%=============================================================================
% APPENDIX X: NEUTRINO MASS SPECTRUM FROM DFD MICROSECTOR
% Complete closed-form derivation: TBM + S₂ lock + microsector-normalized hierarchy
% June 2026: DFD-closed at zero continuous parameters (one data-selected branch bit; see Revised Statement) + EXTERNAL VERIFICATION (χ² = 0.025 vs NuFIT 6.0)
%=============================================================================

\section{Neutrino Mass Spectrum from DFD Microsector}\label{app:neutrino}

This appendix derives a \emph{complete closed-form} neutrino sector from DFD 
microsector relations. Using tribimaximal (TBM) mixing geometry, a discrete 
$S_2$ residual symmetry, and microsector-normalized $\alpha$-power exponents,
we obtain neutrino mass ratios with \textbf{zero continuous parameters}.

%-----------------------------------------------------------------------------
\subsection{DFD Inputs from the Microsector}\label{app:nu-inputs}
%-----------------------------------------------------------------------------

DFD provides three ingredients:

\begin{enumerate}
\item \textbf{TBM mixing geometry} (Appendix~\ref{app:alpha-derivations}): The 
``neutrinos-at-center'' overlap rule gives the tribimaximal mixing matrix
\begin{equation}
\fiteq{U_{\mathrm{TBM}} = \begin{pmatrix}
\sqrt{2/3} & \sqrt{1/3} & 0 \\
-\sqrt{1/6} & \sqrt{1/3} & \sqrt{1/2} \\
\sqrt{1/6} & -\sqrt{1/3} & \sqrt{1/2}
\end{pmatrix}.}
\end{equation}

\item \textbf{Heavy Majorana scale} (Appendix~\ref{app:P_kalpha_MR}):
\begin{equation}
\fiteq{M_R = M_P \alpha^3 \approx 4.7 \times 10^{12}~\text{GeV}.}
\end{equation}

\item \textbf{Electroweak scale} (Section~\ref{sec:quantum}):
\begin{equation}
\fiteq{v = M_P \alpha^8 \sqrt{2\pi} \approx 246~\text{GeV}.}
\end{equation}
\end{enumerate}

TBM fixes the eigenvectors but not the eigenvalues $(m_1, m_2, m_3)$. The 
question is: can the residual symmetry structure fix the mass ratios without 
continuous parameters?

%-----------------------------------------------------------------------------
\subsection{Why $S_3$ Invariance Cannot Split the Doublet}\label{app:nu-s3}
%-----------------------------------------------------------------------------

Let three generations carry the permutation representation of $S_3$. The 
$S_3$-invariant endomorphisms are spanned by $I_3$ and $J = \mathbf{1}\mathbf{1}^T$.

The representation decomposes as $\mathbf{3} \cong \mathbf{1} \oplus \mathbf{2}$, 
where $\mathbf{1} = \mathrm{span}(1,1,1)$ is the singlet and 
$\mathbf{2} = \{x_1 + x_2 + x_3 = 0\}$ is the doublet.

On the doublet, $J$ acts as zero (since $Jx = (x_1+x_2+x_3)\mathbf{1} = 0$), so 
any $S_3$-equivariant operator restricted to the doublet is proportional to 
the identity:
\begin{equation}
\fiteq{A|_{\mathbf{2}} = a\,I_{\mathbf{2}} \quad \Rightarrow \quad \text{degenerate eigenvalues}.}
\end{equation}

\noindent
\begin{tcolorbox}[colback=yellow!5!white, colframe=yellow!60!black]
\textbf{Key insight:} Any $m_2/m_1 \neq 1$ requires breaking $S_3$ to a proper 
subgroup. This is not a bug—it is the mechanism.
\end{tcolorbox}

%-----------------------------------------------------------------------------
\subsection{TBM Selects a Canonical Residual $S_2$}\label{app:nu-s2}
%-----------------------------------------------------------------------------

TBM naturally singles out the $\mu \leftrightarrow \tau$ transposition as 
residual symmetry:
\begin{equation}
\fiteq{S_{\mu\tau} = \begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{pmatrix}.}
\end{equation}

Its eigenvectors in the $\mu$--$\tau$ plane are the even and odd parity axes:
\begin{equation}
\fiteq{v_+ = \frac{1}{\sqrt{2}}(0,1,1), \qquad v_- = \frac{1}{\sqrt{2}}(0,1,-1),}
\end{equation}
with $S_{\mu\tau} v_\pm = \pm v_\pm$.

The third TBM column is exactly $v_+$. Thus TBM motivates a canonical residual 
transposition subgroup $S_2 = \langle S_{\mu\tau} \rangle$.

%-----------------------------------------------------------------------------
\subsection{Microsector-normalized residual-$S_2$ spurion}\label{app:nu-minimal}
%-----------------------------------------------------------------------------

The rigid choice $O = I_3 + P_-$ (which enforces $m_2/m_1=2$ exactly) is the
\emph{minimal-integer} deformation of the identity consistent with residual
$\mu\leftrightarrow\tau$ symmetry. Here we replace that rigidity by a
\emph{microsector-normalized} coefficient that is still knob-free: the coefficient
is fixed as a discrete channel-fraction exponent of $\alpha$ determined by the
already-locked microsector integers.

\paragraph{Setup.}
Let $P_-$ be the rank-1 projector onto the odd axis $v_-$ as before, and define
the residual-$S_2$ spurion family
\begin{equation}
\fiteq{O(\kappa):= I_3 + \kappa\,P_-,}
\end{equation}
so that on parity eigenstates,
\begin{equation}
\fiteq{O(\kappa)\,v_- = (1+\kappa)\,v_-,\qquad
O(\kappa)\,v_+ = 1\cdot v_+.}
\end{equation}
Thus the doublet mass splitting is
\begin{equation}
\fiteq{\boxed{\;\frac{m_2}{m_1} = 1+\kappa\;}\,.}
\end{equation}

\paragraph{No-hidden-knobs microsector normalization.}
In the microsector construction, the line-bundle degree is fixed by minimal-padding
to $(a,n)=(9,5)$, and the CP$^1$ Toeplitz truncation used elsewhere in the unified
derivations has canonical channel count
\begin{equation}
\fiteq{d_{\mathrm{CP}^1}(k)=k+4 \quad\Rightarrow\quad d_{\mathrm{CP}^1}(a)=a+4=13.}
\end{equation}
Residual $\mu\leftrightarrow\tau$ splitting is a two-channel deformation (a doublet),
so a canonical knob-free choice is to assign the spurion strength to the \emph{doublet
channel fraction} $2/13$ in the only universal dimensionless base available to DFD,
namely $\alpha$:
\begin{equation}
\fiteq{\boxed{\; \frac{m_2}{m_1} = \alpha^{-2/13}\;}\qquad\Rightarrow\qquad
\boxed{\;\kappa = \alpha^{-2/13}-1\;}.}
\end{equation}

\paragraph{Canonical-shift variant (Branch B).}
A second, equally canonical knob-free option replaces the numerator $2$ (doublet count)
by the CP$^2$ canonical shift $3$ (the $K^{-1}$ degree on CP$^2$), while the denominator
is fixed by the CP$^1$ channel count induced by the microsector dimension $\dim(\mathbb{CP}^2\times S^3)=7$:
\begin{equation}
\fiteq{d_{\mathrm{CP}^1}(\dim M)=\dim M + 4 = 7+4 = 11,}
\end{equation}
yielding the alternative
\begin{equation}
\fiteq{\boxed{\; \frac{m_2}{m_1} = \alpha^{-3/11}\;}\qquad\Rightarrow\qquad
\boxed{\;\kappa = \alpha^{-3/11}-1\;}.}
\end{equation}

\paragraph{Singlet-doublet hierarchy (microsector-normalized).}
Replace the rigid $r=\alpha^{-1/3}$ ansatz by a microsector-normalized hierarchy
built from locked integers $\dim M=7$ and $n=5$:
\begin{equation}
\fiteq{\boxed{\;\frac{m_3}{m_2} = r:= \alpha^{-\dim M/(4n)} = \alpha^{-7/20}\;}.}
\end{equation}


%-----------------------------------------------------------------------------
\subsection{Combined mass pattern (microsector-normalized)}\label{app:nu-pattern}
%-----------------------------------------------------------------------------

With either choice for $m_2/m_1$ above and the microsector-normalized $r$, the mass
pattern is fixed up to one overall scale:
\begin{equation}
\fiteq{\begin{gathered}
m_1: m_2: m_3 = 1: k: kr, \\
k \in \{\alpha^{-2/13},\,\alpha^{-3/11}\}, \quad r = \alpha^{-7/20}.
\end{gathered}}
\end{equation}


%-----------------------------------------------------------------------------
\subsection{Parameter-free oscillation invariant (discriminator)}\label{app:nu-invariant}
%-----------------------------------------------------------------------------

Fix the overall scale by matching $\Delta m^2_{21}$, so that
\begin{equation}
\fiteq{m_1^2 = \frac{\Delta m^2_{21}}{k^2-1},\qquad
m_2 = k\,m_1,\qquad
m_3 = r\,m_2.}
\end{equation}
Then the dimensionless oscillation invariant becomes a pure $\alpha$-function:
\begin{equation}
\fiteq{\boxed{\;\frac{\Delta m^2_{32}}{\Delta m^2_{21}}
= \frac{(k^2 r^2 - k^2)}{(k^2-1)}\;}
\qquad
(k,r \ \text{as above}).}
\end{equation}


%-----------------------------------------------------------------------------
\subsection{Complete numerical predictions}\label{app:nu-numbers}
%-----------------------------------------------------------------------------

Using $\alpha^{-1}=137.035999084$ and $\Delta m^2_{21}=7.49\times10^{-5}\,\mathrm{eV}^2$
(NuFIT 6.0), the two branches give:

\begin{table}[h]
\centering
\caption{Neutrino mass branch predictions.}\label{tab:neutrino-branches}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lcccccc}
\toprule
Branch & $k$ & $r$ & $(m_1,m_2,m_3)$ [meV] & $\Sigma m_\nu$ [meV] & $\Delta m^2_{32}$ [$10^{-3}$eV$^2$] & $\Delta m^2_{32}/\Delta m^2_{21}$\\
\midrule
A & $\alpha^{-2/13}$ & $\alpha^{-7/20}$ & $(4.60,\ 9.80,\ 54.84)$ & 69.24 & 2.911 & 38.87\\
B & $\alpha^{-3/11}$ & $\alpha^{-7/20}$ & $(2.34,\ 8.96,\ 50.16)$ & 61.46 & 2.437 & 32.54\\
\midrule
\multicolumn{3}{l}{NuFIT 6.0 (NO):} & --- & --- & $2.438\pm0.020$ & $32.55$\\
\bottomrule
\end{tabular}%
}
\end{table}

\textbf{Branch B matches NuFIT 6.0 to $< 0.1\sigma$.}

\begin{remark}[Revised Statement (branch selection) --- June 2026]
Branch A ($k=\alpha^{-2/13}$, doublet count $2$ over $d_{\mathrm{CP}^1}(a)=13$) and Branch B ($k=\alpha^{-3/11}$, canonical shift $3$ over $d_{\mathrm{CP}^1}(\dim M)=11$) are \emph{equally canonical} knob-free constructions; the present axioms do not force the choice between them. The selection of Branch B is made by the oscillation data: Table~\ref{tab:neutrino-branches} shows Branch A's $\Delta m^2_{32}$ missing NuFIT~6.0 by $\approx 24\sigma$, while Branch B ($2.437\times10^{-3}\,\mathrm{eV}^2$) matches to $<0.1\sigma$. The sector is therefore closed at \emph{zero continuous parameters} with \emph{one binary choice anchored empirically}; downstream of that bit, both exponents, the dimensionless oscillation invariant, and the absolute-scale formula $m_3 = \tfrac{14}{13}\pi M_P\alpha^{14}$ are $\alpha$-locked with no further reference to data. The quoted goodness-of-fit ($\chi^2 = 0.025$, $p = 0.99$) is that of the \emph{selected} branch; for a one-of-two discrete selection the look-elsewhere correction is at most a factor of two and does not alter the conclusion. We note --- new in this revision --- that the data-free priming scale of App.~\ref{app:nu-abs-scale} reproduces the $\Delta m^2_{21}$-anchored Branch-B spectrum to $0.1\%$, whereas the same scale under Branch A misses the observed $\Delta m^2_{21}$ by $\approx 6\sigma$: a second, correlated empirical concordance conditioned on Branch B. A candidate topological forcing argument for Branch B exists in the companion volume (T56: integer $11 = \dim H^0(\mathbb{C}P^1,\mathcal{O}(10))$, integer $20 = k_{\max}/3$); if verified at theorem grade it would upgrade this selection from data-selected to derived --- the conservative data-selected form is stated here and in the abstract. This is the same status discipline applied to the CKM apex branch ambiguity before its Euler-projection closure (Appendix~\ref{app:ckm-update}): the binary selection is recorded openly rather than silently absorbed.
\end{remark}

In TBM (with $U_{e3}=0$), the beta-decay and $0\nu\beta\beta$ effective masses are
\begin{align}
m_\beta &= \sqrt{\tfrac{2}{3}m_1^2+\tfrac{1}{3}m_2^2}, \\
m_{\beta\beta} &\in \left[\left|\tfrac{2}{3}m_1-\tfrac{1}{3}m_2\right|,\, \tfrac{2}{3}m_1+\tfrac{1}{3}m_2\right].
\end{align}
For Branch B this yields
\begin{equation}
\fiteq{\boxed{\;m_\beta \approx 5.52\ \mathrm{meV},\qquad
m_{\beta\beta} \in [1.43,\ 4.55]\ \mathrm{meV}\;}}
\end{equation}
with $\Sigma m_\nu \approx 61.5\ \mathrm{meV}$.

\paragraph{Structural identity.}
For the Branch B pair $(k,r)=(\alpha^{-3/11},\alpha^{-7/20})$ one has
\begin{equation}
\fiteq{\boxed{k^2 r^2 = \alpha^{-(6/11+7/10)}=\alpha^{-137/110}}}
\end{equation}
so the combined hierarchy exponent contains the canonical $\alpha^{-1}$ numerator $137$
as an arithmetic consequence of the locked rational channel fractions.

%-----------------------------------------------------------------------------
\subsection{Absolute-scale closure for Branch B from finite-$d$ priming}\label{app:nu-abs-scale}
%-----------------------------------------------------------------------------

This subsection replaces the $\Delta m^2_{21}$ anchoring step with a
DFD-internal absolute-scale closure. The key input is a \emph{forced} finite-dimensional
normalization factor from the same Toeplitz truncation and determinant priming used
in the $\alpha$-locking derivation.

\paragraph{Bundle-degree bookkeeping (no knobs).}
The microsector bundle decomposition is $E=\mathcal{O}(a)\oplus\mathcal{O}^{\oplus n}$ with minimal-padding
$(a,n)=(9,5)$.
The Toeplitz truncation on $\mathbb{C}P^1\subset\mathbb{C}P^2$ carries the Spin$^c$ determinant shift
$L_{\det}=K^{-1}=\mathcal{O}(3)$.
For a Yukawa/Dirac vertex, one inserts the Higgs hyperplane factor: generation
wavefunctions are holomorphic sections of $\mathcal{O}(1)$, so the Dirac overlap lives in
\begin{equation}
\fiteq{\mathcal{O}(a)\;\otimes\;\mathcal{O}(3)\;\otimes\;\mathcal{O}(1)\;\cong\;\mathcal{O}(a+4)
\qquad\text{on}\qquad \mathbb{C}P^1.}
\end{equation}
Thus the Toeplitz level is \emph{forced} to be $m_\nu=a+4$ for the neutrino Dirac sector.

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black]
\textbf{Lemma (Forced finite dimension).}
With $m_\nu=a+4$, the truncated holomorphic state space has dimension
\begin{equation}
\fiteq{d_\nu = \dim H^0(\mathbb{C}P^1,\mathcal{O}(m_\nu)) = m_\nu+1 = a+5.}
\end{equation}
For $a=9$: $\boxed{d_\nu=14}$.
\end{tcolorbox}

\paragraph{Why $d/(d-1)$ appears (not a fit).}
The primed determinant prescription removes the null channel from the finite-dimensional spectrum.
At the level of normalized traces, passing from an unprimed average over $d$ channels to a
primed average over $d-1$ nonzero channels multiplies the normalization by $d/(d-1)$.

Define the \textbf{neutrino finite-$d$ priming factor}:
\begin{equation}
\fiteq{F_\nu:= \frac{d_\nu}{d_\nu-1} = \frac{14}{13}.}
\end{equation}

\paragraph{DFD absolute-scale closure.}
The seesaw closure gives $m_3\propto \pi M_P \alpha^{14}$.
The finite-$d$ priming factor lifts this to:
\begin{equation}\label{eq:m3-branchB-abs}
\fiteq{\boxed{
m_3 = F_\nu\,\pi M_P\,\alpha^{14} = \frac{14}{13}\,\pi M_P\,\alpha^{14}.
}}
\end{equation}

With Branch B ratios $k=\alpha^{-3/11}$, $r=\alpha^{-7/20}$, we get $m_2=m_3/r$ and $m_1=m_3/(kr)$.
Using $\alpha^{-1}=137.036$:

\noindent
\begin{tcolorbox}[colback=red!5!white, colframe=red!60!black, title=DFD-Closed Neutrino Predictions (Zero Continuous Parameters)]
\centering
\scriptsize
\setlength{\tabcolsep}{2pt}
\fittab{%
\begin{tabular}{lcc}
\toprule
Quantity & DFD & NuFIT 6.0 \\
\midrule
$m_1$ & 2.34 meV & --- \\
$m_2$ & 8.96 meV & --- \\
$m_3$ & 50.16 meV & --- \\
$\sum m_\nu$ & \textbf{61.46 meV} & --- \\
\midrule
$\Delta m^2_{21}$ & $7.48\!\times\!10^{-5}$ eV$^2$ & $(7.49 \pm 0.19)\!\times\!10^{-5}$ \\
$\Delta m^2_{31}$ & $2.51\!\times\!10^{-3}$ eV$^2$ & $(2.513 \pm 0.020)\!\times\!10^{-3}$ \\
\midrule
$m_{\beta\beta}$ & 4.55 meV & --- \\
$m_\beta$ & 5.51 meV & --- \\
\bottomrule
\end{tabular}%
}

\smallskip
\textbf{Both splittings match NuFIT 6.0 to $< 0.2\sigma$ at zero continuous parameters (Branch B).}
\end{tcolorbox}

\paragraph{What was used.}
The absolute-scale closure uses only DFD inputs already present:
\begin{enumerate}[nosep]
\item Minimal-padding microsector integer $a=9$
\item Spin$^c$ determinant shift $+3$
\item Higgs hyperplane factor $\mathcal{O}(1)$
\item Primed-channel prescription
\end{enumerate}
No continuous tuning is introduced. The 14/13 factor is forced by $(a,n)=(9,5)$.

%-----------------------------------------------------------------------------
\subsection{The explicit mass matrix (TBM eigenbasis)}\label{app:nu-matrix}
%-----------------------------------------------------------------------------

With TBM eigenvectors and the microsector-normalized hierarchy, the mass spectrum
is
\begin{equation}
\fiteq{m_1,\qquad m_2 = k\,m_1,\qquad m_3 = k r\,m_1,}
\end{equation}
where
\begin{equation}
\fiteq{k\in\{\alpha^{-2/13},\,\alpha^{-3/11}\},\qquad r=\alpha^{-7/20}.}
\end{equation}

Thus the neutrino mass matrix is
\begin{equation}
\fiteq{\boxed{
M_\nu = m_1\,P_1 + (k m_1)\,P_2 + (k r m_1)\,P_3
}}
\end{equation}
in terms of the TBM projectors $P_i=c_i c_i^T$.

%-----------------------------------------------------------------------------
\subsection{Falsification criteria}\label{app:nu-falsify}
%-----------------------------------------------------------------------------

The DFD-closed Branch B (with absolute scale from finite-$d$ priming) gives
concrete predictions summarized below (normal ordering).

\begin{table}[h]
\centering
\caption{Falsification criteria for DFD neutrino sector.}\label{tab:nu-falsify}
\scriptsize
\setlength{\tabcolsep}{2pt}
\fittab{%
\begin{tabular}{lll}
\toprule
Observable & DFD prediction & Falsification \\
\midrule
$\Delta m^2_{21}$ & $7.48\!\times\!10^{-5}$\,eV$^2$ & NuFIT $>3\sigma$ \\
$\Delta m^2_{31}$ & $2.51\!\times\!10^{-3}$\,eV$^2$ & NuFIT $>3\sigma$ \\
$\sum m_\nu$ & 61.46 meV & $<45$ or $>80$ meV \\
$m_{\beta}$ (TBM) & 5.51 meV & $\beta$-decay incomp. \\
$m_{\beta\beta}$ (TBM) & 4.55 meV & $0\nu\beta\beta < 2$ meV \\
Ordering & Normal & Inverted confirmed \\
\bottomrule
\end{tabular}%
}
\end{table}

%-----------------------------------------------------------------------------
\subsection{External global-fit verification}\label{app:nu-verification}
%-----------------------------------------------------------------------------

\paragraph{NuFIT 6.0 Table 1 check (conservative Gaussian).}
NuFIT 6.0 publishes best-fit values and $1\sigma$ uncertainties for the mass-squared 
splittings~\cite{nufit60}. Using the ``IC24 with SK-atm'' Normal Ordering line in Table~1 of their 
JHEP update:
\begin{align}
\Delta m^2_{21} &= (7.49\pm0.19)\times10^{-5}\,\mathrm{eV}^2, \\
\Delta m^2_{3\ell} &= (2.513^{+0.021}_{-0.019})\times10^{-3}\,\mathrm{eV}^2.
\end{align}
Symmetrizing the second uncertainty to $\sigma_{3\ell}=0.020\times10^{-3}\,\mathrm{eV}^2$, 
the normalized pulls for the DFD Branch B predictions are:
\begin{align}
\mathrm{pull}_{21} &= \frac{7.48-7.49}{0.19} = -0.053\,\sigma, \\
\mathrm{pull}_{3\ell} &= \frac{2.51-2.513}{0.020} = -0.15\,\sigma.
\end{align}
The conservative uncorrelated Gaussian statistic is:
\begin{equation}
\fiteq{\boxed{\chi^2_{\mathrm{Gauss}} = \mathrm{pull}_{21}^2+\mathrm{pull}_{3\ell}^2 \approx 0.025}}
\end{equation}
with 2 degrees of freedom.
corresponding to a $p$-value of $0.99$. Branch~B lands essentially on the published 
global-fit best point.

\paragraph{Including realistic $|U_{e3}|^2$ (discrete $0\nu\beta\beta$ comb).}
Because the corpus imposes $\mu$--$\tau$ reflection symmetry to lock
$\theta_{23}=\pi/4$ and $\delta_{\rm CP}=-\pi/2$ (Thm.~\ref{thm:ay-tm1}), the
\emph{same} symmetry is the Takagi/Autonne reality condition $S\,M_\nu^*\,S=M_\nu$
($S$ = 2--3 reflection) on the Majorana mass matrix, and forces \emph{both} Majorana
phases to the CP-conserving values $\alpha_{21},\alpha_{31}\in\{0,\pi\}$
(Grimus--Lavoura). The Majorana phases are therefore \emph{not} free, and the
$0\nu\beta\beta$ effective mass does \emph{not} scan a continuous band; with the
locked TM1 weights $|U_{e1}|^2=\tfrac23$, $|U_{e2}|^2=\tfrac13-3\alpha$,
$|U_{e3}|^2=3\alpha$ and the Branch-B spectrum it collapses to
$m_{\beta\beta}=\big|\sum_i\eta_i\,|U_{ei}|^2 m_i\big|$, $\eta_i\in\{\pm1\}$:
\begin{equation}
\fiteq{m_{\beta\beta}\in\{0.13,\;2.33,\;3.26,\;5.45\}\,\mathrm{meV}\quad(\text{4 CP-conserving points, not a band}).}
\end{equation}
The kinematic mass is phase-free, $m_\beta=\sqrt{\sum_i|U_{ei}|^2 m_i^2}=9.15\,\mathrm{meV}$,
and $\Sigma m_\nu=61.5\,\mathrm{meV}$. Which of the four parities is physically
realized is fixed by the (deferred) sign structure of $M_\nu$; the symmetry-forced,
theorem-grade content is the \emph{four-point discreteness} and the phase-free
$m_\beta$. All four points lie 1--2 orders of magnitude below current laboratory
$0\nu\beta\beta$ limits, so no datum is consulted to obtain them. (The earlier
$c_{13}^2$-scaled-TBM scan reported a continuous band $[0.29,5.55]$~meV; its
endpoints are just the all-aligned/anti-aligned CP-conserving points, so the
symmetry replaces that band with the four discrete teeth above.)

\paragraph{Reproducibility.}
A helper script \texttt{scripts/\allowbreak{}scripts\_\allowbreak{}nufit\_\allowbreak{}table1\_\allowbreak{}gaussian\_\allowbreak{}eval.py} reproduces 
this conservative check:
{\footnotesize
\begin{verbatim}
python3 scripts/scripts_nufit_table1_gaussian_eval.py \
 --dm21 7.48e-5 --dm3l 2.51e-3
\end{verbatim}
}

\paragraph{Optional: profile-level $\Delta\chi^2$ evaluation.}
The Gaussian check above is intentionally conservative (it uses only the published Table~1
central values and $1\sigma$ widths). NuFIT additionally publishes 1D $\Delta\chi^2$ profiles
for each oscillation parameter. To evaluate the DFD prediction against those profiles,
we include \texttt{scripts/\allowbreak{}scripts\_\allowbreak{}nufit\_\allowbreak{}chi2\_\allowbreak{}eval.py}, which (i) loads the NuFIT profile
tables, (ii) interpolates $\Delta\chi^2(x)$, and (iii) reports the total $\chi^2$ for the
predicted parameter vector under the chosen ordering/data set.

We do not hard-code the profile files here (NuFIT periodically updates file names), but the
script documents the expected plain-text format and directory layout.

%-----------------------------------------------------------------------------
\subsection{Summary: DFD-closed neutrino sector (zero continuous parameters)}\label{app:nu-summary}
%-----------------------------------------------------------------------------

\begin{theorem}[Conditional neutrino-sector closure]\label{thm:nu-closure}
Conditional on a single data-selected binary input---the choice of Branch~B over
Branch~A (the doublet-ratio bit, fixed empirically by $\Delta m^2_{32}$; see the
Revised Statement of \S\ref{app:nu-numbers})---the entire light-neutrino sector is
fixed at \emph{zero continuous parameters}:
\begin{enumerate}[nosep]
\item \textbf{Ordering is structural (normal).} The microsector exponents
$k=\alpha^{-3/11}>1$ and $r=\alpha^{-7/20}>1$ force $m_1<m_2<m_3$; inverted
ordering is impossible.
\item \textbf{Absolute scale is $\alpha$-locked.}
$m_3=\tfrac{14}{13}\pi M_P\alpha^{14}=50.16$~meV (Eq.~\ref{eq:m3-branchB-abs}),
with the finite-$d$ priming factor $14/13$ obtained two independent ways
($14/13=1+\tfrac{1}{N_{\rm gen}}\sin^2\theta_W$ with $\sin^2\theta_W=3/13$, and
$d_\nu/(d_\nu-1)$ with $d_\nu=a+5=14$), giving
$(m_1,m_2,m_3)=(2.34,8.96,50.16)$~meV and $\Sigma m_\nu=61.46$~meV.
\item \textbf{Mixing is parameter-free (TM1).}
$\cos^2\theta_{12}\cos^2\theta_{13}=2/3$ exactly and
$\sin^2\theta_{12}=(1-9\alpha)/\big(3(1-3\alpha)\big)=0.3184$
(Thm~\ref{thm:ay-tm1}).
\end{enumerate}
The resulting splittings match NuFIT~6.0 to $<0.2\sigma$ ($\chi^2=0.025$, $p=0.99$)
with no continuous fit parameter. The single branch bit is the only empirical input;
a candidate topological forcing of that bit (T56) is recorded in the companion
volume and, if verified at theorem grade, would render the sector fully derived.
\end{theorem}

\noindent
\begin{tcolorbox}[colback=green!5!white, colframe=green!60!black, title=Neutrino Sector Summary (ZERO CONTINUOUS PARAMETERS \& VERIFIED)]
\textbf{Derivation chain (zero continuous parameters; one data-selected binary branch choice, see the Revised Statement in App.~\ref{app:nu-numbers}):}
\begin{enumerate}[nosep]
\item TBM from ``neutrinos-at-center'' $\to$ $\mu\leftrightarrow\tau$ residual $S_2$
\item Microsector integers $(a,n)=(9,5)$, $\dim M=7$ lock channel fractions
\item $\boxed{k = m_2/m_1 = \alpha^{-3/11}}$ (Branch B; selected over Branch A by the oscillation data)
\item $\boxed{r = m_3/m_2 = \alpha^{-7/20}}$ (from $\dim M/(4n)$)
\item Dirac overlap in $\mathcal{O}(a+4)$ $\to$ $d_\nu = a+5 = 14$
\item Finite-$d$ priming factor $F_\nu = 14/13$
\item $\boxed{m_3 = \frac{14}{13}\,\pi\,M_P\,\alpha^{14}}$ (absolute scale)
\end{enumerate}

\textbf{Striking arithmetic identities:}
\[
\fiteq{\begin{aligned}
k^2 r^2 &= \alpha^{-137/110} \quad \text{(numerator $= \alpha^{-1}$)}\\
m_3 &= \frac{14}{13}\,\pi\,M_P\,\alpha^{14} \quad \text{($14 = a+5$, $13 = a+4$)}
\end{aligned}}
\]

\textbf{DFD predictions vs NuFIT 6.0:}
\begin{center}
\scriptsize
\setlength{\tabcolsep}{2pt}
\fittab{%
\begin{tabular}{lccc}
\toprule
Observable & DFD & NuFIT 6.0 & Pull \\
\midrule
$\Delta m^2_{21}$ & $7.48\!\times\!10^{-5}$ & $(7.49 \pm 0.19)\!\times\!10^{-5}$ & $-0.05\sigma$ \\
$\Delta m^2_{31}$ & $2.51\!\times\!10^{-3}$ & $(2.513 \pm 0.020)\!\times\!10^{-3}$ & $-0.15\sigma$ \\
\bottomrule
\end{tabular}%
}
\end{center}
\textbf{Combined: $\chi^2 = 0.025$ (2 dof), $p = 0.99$.}

\textbf{Complete predictions:}
\begin{itemize}[nosep]
\item $(m_1, m_2, m_3) = (2.34, 8.96, 50.16)$ meV
\item $\sum m_\nu = 61.46$ meV
\item $m_\beta = 5.51$ meV (TBM), $9.22$ meV (with $\theta_{13}$)
\item $m_{\beta\beta} = 4.55$ meV (TBM), $[0.29, 5.55]$ meV (with phases)
\end{itemize}

\textbf{Status: DFD-CLOSED AT ZERO CONTINUOUS PARAMETERS \& EXTERNALLY VERIFIED.} No continuous empirical input; one binary branch choice (B over A) is selected by the oscillation data.
Every number then derives from $\alpha$, $M_P$, and locked microsector integers.
The prediction matches NuFIT 6.0 with $\chi^2 = 0.025$.
\end{tcolorbox}
