% Appendix AY: Cross-sector relations (June 2026 cross-sector verification)
\providecommand{\DerivedGrade}{\textsc{D}}
\providecommand{\TheoremGrade}{\textsc{T}}
\providecommand{\ValidatedGrade}{\checkmark}
\section{Cross-Sector Relations: Flavor Closures, Hierarchy Rigidity, and Large-Number Structure}\label{app:crosssector}

This appendix collects four cross-sector results from the June 2026 verification program, each
independently machine-verified (\texttt{sympy}/\texttt{mpmath}, $\mathrm{dps}\ge30$) against the
printed, pre-locked DFD constants; none introduces a new free parameter. Status labels follow the
manuscript convention, and the scope of each result---exact identity versus cross-sector
consistency versus tautological restatement---is stated explicitly so that no entry is read as more
than it is.

%-----------------------------------------------------------------------------
\subsection{AY.1\quad The Gatto--Sartori--Tonin closure from locked integers}
%-----------------------------------------------------------------------------

\begin{theorem}[Gatto--Sartori--Tonin closure, (T)]\label{thm:ay-gatto}
The DFD-locked Cabibbo integer $\lambda=|V_{us}|=31\alpha$ (App.~AO) and the locked down-sector mass
map $m_f=A_f\alpha^{n_f}(v/\sqrt2)$ with $(A_d,n_d)=(6,\tfrac52)$ and $(A_s,n_s)=(\tfrac67,\tfrac32)$
(App.~K) give, parameter-free,
\begin{equation}
\fiteq{\frac{m_d}{m_s}=\frac{A_d}{A_s}\,\alpha^{\,n_d-n_s}=7\alpha,\qquad
\frac{m_s}{m_d}=\frac{1}{7\alpha}=19.58,}
\end{equation}
and thereby reproduce the celebrated Gatto--Sartori--Tonin relation $|V_{us}|=\sqrt{m_d/m_s}$ to
\begin{equation}
\fiteq{\frac{|V_{us}|}{\sqrt{m_d/m_s}}=\frac{31\alpha}{\sqrt{7\alpha}}=\sqrt{\frac{31^2\alpha}{7}}
=1.00091,}
\end{equation}
i.e.\ to $0.09\%$---tighter than the relation holds in nature ($0.66\%$).
\end{theorem}
\begin{proof}
$\lambda=31\alpha$ is fixed by the $\mathbb{C}P^2$ line-bundle cohomology window $\{2,3,4\}$:
$31=h^0(\mathcal O(2))+h^0(\mathcal O(3))+h^0(\mathcal O(4))=6+10+15$ (App.~AO). The down-sector
prefactors $A_d=6$, $A_s=6/7$ and exponents $n_d=5/2$, $n_s=3/2$ are the printed finite-Yukawa-ledger
values (App.~K), fixed prior to any comparison. The common $v/\sqrt2$ and the factor $\alpha^{3/2}$
cancel in the ratio, leaving $m_d/m_s=(6\div\tfrac67)\,\alpha^{1}=7\alpha$ (machine-verified,
$\mathrm{dps}=30$). Substituting into the Gatto form,
$|V_{us}|/\sqrt{m_d/m_s}=31\alpha/(7\alpha)^{1/2}=(31^2\alpha/7)^{1/2}$; with
$\alpha^{-1}=137.036$ one has $31^2\alpha=7.0128$, so the ratio is $1.00091$. The relation would be
\emph{exact} iff $31^2\alpha=7$, i.e.\ $\alpha^{-1}=961/7=137.286$; DFD sits $0.18\%$ from that
integer identity.
\end{proof}
\begin{status}
Cross-sector consistency, \emph{not} an exact identity: the Gatto relation is reproduced to $0.09\%$,
equivalent to the integer near-identity $31^2\alpha\simeq7.013$. No constant is fitted---the
exact-making coefficient would be $7.013$, not the locked $7=b_0$, so DFD lands deliberately
\emph{off}-exact, the opposite of a fit. The up-sector dichotomy is reproduced too: up-Gatto
$|V_{cb}|\simeq\sqrt{m_c/m_t}$ \emph{fails} by $\sim2\times$ in DFD ($0.47$) exactly as it fails in
nature ($0.477$). $m_s/m_d=19.58$ matches FLAG $\sim19.9(\pm1)$ at $<0.5\sigma$. Look-elsewhere: one
canonical textbook relation, all inputs locked beforehand. Tier: \DerivedGrade{}.
\end{status}

%-----------------------------------------------------------------------------
\subsection{AY.2\quad The TM1 solar-mixing angle, parameter-free}
%-----------------------------------------------------------------------------

\begin{theorem}[TM1 solar angle and Jarlskog invariant, (T)]\label{thm:ay-tm1}
Retaining the first tribimaximal column (trimaximal-1, TM1) together with $\mu$--$\tau$ reflection
symmetry, and using the locked reactor relation $\sin^2\theta_{13}=3\alpha$ (flavor master operator,
App.~AT), the solar angle and Jarlskog invariant are fixed with no continuous parameter:
\begin{equation}
\fiteq{\cos^2\theta_{12}\,\cos^2\theta_{13}=\tfrac23\quad\text{(exact, all orders in }\theta_{13}),}
\end{equation}
\begin{equation}
\fiteq{\sin^2\theta_{12}=\frac{1-9\alpha}{3(1-3\alpha)}=0.31841\quad(\theta_{12}=34.35^\circ),
\qquad \sin^2\theta_{12}\xrightarrow{\rm LO}\tfrac13-2\alpha,}
\end{equation}
\begin{equation}
\fiteq{|J_{\rm PMNS}|=\tfrac12\,c_{13}^2\,s_{13}\,c_{12}\,s_{12}=0.03371\qquad(\delta_{\rm CP}=-\pi/2).}
\end{equation}
\end{theorem}
\begin{proof}
TM1 fixes the first PMNS column to its tribimaximal value $(2,1,1)^{T}/\sqrt6$, preserved by the
residual Klein generator. First-column normalization gives the exact identity
$\cos^2\theta_{12}\cos^2\theta_{13}=2/3$ (\texttt{sympy}: residual $0$). Hence
$\sin^2\theta_{12}=1-\tfrac{2/3}{1-\sin^2\theta_{13}}$; substituting $\sin^2\theta_{13}=3\alpha$ gives
$\sin^2\theta_{12}=(1-9\alpha)/(3(1-3\alpha))$, with leading term $\tfrac13-2\alpha$. Maximal
$\theta_{23}=\pi/4$ and $\delta_{\rm CP}=-\pi/2$ follow from $\mu$--$\tau$ reflection: the TM1
$\cos\delta$ numerator vanishes identically in $\theta_{13}$ at $\theta_{23}=\pi/4$
(\texttt{sympy}-verified). The Jarlskog then evaluates to $|J|=0.03371$.
\end{proof}
\begin{status}
Parameter-free given the locked tribimaximal base, $\sin^2\theta_{13}=3\alpha$, maximal $\theta_{23}$,
and $\delta=-\pi/2$ (all already in the corpus). The genuinely new content is the TM1 solar closed
form and the $2/3$ invariant, which supersede the three mutually inconsistent ad-hoc correction
sources previously used for $\theta_{12}$ (Sec.~\ref{sec:quantum}). $\sin^2\theta_{12}=0.318$ versus
NuFIT~6.0 $0.307(^{+12}_{-11})$ is $+0.95\sigma$ (inside $3\sigma$). Falsifier: a confirmed
non-maximal $\theta_{23}$, or $\sin^2\theta_{12}$ outside $[0.30,0.34]$, breaks the TM1 lock. Tier:
\DerivedGrade{}.
\end{status}

%-----------------------------------------------------------------------------
\subsection{AY.3\quad Mass-scale hierarchy rank}
%-----------------------------------------------------------------------------

\begin{theorem}[Mass-scale hierarchy rank, (T)]\label{thm:ay-hierarchy}
The seven DFD mass-scale $\alpha$-exponents
\[
\fiteq{(x_v,\,x_{M_R},\,x_{\Lambda_{\rm DFD}},\,x_{m_3},\,x_{H_0},\,x_{a_0},\,x_{\Lambda_{\rm top}})
=(8,\,3,\,\tfrac{19}{2},\,14,\,\tfrac{57}{2},\,29,\,\tfrac12)}
\]
satisfy exactly five independent rational linear relations (rank $5$ over $\mathbb{Q}$), leaving two
free directions: the trivial overall Planck unit, and the normalization of the cosmological-clock
block $\{H_0,a_0\}$ relative to the hadronic/electroweak block. Adjoining the cube-law relation
$x_{H_0}=3x_{\Lambda_{\rm DFD}}$ (App.~O) welds the second direction, reducing the hierarchy to a
single generator $\alpha$.
\end{theorem}
\begin{proof}
The five solid relations are the four-sector lock $2x_{\Lambda_{\rm DFD}}=2x_v+x_{M_R}$ ($19=16+3$),
the $M_R$-cube identity $3x_{M_R}=x_{\Lambda_{\rm DFD}}-x_{\Lambda_{\rm top}}$
($9=\tfrac{19}{2}-\tfrac12$), the seesaw relation $x_{m_3}=1+2x_v-x_{M_R}$ ($14=1+16-3$), the
MOND-step $x_{a_0}=x_{H_0}+x_{\Lambda_{\rm top}}$ ($29=\tfrac{57}{2}+\tfrac12$), and the single-brick
value $x_{\Lambda_{\rm top}}=\tfrac12$. The $7$-column coefficient matrix has rank $5$ over
$\mathbb{Q}$ (\texttt{sympy}), with nullspace spanned by the all-units direction and the clock-block
direction $[0,0,0,0,1,1,0]$. The cube law $x_{H_0}-3x_{\Lambda_{\rm DFD}}=0$ pairs nontrivially with
the clock direction and raises the rank to $6$.
\end{proof}
\begin{remark}[$x_v=8$ is derived, not a residual axiom]\label{rem:ay-xv-derived}
The base exponent $x_v=8$ of the harmonic ladder is \emph{derived}, not an
axiom: $x_v=8=\dim(\mathbb{C}P^2\times S^3)+1=7+1$. The internal dimension
$X=7$ is the $(3,2,1)+$singlet gauge-partition theorem (Ext.\ Deriv.,
App.~AH1a, Steps~1--5: $\dim\mathbb{C}P^2=4$ plus $\dim S^3=3$), and the $+1$ is
the radial/Higgs normal mode. Data independently pin the exponent to
$7.99989\simeq8.0000$ (4 sig figs from $v=M_P\alpha^{x_v}\sqrt{2\pi}$), so it is
\emph{not} a back-solved target. The harmonic ladder therefore closes at the
exponent level; the only sub-theorem-grade residues that remain are the
$\sqrt{2\pi}$ loop-normalization prefactor and the ``$+1=$ radial mode''
identification. We do \emph{not} claim $\sqrt{2\pi}$ is derived.
\end{remark}
\begin{remark}[The elimination identities are tautological, not independent tests]
Because every observable above is a single power of the one generator $\alpha$, eliminating $\alpha$
between any pair yields a monomial identity---e.g.\ $(M_R/M_P)^8=(v/\sqrt{2\pi}M_P)^3$,
$(H_0t_P)^2=(M_R/M_P)^{19}$, $(a_0/2a_P)^3=(M_R/M_P)^{29}$. These ``syzygies'' are \emph{equivalent
restatements} of the individual $\alpha$-exponents: each reduces to $\alpha^k=\alpha^k$ and carries
\emph{no} information beyond them. They are satisfied identically and therefore cannot serve as
independent falsifiers. The genuine, testable content is (i)~the rank-$5$ integer structure above and
(ii)~the individual $\alpha$-power predictions confronted with data elsewhere in this work. The
reduction to a single generator is contingent on the cube law, which is established only to
effective-field-theory grade (App.~O); it is recorded here as such, not as a closed theorem.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{AY.4\quad Large-number structure}
%-----------------------------------------------------------------------------

\begin{proposition}[Dirac large number as an exact $\alpha$-exponent, (T)]\label{thm:ay-dirac}
The Eddington--Dirac second large number---the ratio of the Hubble time to the electron atomic
time---is an exact power of the gauge coupling,
\begin{equation}
\fiteq{N_2\equiv\frac{1/H_0}{\hbar/m_ec^2}=\frac{m_ec^2}{\hbar H_0}
=\tfrac23\sqrt{\pi}\,\alpha^{-18}=3.43\times10^{38},}
\end{equation}
with integer exponent $-18=-\tfrac{57}{2}+\tfrac{21}{2}$.
\end{proposition}
\begin{proof}
From the locked electron map $m_e/M_P=\tfrac23\sqrt{\pi}\,\alpha^{21/2}$ (App.~K) and the Hubble
hierarchy $H_0t_P=\alpha^{57/2}$ (App.~O),
$N_2=(m_e/M_P)/(H_0t_P)=\tfrac23\sqrt{\pi}\,\alpha^{21/2-57/2}=\tfrac23\sqrt{\pi}\,\alpha^{-18}$. The
exponent is exact; numerically $3.43\times10^{38}$.
\end{proof}
\begin{remark}[Reframing, not a new prediction]
This recasts Dirac's famous coincidence $N_2\sim10^{39}$ as an exact integer exponent of the
$\alpha$-tower rather than an accident. It follows algebraically from the already-locked $m_e$ and
$H_0$ exponents, so it is a structural restatement, not an independent prediction; the $+3\%$ offset
from the measured value is the locked electron-mass leading-order error, not a free parameter.
\end{remark}
