Reported heat and two-dimensional resolvent checks

The supplied draft reports the cases

together with $\beta =0.3$ at $\mu =\pi /2$, and $\beta =\mu /(2\pi )$ at $\mu =1$. The last twist is of the geometric-helicity type up to orientation.

For the angular heat identity (eq:Sommerfeld), the reported sample set is

giving fifteen parameter points across the five cases. The reported maximum absolute mode-versus-closed discrepancy is

At $\mu =\pi ,\beta =0$, the reported diffraction contribution is at most $4\times 10^{-33}$. The exact vanishing in that case is Corollary cor:structure, not the decimal test.

For the two-dimensional resolvent, the sample parameters are

with reported maximum absolute discrepancy

over ten points. Raw paired values were not printed in the supplied summary; these maxima are therefore retained as summaries, not reconstructed logs.

The independent mode formula underlying such a check is

For $r\ne r^{\prime }$ it is absolutely convergent. Other pairs require the usual kernel interpretation or a convergence prescription.