R_{\rm coord},}
\]
not raw coordinate radii. An independent integration (with a GR--TOV control
reproducing $M_{\max}^{\rm GR}=1.637\,M_\odot$) confirms the quoted values:
$M_{\max}^{\rm DFD}=1.474\,M_\odot=0.900\,M_{\max}^{\rm GR}$ at
$\rho_c\simeq1.4\times10^{15}\,\mathrm{g\,cm^{-3}}$, with $R_{\rm circ}=11.68$~km.

\begin{corollary}[Equation-of-state family and causal ceiling, (T/P)]\label{thm:new-eosfamily}
Across standard nuclear equations of state the DFD maximum mass is suppressed by
$13$--$23\%$ relative to GR ($M_{\max}^{\rm DFD}/M_{\max}^{\rm GR}=0.77$--$0.87$;
e.g.\ SLy $0.80$, AP4 $0.77$, MPA1 $0.78$, MS1 $0.79$), and the tidal deformability
is correspondingly reduced (with a residual circumferential-vs-coordinate radius
convention dependence). Imposing the maximally stiff causal core $c_s=c$ above a
transition density $\rho_t$ on a soft crust yields the absolute bound
\begin{equation}
\fiteq{M_{\max}^{\rm DFD}\le 3.03\,M_\odot\quad(\text{any causal EOS}),}
\end{equation}
versus the GR causal ceiling $\approx4.05\,M_\odot$. A soft base plus a causal core
with $\rho_t\in[1.44,\,1.80]\,\rho_{\rm nuc}$ jointly satisfies the
$M_{\max}\ge2.0\,M_\odot$, $\Lambda_{1.4}$, and $R(2.07\,M_\odot)$ constraints.
\end{corollary}
\begin{falsifier}\label{fals:new-maxmass}
Any neutron star confirmed above $3.03\,M_\odot$ falsifies DFD outright (GR tolerates
up to $\approx4\,M_\odot$). If the $2.59\,M_\odot$ secondary of GW190814 is
established as a neutron star, DFD requires $\rho_t\lesssim1.45\,\rho_{\rm nuc}$,
forcing $\Lambda_{1.4}$ and $R(2.07\,M_\odot)$ jointly into tension with
GW170817$+$NICER -- a near-term decisive test from existing data.
\end{falsifier}

%-----------------------------------------------------------------------------
\subsection{Deep-sector conformal theorems}\label{app:new-cft}
%-----------------------------------------------------------------------------

In the deep-field regime the energy functional is the cubic
$E=\tfrac{c^4}{16\pi G a_\star}\!\int\!\tfrac13|\nabla\psi|^3 - \tfrac{c^2}{2}\!\int\!\rho\psi$.

\begin{theorem}[Conformal structure and the exact $M$--$\sigma$ relation, (T)]\label{thm:new-msigma}
The deep-field functional is invariant under the full conformal group of $\mathbb R^3$
plus a $\psi$-shift (11 generators) and has identically traceless stress tensor. A
point mass carries dilatation (anomaly) charge $Q(m)=\tfrac23\sqrt{G a_0}\,m^{3/2}$,
the two-body force is
$F=\tfrac23(\sqrt{Ga_0}/\ell)\,[(m_1+m_2)^{3/2}-m_1^{3/2}-m_2^{3/2}]\propto 1/\ell$,
and any stationary isolated deep system obeys the exact isotropic relation
\begin{equation}
\fiteq{\sigma_{\rm los}^4 = \frac{4}{81}\,G M a_0,}
\end{equation}
together with the baryonic Tully--Fisher law $v_f^4 = G M a_0$ with coefficient
exactly unity.
\end{theorem}
\begin{proof}
Tracelessness $\sigma_{ii}=0$ off-shell and conservation $\partial_j\sigma_{ij}=0$
on-shell are verified directly for the cubic Lagrangian; the dilatation current of
the conformally invariant $3$-Laplacian integrates to the stated point charge. The
virial identity $\sum_i \mathbf x_i\!\cdot\!\mathbf F_i = \sum_i Q(m_i)-Q(M)$ for a
self-bound system gives $2T=\tfrac23\sqrt{Ga_0}\,M^{3/2}$, hence the isotropic
$\sigma_{\rm los}^4=(4/81)GMa_0$; and $v_f^2=dQ/dM=\sqrt{GMa_0}$ gives BTFR with unit
coefficient. (This corrects the body's order-of-magnitude $\sigma^4\simeq GMa_0$ to
its exact coefficient $4/81$.) Parameter-free dwarf-spheroidal checks: Leo~I
$\sigma_{\rm los}=8.55$ (obs $\approx9.2$), Fornax $12.18$ (obs $\approx11.7$).
\end{proof}

\begin{corollary}[Golden-ratio phantom mass, (T)]\label{thm:new-phantom}
At the MOND radius $r_M=\sqrt{GM/a_0}$ (where $g_{\rm bar}=a_0$), the radial
acceleration relation $g_{\rm obs}=\tfrac12[g_{\rm bar}+\sqrt{g_{\rm bar}^2+4 g_{\rm bar}a_0}]$
forces the enclosed dynamical mass to be exactly the golden ratio times the baryonic:
$M_{\rm dyn}(<r_M)/M = \varphi = 1.6180\ldots$, with phantom fraction $1/\varphi$.
\end{corollary}
\begin{proof}
At $g_{\rm bar}=a_0$ the relation gives $g_{\rm obs}=a_0(1+\sqrt5)/2 = \varphi\,a_0$;
since $M_{\rm dyn}\propto g_{\rm obs}$ and $M\propto g_{\rm bar}$ at fixed $r_M$, the
ratio is $\varphi$ identically.
\end{proof}
The Kuzmin-disk family solves the \emph{full} nonlinear DFD field equation exactly,
$g=\nu(|g_N|/a_0)\,g_N$ with $\nu(y)=\tfrac12[1+\sqrt{1+4/y}]$ and
$v_{\rm flat}^4=GMa_0$, providing an integrable thin-disk benchmark.

%-----------------------------------------------------------------------------
\subsection{Strong-CP protection and the neutron electric dipole}\label{app:new-nedm}
%-----------------------------------------------------------------------------

\begin{theorem}[All-orders determinant-phase protection, (T)]\label{thm:new-detphase}
Let $H(\xi)=H_0+i\xi T$ with $H_0$ real symmetric and $T$ real antisymmetric
(the strengthened-branch conjugation-odd offset, App.~\ref{app:ckm-update}). Then
$\arg\det H(\xi)=0$ for all real $\xi$ (and all finite orders), so the offset that
supplies the weak CP phase never regenerates the strong-CP angle.
\end{theorem}
\begin{proof}
$H_0^{\sf T}=H_0$ and $T^{\sf T}=-T$ give
$H(\xi)^{\sf T}=H_0-i\xi T=\overline{H(\xi)}$ (overbar = complex conjugate, since
$H_0,T,\xi$ real). Hence $\det H=\det H^{\sf T}=\det\overline{H}=\overline{\det H}$,
so $\det H$ is real and $\arg\det H=0$ wherever it is positive. This holds to all
orders in $\xi$, not merely the first-order (trace-form) statement of
App.~\ref{app:ckm-update}.
\end{proof}
\begin{corollary}[Neutron-EDM ledger and falsifier, (P)]\label{thm:new-nedm}
With $\bar\theta=0$ exact (Theorem~\ref{thm:new-detphase}; App.~\ref{app:strongcp_selection_rule}),
the DFD neutron EDM is the CKM long-distance background scaled by
$J_{\rm DFD}/J_{\rm SM}=0.94$, giving $d_n\sim(0.9$--$5.6)\times10^{-32}\,e\,$cm; the
residual $\theta$-channel is $<1.5\times10^{-35}\,e\,$cm. DFD predicts \emph{null} at
all foreseeable sensitivities, and any measured $d_n>10^{-30}\,e\,$cm falsifies the
det-orthogonal architecture (margin $\sim10^6$ to the current bound). The same
even-mapping-torus structure that forces $\bar\theta=0$ is silent on the CP-even
topological susceptibility $\chi_t$, consistent with a nonzero $\chi_t$ and the
$\eta'$ anomaly channel.
\end{corollary}

%-----------------------------------------------------------------------------
\subsection{New parameter-free predictions}\label{app:new-predictions}
%-----------------------------------------------------------------------------

The following are genuine predictions (not reformulations): each is a parameter-free
DFD value confronted with data. Each carries a look-elsewhere note; entries reuse
only locked catalog integers.

\begin{table}[h]
\centering
\caption{New parameter-free DFD predictions. ``rung'' denotes the scheme scale
$\Lambda_{\rm DFD}=61.20$~MeV or $\Lambda_{\overline{\rm MS}}^{(5)}=216.95$~MeV
(App.~\ref{app:complete-params}). The $m_t$ row evaluates with the topological
$v=246.09$~GeV of this appendix; the muon-decay vev $v=246.22$~GeV gives
$m_t=(1-\alpha)v/\sqrt2=172.83$~GeV ($0.88\sigma$).}\label{tab:new-predictions}
\renewcommand{\arraystretch}{1.25}
\fittab{%
\begin{tabular}{lcccc}
\toprule
Observable & DFD form & Predicted & Measured & Dev. \\
\midrule
$F_\pi$ & $\tfrac35\Lambda_{\overline{\rm MS}}$ & $130.17$ MeV & $130.2(8)$ MeV & $0.03\%$ \\
$\eta'$ & $\sqrt{39/2}\,\Lambda_{\overline{\rm MS}}$ & $958.0$ MeV & $957.78$ MeV & $0.02\%$ \\
$\omega$ & $\sqrt{13}\,\Lambda_{\overline{\rm MS}}$ & $782.2$ MeV & $782.66$ MeV & $0.06\%$ \\
$\Delta$ & $\sqrt{32}\,\Lambda_{\overline{\rm MS}}$ & $1227$ MeV & $1232$ MeV & $0.4\%$ \\
$\Omega^-$ & $\sqrt{59}\,\Lambda_{\overline{\rm MS}}$ & $1666$ MeV & $1672$ MeV & $0.4\%$ \\
$m_{0^{++}}/\sqrt\sigma$ & $\sqrt{4\pi}$ & $3.545$ & $3.529(97)$ & $0.16\sigma$ \\
$m_t$ & $(1-\alpha)v/\sqrt2$ & $172.74$ GeV & $172.57(30)$ & $0.57\sigma$ \\
$a_0$ & $2\alpha^{29}c/t_P$ & $1.1966$\,e$-$10 & $1.20(2)$\,e$-$10 & $0.17\sigma$ \\
$A_s$ & $32\pi\alpha^5$ & $2.080$\,e$-$9 & $2.105(30)$\,e$-$9 & $0.6\sigma$ \\
$\sin^2\theta_{13}$ & $3\alpha$ & $0.0219$ & $0.02195(58)$ & $0.1\sigma$ \\
\bottomrule
\end{tabular}%
}
\end{table}

\paragraph{Baryon decuplet (T/P).}
The $J=3/2$ decuplet follows the exact integer arithmetic progression
$m^2 = (32 + 9 n_s)\,\Lambda_{\overline{\rm MS}}^2$ in the number of strange quarks
$n_s$: $\Delta,\Sigma^*,\Xi^*,\Omega^-$ at $n_s=0,1,2,3$ all match within $0.4\%$.
The progression structure (intercept $32$, slope $9$), not the individual hits, is the
non-accidental content.

\paragraph{Baryon octet, incl.\ the nucleon (P).}
The $J=1/2$ octet follows the companion progression
\begin{equation}\label{eq:baryon-octet}
m^2 \;=\; (19 + 9\,n_s)\,\Lambda_{\overline{\rm MS}}^2 ,
\end{equation}
with the \emph{same} strange slope $9$ as the decuplet and intercept
$19 = h^0(\mathcal{O}(1)) + h^0(\mathcal{O}(2)) + h^0(\mathcal{O}(3)) = 3+6+10$, the
$\mathbb{C}P^2$ line-bundle cohomology sum that also fixes $\Lambda_{\rm QCD} = M_P\alpha^{19/2}$
(App.~\ref{app:new-identities}) and the CKM apex $\bar\rho = 19\alpha$. The nucleon is then a
\emph{forced tower state}, not an input:
\[
\fiteq{M_N = \sqrt{19}\,\Lambda_5 = \sqrt{19}\,\sqrt{4\pi}\,M_P\,\alpha^{19/2} = 946~\mathrm{MeV}
\quad(\text{vs } 939,\ +0.7\%),}
\]
the integer $19$ appearing in \emph{both} the prefactor and the exponent. Endpoints match cleanly
--- $N$ ($n_s=0$) at $+0.7\%$ and $\Xi$ ($n_s=2$) at $+0.1\%$ --- while the $n_s=1$ value
$\sqrt{28}\,\Lambda_5 = 1148$~MeV is the $\Lambda$--$\Sigma$ \emph{average} (within $0.5\%$ of
$(\Lambda+\Sigma)/2$); the individual $\Lambda(1116)$--$\Sigma(1193)$ splitting requires the
additional isospin/Gell-Mann--Okubo term absent from the pure $n_s$-linear tower, which is why the
spin-$1/2$ octet ($+0.7\%$) is intrinsically less clean than the spin-aligned decuplet ($0.4\%$).
\emph{Grade (P):} the intercept $19$ is forced (the same cohomology integer as the QCD scale),
removing $M_N$ from the input list; the closer numerical fit $(M_N/\Lambda_5)^2 = 18.73 \approx 75/4$
is \emph{not} a forced DFD quantity and is deliberately not adopted. The decuplet--octet intercept gap
$32-19 = 13$ tracks the measured $\Delta$--$N$ mass-squared splitting ($13.5\,\Lambda_5^2$) to $\sim 4\%$.

\paragraph{Nucleon $\sigma$-term and the central nuclear coupling $c_1$ (conditional).}
The deep-central scalar-isoscalar pion--nucleon coupling $c_1$---the one nuclear low-energy constant
\emph{not} already fixed by the forced relativistic spin--orbit mechanism, the saturation density
$\rho_0$, or the forced $\rho/\omega$ and $\Delta$ tower states---can be closed \emph{conditionally},
by the same mass-squared-tower logic that forces the intercept $19$. The hinge, stated in the open:
\emph{in DFD the light-quark scalar source is not an independent chiral counterterm but the
$U(2)$-equivariant tangent-projector perturbation of the baryon mass-squared tower.} Granting that
rule, the light scalar spurion ($\langle\chi_+\rangle = 4m_\pi^2$ for $N_f=2$) can enter $M_N^2$ only
through the projector onto the coset tangent $T\mathbb{C}P^2$ of $\mathbb{C}P^2 = SU(3)/U(2)$ (Schur /
$U(2)$-equivariance), and the index trace runs over its two complex directions,
$\dim_{\mathbb C}T\mathbb{C}P^2 = 2$, fixing the dimensionless local slope
\begin{equation}\label{eq:massslope-8}
\fiteq{\begin{aligned}
\frac{\partial M_N^2}{\partial m_\pi^2}
\;&=\; \frac{\operatorname{Tr}_{T\mathbb{C}P^2}(\chi_+)}{m_\pi^2}
\;=\; \frac{2\,\langle\chi_+\rangle}{m_\pi^2} \;=\; 8
\\
&\qquad (8 = 2\times4,\ \text{tangent rank}\times\text{scalar trace}).
\end{aligned}}
\end{equation}
Since $M_N^2 = M_0^2 - 8\,M_N c_1\,m_\pi^2 + \cdots$ in chiral perturbation theory, the forced slope
$8$ fixes
\[
c_1 \;=\; -\frac{1}{M_N} \;=\; -\frac{1}{\sqrt{19}\,\Lambda_5} \;=\; -1.057~\mathrm{GeV}^{-1},
\]
whence $\sigma_{\pi N} = -4c_1 m_\pi^2 - \dfrac{9 g_A^2 m_\pi^3}{64\pi f_\pi^2} + \cdots
\simeq 56~\mathrm{MeV}$ (the leading $-4c_1 m_\pi^2 = 4m_\pi^2/M_N = 77$~MeV minus the forced
$21$~MeV chiral loop), inside the observed band $45$--$59$~MeV. The central nuclear binding---and
hence the magic-number ordering at $\rho_0$---then closes with no free coupling.
\emph{Grade (conditional --- strictly weaker than the forced tower).} Unlike the intercept
$19$ and the strange slope $9$ (forced index/cohomology counts already carried by
Eq.~\eqref{eq:baryon-octet}), the slope $8$ is \emph{not} yet load-bearing in the corpus: the tower
forces $M_N=\sqrt{19}\,\Lambda_5$ but does not derive the $m_\pi^2$-response as a projector trace, and
chiral symmetry \emph{explicitly permits} the independent singlet counterterm
$c_1\,\bar N\langle\chi_+\rangle N$. The closure therefore requires a rule \emph{stronger than
symmetry}---``baryon mass-squared perturbations are internal $\mathbb{C}P^2\times S^3$ projector
traces, with no free IR counterterm''---which the present corpus does not derive from a deeper axiom.
With that rule the nucleus is parameter-free at leading chiral order; without it, $c_1 = -1/M_N$ is a
natural value, not a theorem. This is a \emph{heavier} insertion than the baryogenesis Berry-holonomy
closure (Extended Derivations, App.~AH1b), which only \emph{names} a clock inside the fermion kernel
DFD already uses --- here a symmetry-allowed counterterm must be actively \emph{forbidden}. The
empirical slope ($7.2$--$8.8$, for $\sigma_{\pi N}=45$--$59$) brackets $8$, so the integer alone is
not the content; the content is the tangent-projector rule that singles it out. Notably $8$ \emph{is} a
genuine $\mathbb{C}P^2$ cohomology dimension in its own right --- $\dim H^0(\mathbb{C}P^2,T\mathbb{C}P^2)
= \dim\mathrm{Aut}(\mathbb{C}P^2) = \dim\mathfrak{sl}(3,\mathbb{C}) = 8$, the holomorphic tangent
sections, the same \emph{kind} of Dolbeault index as the intercept $19$ --- which strengthens that $8$
is the right \emph{type} of object. But the $\mathbb{C}P^2$ structure admits \emph{more than one} route
to $8$ (the tangent-rank product $2\times4$ and the holomorphic-section count $\dim H^0 = 8$ are
distinct bookkeepings, $2$ directions $\times\,4m_\pi^2$ versus $8$ sections $\times\,m_\pi^2$); the
geometry does not uniquely select one mechanism. That over-determination is precisely why the forced
content is the no-counterterm \emph{rule}, not the integer.

\paragraph{Quark--lepton complementarity (P).}
The PMNS reactor angle and the CKM Cabibbo angle are locked by
$\sin^2\theta_{13}^{\rm PMNS} = \tfrac{3}{31}\sin\theta_{12}^{\rm CKM}$, since
$\sin^2\theta_{13}=3\alpha$ ($3=\chi(\mathbb{C}P^2)$) and $\sin\theta_{12}^{\rm CKM}=31\alpha$;
$3/31=0.0968$ vs data $0.0976$ ($0.8\%$).

\paragraph{CP-sector gauntlet (P).}
The exact DFD CKM matrix (App.~\ref{app:ckm-update}) yields, at zero free parameters:
$\sin2\beta=0.719$ (vs $0.708\pm0.011$, $+1.0\sigma$);
$\phi_s=-2.14^\circ$ (vs $-2.2\pm0.9^\circ$, $0.1\sigma$);
$\mathrm{Br}(B_s\!\to\!\mu\mu)=3.40\times10^{-9}$ (vs $3.34\pm0.27$, $0.2\sigma$ -- where
the global-fit value sits $1.2\sigma$ high); and an $\varepsilon_K$ deficit of
$12$--$17\%$ ($\sim-2\sigma$), a falsifiable signature that lands on the
exclusive-$|V_{cb}|$ side of the standing $\varepsilon_K$ tension.

\paragraph{Look-elsewhere summary.}
$F_\pi=\tfrac35\Lambda_{\overline{\rm MS}}$ is the unique expression in its window over
both rungs, both functional forms, and denominators $\leq10$; $\eta'=\sqrt{39/2}$ and
$a_0=2\alpha^{29}c/t_P$ are likewise the only candidates in their windows;
$A_s=32\pi\alpha^5$ is $1$ of $\sim225$ grammar candidates within the Planck
$1\sigma$ band. The vector-meson and decuplet entries sit on a denser integer grid;
their evidential weight is in the progression structure, scored separately.

\paragraph{$A_s$ coefficient status: forced power, asserted prefactor.}
We grade $A_s=32\pi\alpha^5$ (Tab.~\ref{tab:new-predictions}) as \emph{forced in its
power, fitted in its coefficient}. In the de~Sitter dictionary
$A_s=H_\star^2/(8\pi^2\varepsilon_W\bar M_P^2)$ with the inflationary Hubble on the locked
Majorana rung, $H_\star\propto M_R=M_P\alpha^3$ (App.~\ref{app:new-identities}), and the
QED one-loop measure $\varepsilon_W=\alpha/(4\pi)$, the exponent collapses \emph{exactly}
to $\alpha^5=(\alpha^3)^2/\alpha^1$ (sympy residual $0$); this power is rigid. The numeric
coefficient $32\pi$ is \emph{not} derived from DFD geometry: it is fixed entirely by the
$H_\star$--$M_R$ normalization, for which no first-principles argument (de~Sitter horizon,
$S^3/\mathbb{C}P^2$ internal volume, spectral action, or Friedmann/Einstein coupling)
exists in the corpus.

\emph{Planck-mass bookkeeping.} DFD fixes
$M_R=M_P\alpha^3=4.74\times10^{12}$~GeV using the \emph{non-reduced}
$M_P=1.22\times10^{19}$~GeV, whereas the standard amplitude formula carries the
\emph{reduced} $\bar M_P=M_P/\sqrt{8\pi}=2.44\times10^{18}$~GeV. With conventions made
consistent the ratio that reproduces the amplitude is
$H_\star/M_R=\sqrt{8\pi}=5.013=M_P/\bar M_P$ (verified: $H_\star=\sqrt{8\pi}\,M_R$ in the
reduced-mass formula gives $A_s=32\pi\alpha^5=2.080\times10^{-9}$, $-0.82\sigma$). The
catalog's $H_\star=8\pi\,M_R$ writes the \emph{same} number only because it evaluates the
reduced-mass formula with the non-reduced $M_P$, so $M_P$ cancels; the ``$8\pi$'' is then
$\sqrt{8\pi}$ of genuine reduced/non-reduced conversion times $\sqrt{8\pi}$ of an
\emph{underived} Hubble--Majorana ratio.

\emph{Convention history.} An earlier deposit wrote $H_\star=\sqrt{8\pi}\,M_R$ and reported
$A_s=4\alpha^5=8.28\times10^{-11}$ ($-67\sigma$); that figure is an artifact of the same
mismatch (the non-reduced $M_P$ used inside the reduced-mass formula), \emph{not} evidence
against $\sqrt{8\pi}$. Under consistent conventions $H_\star=\sqrt{8\pi}\,M_R$ already lands
in the Planck band, and the relabelling to $8\pi$ double-counts the Planck-mass conversion.
Either way the residual freedom is a single $\sqrt{8\pi}$ in $H_\star/M_R$ that is not
derived; we therefore present $A_s$ as \textbf{forced power $\alpha^5$, asserted
coefficient}, pending a first-principles $H_\star/M_R=\sqrt{8\pi}$.

%-----------------------------------------------------------------------------
\subsection{Status and the remaining open lemma}\label{app:new-status}
%-----------------------------------------------------------------------------

\begin{status}\label{stat:new-results}
The identities of \ref{app:new-identities} and the closed forms of
\ref{app:new-strongfield}--\ref{app:new-nedm} are exact (machine-verified) and reuse
no new integers; their role is structural rigidity. The entries of
\ref{app:new-predictions} are parameter-free predictions at the stated grades. The
$\alpha^{19/2}$ confinement exponent that underlies several of these is reduced to a
single open lemma: writing $\tfrac{19}{2} = (k_{\max}-N_{\rm gen})/(2N_{\rm gen}) = 57/6$,
the exponent is rigidity-selected (the unique catalog form in the empirical window),
but its derivation is contingent on the cube law of
Proposition~\ref{thm:new-qcdhubble}, which is not independently proved here. We
therefore present $\tfrac{19}{2}$ as rigidity-selected, not as derived.
\end{status}
