% 344_Addendum_FlavorRefutations.tex
% Three flavor-sector predictions of Paper 01 recorded as refuted:
% tri-bimaximal theta_13 = 0, delta_CP ~ 240 deg, and the topological
% strong-CP argument. The abstract of Paper 01 was reframed to flag these
% for the registry (Paper 40); this addendum makes that citation true by
% recording the three as refuted. Verifier: verify_P344.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{booktabs}
\usepackage{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\title{Addendum 344: Flavor-Sector Refutations\\
\large Tri-bimaximal $\theta_{13}$, $\delta_{\mathrm{CP}}$, and the strong-CP claim of Paper 01}
\author{TOE Corpus}
\date{2026-06-15}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} REFUTATION (three predictions recorded refuted) \quad|\quad
\textbf{Addresses:} Paper 01 flavor-sector predictions \quad|\quad
\textbf{Verifier:} \texttt{verify\_P344.py} --- all PASS

\begin{abstract}
Paper 01 advanced three flavor-sector predictions that current data do
not support: the tri-bimaximal value $\theta_{13} = 0$ for the reactor
mixing angle, a leptonic CP phase $\delta_{\mathrm{CP}} \approx 240^\circ$,
and a topological resolution of the strong-CP problem
($\theta_{\mathrm{QCD}} = 0$, ``no axion needed''). The paper's abstract
was reframed to mark these as not having survived current data and to flag
them for the Paper 40 registry. This addendum makes that citation true.
The reactor angle is measured at $\theta_{13} \approx 8.5^\circ$
($\sin^2\theta_{13} \approx 0.022$, NuFIT-6.0), nonzero at more than
$5\sigma$ since Daya Bay (2012), so $\theta_{13} = 0$ is excluded. The
current global fit gives $\delta_{\mathrm{CP}} \approx 177^\circ$
$(+19/-20)^\circ$ for normal ordering, consistent with CP conservation
within $1\sigma$, placing $240^\circ$ roughly $3\sigma$ above the central
value; the paper's claim of agreement rested on the superseded
$197^\circ \pm 25^\circ$ fit. The strong-CP argument constrains the wrong
object: setting the instanton winding number $n = 0$ fixes the gauge
sector, not the Lagrangian $\theta$-coupling, which is the physical
strong-CP parameter and an independent input; strong CP remains an open
problem. Three remarks --- P001\_4, P001\_5, P001\_6 --- are added to
Paper 01 adjacent to the claims they qualify, and the three are recorded
\emph{refuted} in the registry. A corpus that records its own refutations
is functioning as designed.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Tri-bimaximal $\theta_{13} = 0$ (P001\_4)}

\begin{proposition}[The reactor angle is nonzero]
\label{prop:th13}
The tri-bimaximal pattern fixes the reactor angle at $\theta_{13} = 0$.
The measured value is $\sin^2\theta_{13} = 0.02224$, equivalently
$\theta_{13} \approx 8.5^\circ$ (NuFIT-6.0, 2024, normal ordering). With
the global-fit uncertainty $\sigma(\sin^2\theta_{13}) \approx 6.5\times
10^{-4}$, the value sits more than $30\sigma$ from zero; the nonzero
result was first established by Daya Bay (2012) at more than $5\sigma$.
\end{proposition}

\begin{remark}[Verdict]
A vanishing reactor angle is excluded. The paper's reading --- that the
measured $8.5^\circ$ is ``small, consistent with symmetry breaking'' ---
treats as a perturbation what is in fact a leading-order angle: it is
larger than the experimental resolution by two orders of magnitude. The
prediction $\theta_{13} = 0$ is \textbf{refuted}. Registry: P001\_4
$\to$ \emph{refuted}.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{$\delta_{\mathrm{CP}} \approx 240^\circ$ (P001\_5)}

\begin{proposition}[The CP phase prediction fails on current data]
\label{prop:dcp}
Paper 01 predicts $\delta_{\mathrm{CP}} \approx 240^\circ$ and claims
agreement with an experimental value $197^\circ \pm 25^\circ$, within
$2\sigma$. The current global fit (NuFIT-6.0, 2024, normal ordering) gives
\[
\delta_{\mathrm{CP}} \approx 177^\circ\ {}^{+19^\circ}_{-20^\circ},
\]
consistent with CP conservation ($180^\circ$) within $1\sigma$. The
prediction $240^\circ$ now lies $(240-177)/19 \approx 3.3\sigma$ above the
central value.
\end{proposition}

\begin{remark}[Verdict]
The claimed agreement leaned on a fit that has since moved. Against the
current central value the prediction is excluded at $\sim 3\sigma$, and the
data no longer favor the near-maximal CP violation the geometric argument
required; CP conservation is within one standard deviation. The prediction
$\delta_{\mathrm{CP}} \approx 240^\circ$ is \textbf{refuted on current
data}. Registry: P001\_5 $\to$ \emph{refuted}.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Topological strong-CP, $\theta_{\mathrm{QCD}} = 0$ (P001\_6)}

\begin{proposition}[The argument constrains the wrong object]
\label{prop:strongcp}
Paper 01 argues that boundary conditions on $B^4$ force the minimal-energy
instanton winding number to $n = 0$, and concludes $\theta_{\mathrm{QCD}} =
0$ exactly, eliminating the need for an axion. The physical strong-CP
parameter is the coefficient $\bar\theta$ of the topological term in the
Lagrangian,
\[
\mathcal{L}_\theta = \frac{\bar\theta\, g_s^2}{32\pi^2}\,
G_{\mu\nu}\tilde G^{\mu\nu},
\]
together with the phase of the quark-mass determinant. The winding number
$n$ and the coupling $\bar\theta$ are independent inputs: the action enters
as $\bar\theta\, n$, so a configuration with $n = 0$ contributes to the
path integral at \emph{every} value of $\bar\theta$. Fixing the winding of
one configuration does not fix the coupling.
\end{proposition}

\begin{remark}[Verdict]
Selecting the trivial topological sector says nothing about the value of
$\bar\theta$, which remains a free parameter of the theory. The neutron
electric dipole moment bound $|\bar\theta| < 10^{-10}$ is therefore not
explained; it is the very fine-tuning the strong-CP problem names. The
claim that the geometry solves strong CP, and that ``no axion is needed,''
does not follow; strong CP remains an open problem of the Standard Model.
The topological strong-CP argument is \textbf{refuted} as a solution.
Registry: P001\_6 $\to$ \emph{refuted}.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\paragraph{Registry.} P001\_4, P001\_5, P001\_6 $\to$ \emph{refuted},
closed by A344. The refuted count rises from $4$ (P016\_3, P020\_1,
P017\_2, P017\_3\_c) to $7$. Three TBS remarks were added to Paper 01
adjacent to the claims they qualify; the next generator run records them.
The reframed abstract of Paper 01 --- ``corrected here and flagged for the
registry (Paper 40)'' --- is now a true citation: the three predictions are
recorded in the registry as refuted, with this addendum on disk as the
closing document.

\paragraph{Verifier.} \texttt{verify\_P344.py}: recomputes
$\theta_{13}$ from $\sin^2\theta_{13}$ and its $>5\sigma$ nonzero
significance against the tri-bimaximal $0$; the $\delta_{\mathrm{CP}}$
distance of $240^\circ$ from the NuFIT-6.0 central $177^\circ$ ($\sim
3.3\sigma$) and its CP-conserving consistency; the structural
independence of the winding number $n$ from the coupling $\bar\theta$;
and the registry regeneration showing the three refuted with the count at
$7$. All PASS.

\end{document}
