Finite Spectral Law for the Full 13-Mode Hecke Operator

This note tests the claim that the Hecke-mode counting function might look like an omega^3 vacuum spectrum.

The operator used is the full 13 x 13 matrix A A^T, where A is the saved 13 x 7 Hecke-eigenvalue matrix. Its nonzero eigenvalues are the same as those of the 7 x 7 Gram matrix G = A^T A, with six additional zero modes.

Exact finite spectrum

• full eigenvalues of A A^T: [-5.0904471116982714e-14, -3.4580211781131825e-14, -1.4897625770551112e-14, 6.906198918007131e-15, 1.8947193150523197e-14, 5.881913447338764e-14, 10.371881134081775, 19.439718777243456, 76.7815405767042, 98.09369154373766, 274.1373051897651, 491.62421002465726, 821.5516527538093]
• positive eigenvalues: [10.371881134081775, 19.439718777243456, 76.7815405767042, 98.09369154373766, 274.1373051897651, 491.62421002465726, 821.5516527538093]
• positive frequencies omega = sqrt(lambda): [3.220540503406497, 4.409049645586162, 8.762507664858514, 9.904225943693817, 16.557092292723535, 22.172600434424854, 28.662722354197435]
• UV cutoff frequency omega_max: 28.662722
• shortest supported wavelength 1/omega_max: 0.034889

Resolvent cutoffs

• nearest pole to x=0: 1/lambda_max = 0.001217
• farthest positive pole: 1/lambda_min = 0.096415

Counting-function fits on the positive branch

• best free power fit: N_+(omega) ≈ 0.496826 * omega^0.819504
• free-fit relative error: 0.101354
• fixed p=1 fit error: 0.160286 with scale 0.275895
• fixed p=2 fit error: 0.426574 with scale 0.010660
• fixed p=3 fit error: 0.557385 with scale 0.000373

Honest conclusion

• The arithmetic model is automatically UV-regulated because the operator is finite-dimensional.
• The full 13-mode counting function has a six-fold zero-mode jump at the origin, so it is not globally power-law.
• Even after discarding the zero modes and fitting only the seven positive frequencies, p = 3 is a poor fit.
• Among the simple laws N_+(omega) ~ C * omega^p, the best-fit exponent is close to 0.82, not 3.
• So the defensible statement is finite spectral cutoff, not an omega^3 vacuum law.