Appendix E: Computational verification

E. Computational verification

The accompanying supplement provides native spectra, control vectors, executable implementations, generated tables and input provenance. The empirical profiles in Section 10 are those of the cross-domain study [23]. Table 3 summarizes checks of their operator representation and the mathematical constructions developed here.

Table 3. Computational checks of the operator identities and closure bound. Errors are numerical residuals. The independent closure fixtures use a separate high-precision implementation.

For the seventeen native readouts, direct ratio counting and the log-commutator representation agree in all $17\times3\times15=765$ tolerance–constant cells. Small non-diagonal matrix fixtures check the commutator spectrum and trace-square identity. Modular normalization is checked for all seventeen readouts, and thirty cases test the closure bound. A separate implementation uses 70-digit arithmetic for eight closure fixtures and an exact non-diagonal modular fixture $\rho=\left(\begin{smallmatrix}2&1\\1&2\end{smallmatrix}\right)/4$, whose ratio spectrum is $\{3,1/3,1,1\}$.

Independent symbolic calculations verify the reciprocal-storage identities, three-coordinate Hessian and susceptibility, weighted Markov carrier and collar boundary energy directly from their defining expressions. The supplement contains specifications, source code, numerical settings, dependencies, random seeds, integrity hashes and actual check outputs.

The spectral arrays and control vectors are the finite inputs accompanying the cross-domain study. The computations above verify their algebraic representations and the numerical identities; they do not repeat image extraction, radial integration, catalogue acquisition or the underlying control campaigns. Execution records and clean-run instructions accompany the supplement. The physical models used in Sections 11, 12 and 13 are defined and derived in the text, so their assessment does not require access to the unpublished reports cited for attribution.