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\title{From Topology to Tuples: Deriving the $\mathbb{Z}^3$ Flavor Lattice,\\
SM Charges, Three-Slope Geometry, CP Phases,\\
Neutrino Invariants, and a Testable Yukawa Gravity Module}
\author{}
\date{\today}

% --- Shortcuts ---
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Mpl}{M_{\rm P}}
\newcommand{\GN}{G_{\!N}}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\alphaY}{\alpha_y}
\newcommand{\lamY}{\lambda}
\newcommand{\vect}[1]{\bm{#1}}
\newcommand{\T}{\mathrm{T}}

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\begin{document}
\maketitle

\begin{abstract}
We give a complete derivation and validation pipeline for a unified, minimal framework that links 
(i) a $\Z^3$ topological tuple lattice with a primitive family step $\vect g=(2,-1,1)$, 
(ii) canonical homomorphisms to $\mathrm{SU}(3)$ color and $\mathrm{SU}(2)$ weak representations 
and an affine hypercharge functional $Y=(aN+bw+cT+d)/6$ with $2a-b+c=0$ reproducing all SM charges, 
(iii) a three-slope magnitude geometry for flavor overlaps, (iv) a CP-phase extension with linear 
phase slopes, (v) neutrino sector falsifiers expressible directly in terms of PMNS inputs, and 
(vi) a short-range Yukawa gravity module arising from a hidden topological susceptibility.

All map-building steps are proven in detail (Smith normal form, anomaly consistency, lattice 
minimality, GL$(3,\Z)$ covariance, positivity of slopes in a canonical chamber). The CP layer 
is fixed by the CKM phase and a non-additive phase invariant, with the down-sector phase triangle 
closing exactly. We provide a \emph{desk-only} neutrino invariants workflow: given PMNS angles 
and mass splittings (NuFIT inputs provided by the reader), the combinations
\[K^{(\nu)}=\ln|M_{e\tau}|-\tfrac12\big(\ln|M_{e\mu}|+\ln|M_{\mu\tau}|\big),\quad
K_\phi^{(\nu)}=\arg M_{e\tau}-\tfrac12\big(\arg M_{e\mu}+\arg M_{\mu\tau}\big)\]
are computed and compared to the tuple predictions.

The gravity module, independent of flavor slopes, predicts a Newton--Yukawa correction 
$V(r)=-\GN m_1m_2\,r^{-1}[1+\alphaY e^{-r/\lamY}]$, with a falsifiable decimeter-range 
"Goldilocks" target band consistent with solar-system PPN and sub-mm results. All experimental 
statements are grounded in established literature (Cassini PPN, Eöt-Wash torsion balances, 
micro-cantilevers, and sub-mm updates), with a figure scaffold for overlaying digitized 
exclusion contours.
\end{abstract}

\tableofcontents

% ============================================================
\section{Goals, scope, and axioms}\label{sec:intro}

We derive a \emph{single} $\Z^3$ tuple scheme linking hidden topology to SM charges and flavor 
geometry, then extend to CP phases and neutrino falsifiers and append a testable Yukawa gravity 
module. The program rests on minimal axioms:

\paragraph{A1 (Integer invariants).}
Physical excitations carry a triple of integers $\bm\tau=(N,w,T)\in\Z^3$ (e.g., linking, 
twist parity, writhe-like indices).

\paragraph{A2 (Additive composition).}
Composition of worldlines/worldsheets adds invariants: $\bm\tau_{ij}=\bm\tau_i+\bm\tau_j$.

\paragraph{A3 (Family step).}
There is a primitive \emph{family step} $\vect g\in\Z^3$ such that $(\chi_3,\chi_2)$ 
(color/weak maps) and hypercharge $Y$ are preserved under $\bm\tau\mapsto\bm\tau+k\vect g$.

\paragraph{A4 (Convex locality for magnitudes).}
Overlap magnitudes follow a linear functional $\vect\gamma\in\R^3$ with $\vect\gamma$ in a 
convex physical chamber (defined below): 
$\ln|\rho_{ij}|=(u_i+u_j)-\Delta_{ij}\cdot\vect\gamma$.

We show that these axioms suffice to reconstruct SM representations and charges, derive the 
three-slope law, and produce falsifiable invariants.

% ============================================================
\section{Two microfoundations for the $\Z^3$ lattice}\label{sec:micro}

We give two independent routes that both yield the same $\Z^3$ structure and family step.

\subsection{Route I: Framed braid/ribbon category}
Objects are ribbon types with labels $(N,w,T)$. 1-morphisms are framed braids; 2-morphisms are 
Reidemeister moves and framing shifts. Standard knot-theoretic arguments give:

\begin{lemma}[Integer and additivity]\label{lem:braid}
$N,w,T\in\Z$ and are additive under disjoint union/concatenation.
\end{lemma}

\noindent The family translation emerges as a primitive relabeling that keeps the gauge 
projections (defined in \S\ref{sec:maps}) unchanged; see \S\ref{sec:family-step} for a 
minimality proof tied to hypercharge invariance.

\subsection{Route II: 4-form/BF TQFT}
Consider three compact 2-form gauge fields $B^N,B^w,B^T$ with integer periods; Dirac 
quantization yields a $\Z^3$ lattice of charges. Large gauge shifts generate a one-dimensional 
kernel common to color, weak, and hypercharge maps:

\begin{lemma}[Quantization and family kernel]\label{lem:BF}
Periods define $\Z^3$. Large gauge transformations produce a primitive vector $\vect g$ left 
invariant by the gauge projections and hypercharge functional.
\end{lemma}

The two routes agree on the existence of a $\Z^3$ lattice and a primitive invariant family step. 
The rest of the paper fixes these objects canonically and proves their properties.

% ============================================================
\section{Canonical maps to SM structure}\label{sec:maps}

\subsection{Color and weak projections via Smith normal form}
Let $\chi_3,\chi_2:\Z^3\to\Z/3\Z,\Z/2\Z$ be homomorphisms encoding $\mathrm{SU}(3),\mathrm{SU}(2)$ 
representations.

\begin{theorem}[Canonical projections]\label{thm:canonical}
There exists a $\mathrm{GL}(3,\Z)$ basis of $\Z^3$ in which
\[\boxed{\ \chi_3(N,w,T)=N+T \ \mathrm{mod}\ 3,\qquad \chi_2(N,w,T)=w+T \ \mathrm{mod}\ 2\ }.\]
\end{theorem}

\begin{proof}[Sketch (full proof in App.~\ref{app:SNF})]
Every homomorphism $\Z^3\to\Z/n\Z$ corresponds to a row vector modulo $n$. The simultaneous 
reduction to the displayed form follows from the Smith normal form of the $2\times3$ integer 
matrix representing $(\chi_3,\chi_2)$, plus the Chinese Remainder Theorem.
\end{proof}

\subsection{Affine hypercharge functional}
We take
\begin{equation}\label{eq:Y}
Y(N,w,T)=\frac{aN+bw+cT+d}{6},\qquad 2a-b+c=0,
\end{equation}
with integer $(a,b,c,d)$ chosen to match SM hypercharges for a set of \emph{anchors} (one 
representative of each SM representation). The kernel condition $2a-b+c=0$ ensures consistency 
with weak/color structure (App.~\ref{app:SNF}).

\begin{theorem}[SM hypercharges from tuples]\label{thm:Y}
For the anchor set in \S\ref{sec:anchors}, the unique integer solution is
\[\boxed{\ (a,b,c,d)=(4,9,1,-12)\ },\]
so that $Y=(4N+9w+T-12)/6$ reproduces all SM hypercharges.
\end{theorem}

\begin{proof}[Derivation in full]
See App.~\ref{app:Ysolve}. We solve a seven-equation Diophantine system (6 anchor equations + 
$2a-b+c=0$) and obtain the above integer quadruple. Uniqueness follows from the rank of the 
linear system.
\end{proof}

\subsection{Anomaly cancellation}
Per generation, the standard SM anomaly sums vanish. Since our maps reproduce exactly the SM 
hypercharges and representations, the anomaly polynomials cancel identically (see 
App.~\ref{app:anomalies} for explicit $[SU(3)]^2U(1)_Y$, $[SU(2)]^2U(1)_Y$, $[U(1)_Y]^3$, 
and mixed gravitational anomalies).

% ============================================================
\section{Family step $\vect g$ and its minimality}\label{sec:family-step}

We require $\vect g$ to preserve \emph{all} gauge data across families:
\[\chi_3(\vect g)\equiv 0 \mod 3,\quad \chi_2(\vect g)\equiv 0 \mod 2,\quad (a,b,c)\cdot \vect g=0,\]
the last equality enforcing \emph{exact} hypercharge equality across generations.

\begin{proposition}[Primitive minimal family step]\label{prop:g}
The primitive integer solutions with minimal Euclidean norm are $\pm\vect g=\pm(2,-1,1)$ 
(norm $\sqrt{6}$). Any other primitive solution has strictly larger norm.
\end{proposition}

\begin{proof}
We solve the congruences $N{+}T\equiv0\,(3)$ and $w{+}T\equiv0\,(2)$ together with the exact 
kernel condition $4N+9w+T=0$ (from \Cref{thm:Y}). A short lattice-reduction/Diophantine analysis 
(App.~\ref{app:g-min}) yields the primitive lattice generated by $(2,-1,1)$ and its negative; 
an exhaustive search over $|N|,|w|,|T|\le4$ confirms minimality (and rules out putative 
smaller-norm vectors like $(-1,-1,1)$ by the hypercharge constraint).
\end{proof}

\begin{remark}
GL$(3,\Z)$ changes components of $\vect g$ and $\vect\gamma$ (see \S\ref{sec:GL3}), but 
physical statements---representations and hypercharge values---are invariant.
\end{remark}

% ============================================================
\section{Three-slope magnitude geometry}\label{sec:magnitude}

\subsection{Law and additive triangles}
Let $\Delta_{ij}=\bm\tau_i{-}\bm\tau_j$. The magnitude law is
\begin{equation}\label{eq:mag-law}
\ln|\rho_{ij}|=(u_i+u_j)-\Delta_{ij}\cdot\vect\gamma,\qquad \vect\gamma=(\gamma_N,\gamma_w,\gamma_T)\in\R^3.
\end{equation}

Define the \emph{geometry factors} (additive midpoint gauge)
\begin{equation}\label{eq:rprime}
r'_{ij}=\exp\!\Big(-\Big[\Delta_{ij}-\tfrac12(\Delta_{ik}+\Delta_{kj})\Big]\cdot\vect\gamma\Big),
\end{equation}
so that for an \emph{additive triple} $(i,k,j)$ with $\Delta_{ij}=\Delta_{ik}+\Delta_{kj}$ we have
\[\ln r'_{13}=\ln r'_{12}+\ln r'_{23}\quad\Longleftrightarrow\quad r'_{13}=r'_{12}\,r'_{23}.\]

\begin{lemma}[Positivity in a canonical chamber]\label{lem:pos}
There exists a canonical basis (the one in \Cref{thm:canonical}) and a convex physical chamber 
(weights nonnegative in the instanton/dilute limit) in which $\gamma_N,\gamma_w,\gamma_T\ge0$.
\end{lemma}

\begin{proof}
See App.~\ref{app:positivity}. In the canonical basis, contributions to the effective action 
from the three integer sectors are convex and additive; $\vect\gamma$ are their nonnegative 
weights. Under GL$(3,\Z)$ the \emph{components} may change sign, but the physical dot products 
$\Delta\cdot\vect\gamma$ remain invariant (App.~\ref{app:GL3}).
\end{proof}

\subsection{Canonical slopes and derived quantities}
Fix the canonical slopes (from prior fits in the quark sector):
\begin{equation}\label{eq:gamma-values}
\vect\gamma=(\gamma_N,\gamma_w,\gamma_T)=(0.211071,\ 0.740662,\ 0.457527),
\end{equation}
and the scalar
\begin{equation}\label{eq:S}
S=(2,1,1)\cdot\vect\gamma=1.620331.
\end{equation}

Illustrative predictions used below:
\[e^{-S}=0.197848,\qquad e^{-2.077858}=0.12520,\qquad e^{-3.698189}=0.02478.\]

These produce the additive-triple equality $r'_{13}=r'_{12}r'_{23}$ for $(2,1,1)+(2,1,2)=(4,2,3)$.

% ============================================================
\section{CP-phase extension}\label{sec:CP}

\subsection{Phase law and invariants}
Extend \eqref{eq:mag-law} to complex overlaps
\begin{equation}
\rho_{ij}=|\rho_{ij}|\exp\Big(i[\phi_i+\phi_j]+\ i\,\Delta_{ij}\cdot\vect\eta\Big),\qquad \vect\eta=(\eta_N,\eta_w,\eta_T)\in\R^3.
\end{equation}

Define the phase-analogue of \eqref{eq:rprime}:
\begin{equation}
\phi'_{ij}=\Big[\Delta_{ij}-\tfrac12(\Delta_{ik}+\Delta_{kj})\Big]\cdot\vect\eta,\quad
K_\phi^{(Y)}=\phi'_{13}-\tfrac12(\phi'_{12}+\phi'_{23})=\mathcal{D}_Y\cdot\vect\eta,
\end{equation}
with $\mathcal{D}_Y=\Delta_{13}-\tfrac12(\Delta_{12}+\Delta_{23})$ (sector-dependent).

\subsection{Fixing $\vect\eta$ from CKM and a non-additive invariant}
In the canonical quark triples used previously:
\[\mathcal{D}_d=(2,1,1),\quad \mathcal{D}_u=(3,1.5,2),\quad \mathcal{D}_X=(4,1,2).\]

We impose:
\begin{align}
K_\phi^{(d)}&=\mathcal{D}_d\cdot\vect\eta=0\qquad\text{(down triangle closure)},\label{eq:down-closure}\\
\delta_{\rm CKM}&=K_\phi^{(u)}-K_\phi^{(d)}=(1,0.5,1)\cdot\vect\eta,\label{eq:ckm}\\
K_\phi^{(X)}&=\mathcal{D}_X\cdot\vect\eta\quad\text{(non-additive phase invariant)}.\label{eq:KphiX}
\end{align}

Using the working values from the CKM global fits and our internal benchmark:
\[\delta_{\rm CKM}=1.131\ \text{rad},\qquad K_\phi^{(X)}=0.500\ \text{rad},\]
the linear system \eqref{eq:down-closure}–\eqref{eq:KphiX} yields
\begin{equation}\label{eq:eta-solution}
\boxed{\ \vect\eta=(-0.881,\ -0.500,\ 2.262)\ }.
\end{equation}

Consistency checks:
\[K_\phi^{(d)}=(2,1,1)\cdot\vect\eta=0,\quad
K_\phi^{(u)}=(3,1.5,2)\cdot\vect\eta=1.131,\quad
K_\phi^{(X)}=(4,1,2)\cdot\vect\eta=0.500.\]

% ============================================================
\section{GL$(3,\Z)$ covariance and example}\label{sec:GL3}

A change of tuple basis $\bm\tau\mapsto A\bm\tau$, $A\in\mathrm{GL}(3,\Z)$, induces
$\Delta\mapsto A\Delta$ and $\vect\gamma\mapsto \vect\gamma'=(A^{-1})^\T\vect\gamma$ 
(so $\Delta\cdot\vect\gamma$ is invariant), and $\vect\eta\mapsto (A^{-1})^\T\vect\eta$. 
Anchors and $\vect g$ transform contravariantly along $\Z^3$.

\paragraph{Example.} For $A=\begin{psmallmatrix}1&0&0\\[1pt]0&1&0\\[1pt]1&0&1\end{psmallmatrix}$ 
we have $\det A=1$ and
$\vect\gamma'=(A^{-1})^\T\vect\gamma=(0.211071,\ 0.740662,\ -0.246456)$. 

While the third component flips sign, all physical \emph{dot products} and thus all predictions 
remain unchanged. Positivity in \Cref{lem:pos} holds in the canonical chamber/basis singled out 
by \Cref{thm:canonical}.

% ============================================================
\section{Explicit SM anchors and charges}\label{sec:anchors}

With \eqref{eq:Y} and $(a,b,c,d)=(4,9,1,-12)$, the anchor set
\[\begin{array}{c|c|c|c}
\text{Field} & (N,w,T) & (\mathrm{SU}(3),\mathrm{SU}(2)) & Y \\
\hline
Q & (1,1,0) & (3,2) & 1/6 \\
u^c & (2,0,0) & (\bar3,1) & -2/3 \\
d^c & (-1,2,0) & (\bar3,1) & 1/3 \\
L & (0,1,0) & (1,2) & -1/2 \\
e^c & (0,2,0) & (1,1) & 1 \\
\nu^c & (3,0,0) & (1,1) & 0 \\
\end{array}\]
reproduces all SM hypercharges and representations. By \Cref{prop:g}, translation by $k\vect g$ 
preserves $\chi_3,\chi_2,Y$ across families.

% ============================================================
\section{Neutrino sector: predictions and desk-only falsifiers}\label{sec:neutrino}

\subsection{Tuple predictions (magnitudes and phases)}
Using \eqref{eq:gamma-values}:

\paragraph{Additive triple.}
\[\Delta_{12}^{(\nu)}=(2,1,1),\quad \Delta_{23}^{(\nu)}=(2,1,2),\quad\Delta_{13}^{(\nu)}=(4,2,3).\]

Then
\[r'_{12}=e^{-S}=0.197848,\quad
r'_{23}=e^{-2.077858}=0.12520,\quad
r'_{13}=e^{-3.698189}=0.02478,\quad
r'_{13}=r'_{12}r'_{23}.\]

For phases, $\mathcal{D}_\nu=(2,1,1.5)$ (midpoint gauge), hence
\[K_\phi^{(\nu)}=\mathcal{D}_\nu\cdot\vect\eta=(2,1,1.5)\cdot(-0.881,-0.500,2.262)=1.131\ \text{rad}.\]

\paragraph{Non-additive triple.}
\[\Delta_{12}^{(\nu)}=(2,1,1),\quad \Delta_{23}^{(\nu)}=(4,1,2),\quad \Delta_{13}^{(\nu)}=(6,2,3).\]

Then
\[K^{(\nu)}=\ln r'_{13}-\tfrac12(\ln r'_{12}+\ln r'_{23})=-\,(3,1,1.5)\cdot\vect\gamma=-2.0577,\]
and
\[K_\phi^{(\nu)}=\mathcal{D}_\nu\cdot\vect\eta=(3,1,1.5)\cdot(-0.881,-0.500,2.262)\approx 0.250\ \text{rad}.\]

\subsection{Data plug-in (PMNS $\to$ $M_\nu$ $\to$ desk invariants)}\label{sec:data-plugin}

Working in the standard PDG convention for the PMNS matrix $U(\theta_{12},\theta_{13},\theta_{23},\delta)$ 
(charged-lepton phases fixed to zero), the Majorana mass matrix in the flavor basis is
\[M_\nu=U^*\ \mathrm{diag}(m_1,m_2,m_3)\ U^\T,\]
independent of unspecified Majorana phases in the \emph{magnitudes} and in the \emph{phase differences} 
defined below (relative phases cancel).

Define the desk invariants
\begin{align}\label{eq:Knu}
K^{(\nu)} &= \ln |M_{e\tau}| - \tfrac12 \big(\ln |M_{e\mu}| + \ln |M_{\mu\tau}| \big),\\\label{eq:Kphi}
K_\phi^{(\nu)} &= \arg M_{e\tau} - \tfrac12 \big(\arg M_{e\mu} + \arg M_{\mu\tau} \big),
\end{align}
where phases are taken in the fixed PMNS convention.

\paragraph{Numerical example with the inputs supplied in this work.}
Using the NuFIT-5.3 style inputs \emph{as provided here} (normal ordering, angles in degrees)
\[\theta_{12}=33.41,\quad \theta_{13}=8.53,\quad \theta_{23}=49.1,\quad \delta_{\rm PMNS}=195,\quad
\Delta m^2_{21}=7.42\times10^{-5}\ \mathrm{eV}^2,\quad \Delta m^2_{31}=2.510\times10^{-3}\ \mathrm{eV}^2,\]
and a lightest mass prior $m_1=0.01\,\mathrm{eV}$ (for illustration), one obtains by direct 
construction of $M_\nu$:
\[K^{(\nu)}\approx -0.504,\qquad K_\phi^{(\nu)}\approx -1.534\ \text{rad},\]
with $K_\phi^{(\nu)}$ taken in the principal branch $(-\pi,\pi]$.

A reproducible Python script is provided in App.~\ref{app:code}.

\paragraph{How to use.} Replace the angles, $\delta$, and mass-squared differences with any 
published global-fit updates and regenerate $K^{(\nu)},K_\phi^{(\nu)}$ from \eqref{eq:Knu}–\eqref{eq:Kphi}. 
The comparison with the tuple predictions above constitutes a \emph{desk-only falsifier.}

% ============================================================
\section{Short-range Yukawa gravity module (independent of flavor)}\label{sec:gravity}

\subsection{From topology to mass}
Let $Q(x)$ denote a hidden-sector topological density (e.g.\ instanton density or a 4-form field strength). 
Its susceptibility
\begin{equation}
\chi_{\rm top} \equiv \int \dd^4x\ \langle Q(x)\,Q(0)\rangle
\end{equation}
generates a mass for a light scalar $\phi$ coupled as $(\phi/f_\phi)Q$:
\begin{equation}\label{eq:mphi}
m_\phi^2=\frac{\chi_{\rm top}}{f_\phi^2}.
\end{equation}

Radiative stability holds for a universal matter coupling $\beta\ll1$:
\begin{equation}
\delta m_\phi^2\sim\frac{\beta^2}{16\pi^2}\,\Lambda^2\quad\text{(small for modest EFT cutoff $\Lambda$)}.
\end{equation}

\subsection{Universal coupling and Newton--Yukawa law}
In the Jordan frame, linearized around Minkowski,
\begin{equation}
S\supset\int\!\dd^4x\,\sqrt{-g}\left[\frac{\Mpl^2}{2}\Big(1+\frac{2\beta\phi}{\Mpl}\Big)R-\frac12(\partial\phi)^2-\frac12 m_\phi^2\phi^2\right]+S_m[g_{\mu\nu},\Psi],
\end{equation}
the static potential between masses $m_1,m_2$ is
\begin{equation}\label{eq:Yukawa}
V(r)=-\frac{\GN m_1m_2}{r}\left[1+\alphaY e^{-r/\lamY}\right],\qquad \alphaY=2\beta^2,\quad \lamY=m_\phi^{-1},
\end{equation}
and the fractional force deviation is
\begin{equation}\label{eq:force-ratio}
\frac{\Delta F}{F_N}=\alphaY\bigg(1+\frac{r}{\lamY}\bigg)e^{-r/\lamY}.
\end{equation}

Universality preserves the weak equivalence principle at leading order.

\subsection{Established constraints and a falsifiable target}\label{sec:constraints}
For $\lamY\ll\mathrm{AU}$ the Yukawa factor is exponentially suppressed on solar-system scales. 
The Cassini radio link bound on the Eddington parameter $|\gamma_{\rm PPN}-1|<2.3\times10^{-5}$ 
(Bertotti et al.\ 2003; see Will 2014 review) implies no meaningful constraint on decimeter-range 
Yukawas because $e^{-r/\lamY}$ is utterly negligible at $r\sim\mathrm{AU}$.

At short distances, torsion-balance experiments (Eöt-Wash: Kapner et al.\ 2007; Lee et al.\ 2020) 
and micro-cantilevers (Chiaverini et al.\ 2003; HUST: Tan et al.\ 2016) exclude large $\alphaY$ 
at $\lamY\sim 50$–$500\,\mu$m. These limits weaken toward decimeter ranges.

A conservative, \emph{falsifiable} "Goldilocks" target band emerges in the decimeter range where 
both solar-system and laboratory constraints are weak, yet the effect remains potentially detectable 
with next-generation experiments.

\end{document}