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\hypersetup{pdftitle={Em02-B: Computational Evaluation of Simulated Crossed Families},pdfauthor={}}

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\title{\textbf{Em02-B}\\[0.4em]
Computational Evaluation of Simulated Crossed Families\\[0.4em]
\large Weighted atomic cone integrals, localization audits,\\
and one simulated non-tip adjoint}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
This note records the audit in
\texttt{results/tables/audit\_manifest.json},
source identifier \texttt{1eeba3536183}.
Fifteen unit tests passed.
The runner label is \texttt{singular\_diagnostic}; the admission
field is \texttt{not\_claimed}.
A configuration labelled \texttt{physical\_identified} is rejected
by \texttt{require\_simulated}.

For the cutoff in the package,
\(c_R=0.625\) and
\(c_{47}=0.15045103006514954\).
The chart formula agrees to \(1.96\times 10^{-13}\).
At fixed \(\epsilon=0.1\), the \(47\) mesh residual falls from
\(1.34\times 10^{-4}\) at \(21\) nodes to
\(1.12\times 10^{-5}\) at \(81\) nodes, which supports a grid-bias
reading of the earlier stable residual.
The amplitude probe has exact target
\(0.14179675523330723\), independent of \(\epsilon\).

The weighted atomic \(5a\) integral at \(\delta=0.35\) is
\(4.851387241020615\) by a Gauss cap rule and
\(4.845938176179326\) by a scrambled Sobol rule on \(2^{12}\) nodes.
The relative gap is \(1.1\times 10^{-3}\).
These quadratures do not share nodes.
A scalar simulated \(\mathscr P_{5a}\) integral on
\(\gamma\in[-1.2,-0.4]\) equals \(-0.004633651977461527\).
No inherited physical coefficient is claimed.
The subsequent isolated-row run, source \texttt{05485aec7a9a},
builds the specialized vertex and both adjoint constructions.
They agree termwise to \(2.8\times 10^{-14}\).
The seven-dimensional seed integral changes sign from
\(-49.40\) to \(21.20\) to \(-8.25\) as the \(U\)-node count goes
from 5 to 7, so no row value is released.
The later overlap reduction separates that failure from the inner
integral. At \(s_*\) the factored inner value is \(-2428.716\), and a
16-node overlap-box rule agrees to \(0.28\%\). The outer cone rules
do not: Gauss gives \(3.01,-0.15,-2.58\) and Sobol stays near \(-4.3\).
The row remains unreleased.
\end{abstract}

\tableofcontents

\section{Scope}
The physical ledger of Em02-A-v6 remains open.
The package evaluates simulated data only.
The obsolete shared-node value \(34.747\ldots\) is not a result of
this revision: that sum used an inserted null node and a single
panel width. It is not in the manifest.

\begin{convention}
\label{conv:simulated}
Every stored run carries \texttt{smooth\_benchmark},
\texttt{singular\_diagnostic}, or
\texttt{manufactured\_cancellation}.
\texttt{physical\_identified} raises \texttt{RuntimeError}.
\end{convention}

\section{Localization}
The five-cone mass at fixed rescaled resolution, not a refinement
in \(\epsilon\), gives residual \(-4.17\times 10^{-8}\) against
\(c_R\) at \(\epsilon=0.25,0.125,0.0625\).

The \(47\) chart residual at those scales stays near
\(-1.12\times 10^{-5}\) on an \(81\)-node mesh.
Holding \(\epsilon=0.1\) and refining the mesh gives
\begin{center}
\small
\begin{tabular}{lcc}
\toprule
nodes & scaled integral & residual against \(c_{47}\)\\
\midrule
\(21\) & \(0.150316797673\) & \(-1.342\times 10^{-4}\)\\
\(41\) & \(0.150407097916\) & \(-4.393\times 10^{-5}\)\\
\(81\) & \(0.150439836944\) & \(-1.119\times 10^{-5}\)\\
\bottomrule
\end{tabular}
\end{center}
The residual shrinks as the mesh is refined. That supports grid
bias. It does not by itself prove the bias model; it is the check
the previous note lacked.

For \(H=1+0.3\,T\), the scaled probe equals
\(0.3\int_0^\infty t^2 b_0(2t^2)\,\mathrm d t
=0.14179675523330723\)
exactly, before quadrature error.
The trapezoidal values are
\(0.141796588567\), \(0.141796421900\), and \(0.141796088567\).
The drift is consistent with amplification of a small cone-mass
error after subtraction, not with failure of the localization
identity.

\section{Weighted atomic integral}
The passive \(T\)-cone is weighted by \(e^{-T}\), and
\(\int_0^\infty T e^{-T}\,\mathrm d T=1\).
The angular factor on that cone is the sphere probability.
The dynamical integral is
\begin{equation}
 I=\int \frac{S}{4\pi}\,c_\delta(S,\mathbf n)\,\mathrm d\mathbf n\,\mathrm d S,
 \label{eq:atomic}
\end{equation}
with \(c_\delta\) the translated atomic profile and \(U=U_*\).
The site is \(U_*=(1,1,0,0)\), \(V_*=0\), so \(b-a\) is future null.
Route~B is a Gauss rule on the cap containing the mollifier
support: \(24\times 16\times 24\) nodes.
Route~A is a scrambled Sobol rule of length \(2^{12}\) on the same
cap and does not call Route~B.

\begin{center}
\small
\begin{tabular}{lcc}
\toprule
profile & Gauss & Sobol\\
\midrule
\(\rho_\delta\) & \(4.851387241020615\) & \(4.845938176179326\)\\
\(\kappa_\delta\) & \(-24.16497259148322\) & \(-24.198566431394124\)\\
\bottomrule
\end{tabular}
\end{center}
Absolute gaps are \(5.45\times 10^{-3}\) and \(3.36\times 10^{-2}\).
Relative gaps are about \(0.11\%\) and \(0.14\%\).
A coarser Gauss rule, \(12\times 8\times 12\), differs from a
\(20\times 12\times 16\) rule by \(1.5\%\) relative.
This is an independent quadrature check of a closed-form integrand.
It is not the seven-row pairing: the vertex and the component
adjoint are not in \(I\).

\section{Simulated non-tip adjoint}
With \(f=\sigma=1\) and unit scalar \(A\)-factors, the displayed
\(\mathscr P_{5a}\) reduces to
\(\partial_{s^0}^2\partial_{t^0}^2\Psi\) for
\(\Psi=\exp(-((s^0-1.5)^2+(t^0)^2))\).
Derivatives are taken before the chart evaluation.
On \(\gamma\in[-1.2,-0.4]\) and transverse radius at most \(0.8\),
with density \(\beta/(8\pi^2|\gamma|)\) at \(\beta=1.5\),
\texttt{dblquad} returns
\(-0.004633651977461527\),
error estimate \(1.6\times 10^{-15}\).
This is simulated scalar data, not the inherited component map.

\section{Ledger}
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}YY@{}}
\toprule
Result & Status\\
\midrule
\(c_R\), \(c_{47}\), chart twin &
Values of this \(b_0\)\\[0.3em]
Mesh residuals shrinking at fixed \(\epsilon\) &
Supports grid bias\\[0.3em]
Probe against the exact \(\epsilon\)-free target &
Quadrature check\\[0.3em]
Gauss/Sobol atomic integral &
Independent quadrature of S1\\[0.3em]
Scalar non-tip \(\mathscr P_{5a}\) &
Simulated adjoint, one row\\[0.3em]
Seven-row physical pairing, S2, authorization &
Not executed\\
\bottomrule
\end{tabularx}
\end{center}

\paragraph{Verification boundary.}
Numbers are tied to the named manifest and to fifteen passed tests.
No physical admission is claimed.


\section{Package boundary}
\begin{remark}
\label{conv:package}
The Python package that wrote the named manifest is archived from
\url{https://github.com/cheungngo/Em02-B}.
That archive is package evidence.
It is not a theorem of this note, and it is not a certificate.
\texttt{require\_simulated} rejects \texttt{physical\_identified}.
The later file \texttt{row5a\_05485aec7a9a.json} records that the
seven-dimensional seed integral changes sign, so
\texttt{row\_value\_released} stays false.
The Gauss--Sobol status
\texttt{independent\_quadrature\_of\_closed\_form\_atomic\_integrand}
is not that seed integral.
The outer cone rules do not agree.
No physical admission is claimed.
No compilation is asserted.
\end{remark}

\begin{thebibliography}{99}
\bibitem{Em02A}
N.~Cheung, \emph{Em02-A}, version 6. Formula source.
\end{thebibliography}
\end{document}