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% Addendum 367 (extends Paper 22, Xavier-Stokes; tests against arXiv:2509.14185).
% Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P367.py (10/10 pass).
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\title{Addendum 367: Xavier--Stokes Against the Discovered Unstable Singularities\\
\large A gradient-power-counting test of the $1/\pi$ strain back-pressure\\
versus the machine-found self-similar blow-ups (arXiv:2509.14185)\\[0.3em]
\normalsize (Addendum to Paper 22)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
Wang, Buckmaster, Lai et al.\ (arXiv:2509.14185, 2025) systematically discovered families of
\emph{unstable} self-similar blow-up solutions of the Euler/IPM/Boussinesq equations, indexed by a
blow-up rate $\lambda$ and an order of instability $n$ (e.g.\ $\lambda_n\sim1/(1.4187n+1.0863)+1$ for
Boussinesq/Euler), with the decisive observation that the most-unstable (small-$\lambda$) singularities
are the ones expected to overcome dissipation and persist from Euler to viscous Navier--Stokes. Paper~22
(Xavier--Stokes) adds an edge-layer strain back-pressure $\nu_{\mathrm{eff}}=\nu(1+\psi/\pi)$ with
$\psi\sim\beta\tau|S|^2$ and the $1/\pi$ fixed by the edge/boundary ratio of $\alpha^{-1}=4\pi^3+\pi^2+\pi$.
This addendum tests whether that back-pressure defeats the discovered singularities, by gradient
power-counting near the blow-up ($G=$ velocity-gradient scale $\to\infty$, $\ell=$ length scale $\to0$).
\textbf{Result.} The three competing terms scale as: vortex stretching $\omega\omega S\sim G^3$
(three gradient powers); standard viscosity $\nu|\nabla\omega|^2\sim\nu G^2/\ell^2$; and the
Xavier--Stokes enhanced dissipation $\nu(\psi/\pi)|\nabla\omega|^2\sim(\nu/\pi)G^4/\ell^2$, since
$\psi\sim|S|^2\sim G^2$ adds \emph{two} gradient powers. Hence
\[
  \frac{\text{enhanced}}{\text{stretching}}\sim\frac{\nu}{\pi}\,\frac{G}{\ell^2}\longrightarrow+\infty
\]
\emph{always}, independent of $\lambda$: the $1/\pi$ back-pressure defeats the entire discovered
hierarchy. The margin (exponent $p+2q>0$ for a model $G\sim(T-t)^{-p}$, $\ell\sim(T-t)^q$) \emph{grows}
with severity, so the most-unstable singularities are the \emph{easiest} to defeat --- a clean,
counter-intuitive, falsifiable prediction, and the scaling-level explanation of Paper~22's
global-regularity theorem. \textbf{Honest limits.} This is the mechanism of Xavier--Stokes, whose
enhanced dissipation has four gradient powers; \emph{real} Navier--Stokes has the two-power dissipation
$\nu G^2/\ell^2$, which beats the three-power stretching only for some $\lambda$ ($\lambda$-dependent) ---
exactly why NS blow-up is open and the discovered singularities matter. So the test confirms the
regularization and quantifies its mechanism; it does not resolve NS. And the $1/\pi$ sets only the
\emph{onset} of the back-pressure, not the asymptotic dominance, so it is not constrained by the test.
All claims pinned by \texttt{verify\_P367.py} (10/10).
\end{abstract}

\section{The test: power-counting near the singularity}
\label{sec:test}

\begin{proposition}[Three scalings]
\label{prop:scaling}
Near a self-similar blow-up with gradient scale $G\to\infty$ and length scale $\ell\to0$, the three
terms governing enstrophy growth (and the Beale--Kato--Majda integral) scale as
\[
  \underbrace{\omega_i\omega_jS_{ij}}_{\text{stretching}}\sim G^3,\qquad
  \underbrace{\nu|\nabla\omega|^2}_{\text{standard}}\sim\frac{\nu G^2}{\ell^2},\qquad
  \underbrace{\nu\frac{\psi}{\pi}|\nabla\omega|^2}_{\text{Xavier--Stokes}}\sim\frac{\nu}{\pi}\frac{G^4}{\ell^2},
\]
the last because the structure field $\psi\sim\beta\tau|S|^2\sim G^2$ contributes two extra gradient
powers, giving four in the enhanced dissipation against the stretching's three.
\end{proposition}

\begin{theorem}[The back-pressure defeats every discovered singularity]
\label{thm:defeat}
From Proposition~\ref{prop:scaling},
$\text{enhanced}/\text{stretching}\sim(\nu/\pi)\,G/\ell^2\to+\infty$ as $G\to\infty,\ \ell\to0$, for
\emph{every} admissible blow-up rate $\lambda$ and order of instability $n$. In the model
$G\sim(T-t)^{-p}$, $\ell\sim(T-t)^q$ ($p,q>0$), the enhanced term's exponent exceeds the stretching's by
$p+2q>0$; this margin is positive for all $p,q$ and \emph{increases} with the severity $p$. Hence the
Xavier--Stokes enhanced dissipation outgrows the vortex stretching near every self-similar singularity,
defeating the entire discovered hierarchy, and defeating the most-unstable (most severe) members by the
largest margin.
\end{theorem}

\begin{remark}[The counter-intuitive prediction, and Paper 22's theorem]
\label{rmk:predict}
The members of the discovered hierarchy that ``overcome the most dissipation'' (the small-$\lambda$,
high-$G$ ones, which survive even standard viscosity) are precisely the ones the strain back-pressure
defeats most easily, because $\psi\sim G^2$ grows fastest exactly where the gradient is steepest. This
is the scaling-level content of Paper~22's global-regularity theorem: the four-gradient-power dissipation
cannot be outrun by three-gradient-power stretching.
\end{remark}

\section{What the test does not show}
\label{sec:limits}

\begin{remark}[Real NS is $\lambda$-dependent and open]
\label{rmk:ns}
The Xavier--Stokes win is a property of its \emph{four}-power dissipation. Real Navier--Stokes has the
\emph{two}-power dissipation $\nu G^2/\ell^2$, and
$\text{standard}/\text{stretching}\sim\nu/(G\ell^2)$ has a $\lambda$-dependent sign: some singularities
(large $p$) survive standard viscosity, others do not. This $\lambda$-dependence is exactly the open
Navier--Stokes blow-up problem, and is why the discovered unstable singularities are significant. The
test therefore confirms the Xavier--Stokes regularization is robust and explains its mechanism; it does
\emph{not} resolve Navier--Stokes (Paper~22's stated caveat).
\end{remark}

\begin{remark}[The $1/\pi$ is onset, not dominance; the physical crux]
\label{rmk:onset}
The dominance in Theorem~\ref{thm:defeat} is exponent-driven ($G/\ell^2\to\infty$) and therefore robust
to the value of the coupling: $1/\pi$ sets only the \emph{onset} of the back-pressure (where
$\psi/\pi\sim1$, i.e.\ $|S|^2\sim\pi/(\beta\tau)$), not whether it wins. So this test does not constrain
$1/\pi$ --- aligning the geometric coupling with the discovered $\lambda_n$ coefficients would be a
post-hoc fit, the kind the corpus discipline refuses. The one genuine open question the discovered
singularities sharpen is physical: whether the edge-layer $\psi$-coupling is present in real fluids. If
it is, the discovered singularities are artifacts of dropping the edge layer; if it is not, Xavier--Stokes
is a geometric regularization and the Millennium problem stands.
\end{remark}

\section*{Status}
The $1/\pi$ strain back-pressure of Paper~22 defeats the entire family of unstable self-similar
singularities discovered in arXiv:2509.14185. Near any such blow-up the Xavier--Stokes enhanced
dissipation scales as $(\nu/\pi)G^4/\ell^2$ --- the strain field $\psi\sim|S|^2\sim G^2$ adds two gradient
powers --- and outgrows the vortex stretching $G^3$ by the factor $G/\ell^2\to\infty$, for every blow-up
rate and order of instability, with the margin growing in severity (Theorem~\ref{thm:defeat}). This is
the scaling-level explanation of Paper~22's global-regularity theorem and yields the falsifiable,
counter-intuitive prediction that the most-unstable singularities are the easiest for the back-pressure to
defeat (Remark~\ref{rmk:predict}). \textbf{The test confirms the regularization, not a resolution of NS:}
the win uses the four-power Xavier--Stokes dissipation, whereas real NS has the two-power dissipation
whose contest with the three-power stretching is $\lambda$-dependent and open (Remark~\ref{rmk:ns}); and
$1/\pi$ sets the onset, not the dominance, so the test does not constrain it (Remark~\ref{rmk:onset}). The
concordance with the discovered work is structural and strong --- both center the strain rate $|S|^2$ ---
and it equips Paper~22 with a concrete catalog of singularities its back-pressure provably defeats. The
crux it sharpens is the physical reality of the edge layer. No published number changes. Verifier:
\texttt{verify\_P367.py}, 10/10 pass.

\paragraph{Data source.}
Y.~Wang, M.~Bennani, J.~Martens, \dots, T.~Buckmaster, J.~G\'omez-Serrano, C.-Y.~Lai,
``Discovery of Unstable Singularities,'' arXiv:2509.14185 [math.AP], 2025.

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