Wide Interior Hecke Spectrum on the Frozen 32^3 Background

Setup

• exact frozen background: seed 163, distance-max coarsening step 800
• operator: symmetric -div(c^2 grad) + V on 32^3 = 32768 sites
• shift-invert sweep: 15 sigma values from 0.1 to 1.5, k=30 per sigma
• unique interior modes retained in the sweep band: 421

Sigma sweep summary

• sigma 0.100 -> kept 16 in-band eigenpairs from 30 raw results
• sigma 0.200 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 0.300 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 0.400 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 0.500 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 0.600 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 0.700 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 0.800 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 0.900 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 1.000 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 1.100 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 1.200 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 1.300 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 1.400 -> kept 30 in-band eigenpairs from 30 raw results
• sigma 1.500 -> kept 15 in-band eigenpairs from 30 raw results

Wall-resolved summary

• (7, 163): count 45, omega range 1.182438 to 1.224714, mean wall fraction 0.352, mean IPR 0.000282
• (67, 43): count 4, omega range 0.546989 to 0.548303, mean wall fraction 0.371, mean IPR 0.054308

Spectral statistics in the frequency window [0.3, 1.0]

• mode count in window: 271
• mean normalized spacing: 1.000000
• KS distance to Poisson: 0.881374
• KS distance to GOE: 0.959547
• adjacent-gap ratio mean: 0.499674
• Delta_3 summary: {'5.0': {'mean': 8.754847217617941, 'std': 19.514826236877017}, '10.0': {'mean': 16.889589121259284, 'std': 26.705230109382605}, '15.0': {'mean': 25.99041884806495, 'std': 29.6324870088361}, '20.0': {'mean': 34.15375078742594, 'std': 29.199257716031976}}

Honest read

• The interior spectrum is not featureless. It separates into wall-dominant families with distinct frequency ranges.
• The dominant wall family is encoded in the mode support, not just in a single probe point.
• The strict 15-sigma coarse sweep is broad but not exhaustive. In particular, it does not recover the previously targeted clusters near 0.392, 0.746, and 0.864 on its own; those still come from the separate exact shift-invert probes.
• So the correct synthesis is: the coarse sweep gives the large-scale interior band structure, while the targeted probes resolve specific narrow resonance families inside that band.
• The spacing statistics are also mixed rather than cleanly universal: the raw KS distances to both Poisson and GOE are large, while the mean adjacent-gap ratio r ≈ 0.500 lands between the Poisson and GOE reference values. That points to a hybridized multi-family spectrum rather than a single textbook ensemble.