Retained SAE checks

C4 At $\alpha =0.8,\beta =0.3$, the tested orders were $\nu =0.375,0.875$. The source reports small-$\epsilon$ asymptotic relative errors $4.5\times 10^{-3}$ and $2.6\times 10^{-9}$ at $\epsilon =10^{-6}$. For large $\epsilon =10^6$, it reports $2.8\times 10^{-5}$ and $5.8\times 10^{-11}$. It reports $j/F=-2eq$, agreement of leading stress coefficients to 12 printed digits, structural relations to $10^{-29}$, and conservation and trace residuals at most $10^{-30}$. The conjugate-mode swap reportedly agrees in square and stress to $3.5\times 10^{-30}$, with opposite current.

The small-radius tests lie in regimes compatible with the corrected Theorem thm:small; they do not establish the incorrect predecessor remainder uniformly for all $0<\nu <1/2$.

C5 For $\alpha =0.83,\beta =0.45$, and a positive matrix with reported eigenvalues $0.49,2.11$, the source reports conservation $1.5\times 10^{-31}$, trace $7\times 10^{-31}$, current conservation $1.3\times 10^{-32}$, and $j_r=-1.63\times 10^{-5}\ne 0$. The rotation rule was reported exact. The off-diagonal-to-zero limit reportedly agrees with the diagonal result to $4\times 10^{-31}$, and the line family integral with its closed form to $6\times 10^{-30}$.

The SAE text additionally reports large-radius relative error $5\times 10^{-4}$ at $r=10^3$, described as decreasing like $r^{-2{\nu }_0}$, and a small-radius scaled error $2\times 10^{-4}$ at $r=10^{-6}$. These tests are not extended to arbitrary boundary charts.

C7 The reported bound-state diagnostics are:

• after pole subtraction, values $-0.04893705$ on both sides of $p=\kappa (1\pm 10^{-8})$, differing by $1.6\times 10^{-9}$;

• an algebraic $p^{-2}$ tail coefficient $-{\kappa }^2{K}_0{K}_0/(\pi \alpha )$, agreeing to $3\times 10^{-5}$ at $p=200$;

• the Struve identity to $3\times 10^{-26}$, the longitudinal split identity to $5\times 10^{-15}$, and the principal-value Hankel identity below $10^{-25}$;

• the ${\mathsf{E}}_{\lambda }$ bound-state residue matching ${\Vert K_{\nu }\Vert }^{-2}$ to 12 digits;

• logarithmic asymptotic tests at ${\kappa }_{L}r=10^{\pm 3},10^{\pm 6},10^{\pm 12}$, with two-term relative error $2.2\times 10^{-3}$ at $10^{12}$, described as decreasing like ${\log }^{-2}$.

Filgueiras–Moraes report Eight $(\alpha ,{\eta }_0)$ comparisons were reported at most $6\times 10^{-19}$. The energy scale and longitudinal term must be restored when comparing the dimensionless expression with their equation (10).

would strengthen these reports. Supply the actual scripts, exact parameter encodings, dependency versions, raw values, precision schedules, quadrature tolerances, oscillatory truncations, pole prescriptions, certified tails, and image-boundary handling. Until then the retained numbers are comparisons, not rigorous error enclosures or proof of a universal identity.