Em02-B Computational Evaluation of Simulated Crossed Families Weighted atomic cone integrals, localization audits, and one simulated non-tip adjoint

This note records the audit in results/tables/audit_manifest.json, source identifier 1eeba3536183. Fifteen unit tests passed. The runner label is singular_diagnostic; the admission field is not_claimed. A configuration labelled physical_identified is rejected by 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 $ϵ=0.1$, 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 $ϵ$.

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 ${\mathcal{P}}_{5a}$ integral on $\gamma \in [-1.2,-0.4]$ equals $-0.004633651977461527$. No inherited physical coefficient is claimed. The subsequent isolated-row run, source 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 $s_{\ast }$ 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.

The physical ledger of Em02-A-v6 remains open. The package evaluates simulated data only. The obsolete shared-node value $34.747\dots$ 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.

Every stored run carries smooth_benchmark, singular_diagnostic, or manufactured_cancellation. physical_identified raises RuntimeError.

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

The $47$ chart residual at those scales stays near $-1.12\times 10^{-5}$ on an $81$-node mesh. Holding $ϵ=0.1$ and refining the mesh gives

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 $H=1+0.3T$, 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.

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

with $c_{\delta }$ the translated atomic profile and $U=U_{\ast }$. The site is $U_{\ast }=(1,1,0,0)$, $V_{\ast }=0$, 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.

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 $I$.

With $f=\sigma =1$ and unit scalar $A$-factors, the displayed ${\mathcal{P}}_{5a}$ 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 $\beta =1.5$, dblquad returns $-0.004633651977461527$, error estimate $1.6\times 10^{-15}$. This is simulated scalar data, not the inherited component map.

boundary. Numbers are tied to the named manifest and to fifteen passed tests. No physical admission is claimed.

The Python package that wrote the named manifest is archived from 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. require_simulated rejects physical_identified. The later file row5a_05485aec7a9a.json records that the seven-dimensional seed integral changes sign, so row_value_released stays false. The Gauss–Sobol status 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.

Em02A

N. Cheung, Em02-A, version 6. Formula source.