Hecke Eigenvalue Spectrum on the Frozen Distance-Max Background

Setup

• frozen background: seed 163, step 800
• lattice size: 32768 = 32^3
• sparse operator uses the symmetric discretization -div(c^2 grad) + V
• operator nnz: 229376
• symmetry defect: 0.0

Bulk edge

• global formula omega_bulk_min = sqrt(c2_min) * 2pi / 32 = 0.027984
• largest-basin formula (7 basin) = 0.073809

Bound states

Number of modes below the global bulk edge: 0

mode	omega	wall_type	participation_volume	parallel_spread	perp_decay
-	-	none below bulk edge	-	-	-

Probe comparison

• probe 0.196251 -> nearest eigenmode 0.195586 (mode 38, wall (11, 7), rel err 0.003)
• probe 0.392503 -> nearest eigenmode 0.236097 (mode 100, wall (11, 7), rel err 0.398)
• probe 0.745755 -> nearest eigenmode 0.236097 (mode 100, wall (11, 7), rel err 0.683)
• probe 0.863506 -> nearest eigenmode 0.236097 (mode 100, wall (11, 7), rel err 0.727)

Vlam Shift-Invert Follow-Up

The low-100 solve only covered the bottom of the spectrum. A follow-up shift-invert solve on vlam targeted the three larger probe frequencies directly on the same exact 32^3 frozen operator.

• around 0.392503: there is a tight cluster of interior eigenmodes at 0.392240 to 0.393010
• around 0.745755: there is a tight cluster at 0.745681 to 0.745814
• around 0.863506: there is a tight cluster at 0.863457 to 0.863566

So the probe peaks at 0.392, 0.746, and 0.864 are genuine interior eigenmodes of H, not transient artifacts.

The wall typing of those interior modes is:

• the 0.392 cluster is mixed, with one compact (43,11)-dominant mode at 0.392240 (participation volume 120.6, perpendicular spread 3.95) and several broader (11,7) wall modes
• the 0.746 cluster is entirely (11,7)-dominant, broad, and wall-guided
• the 0.864 cluster is also entirely (11,7)-dominant, broad, and wall-guided

Honest read

• On this frozen background there is no strict sub-band below the naive bulk edge. The low spectrum starts around 0.111905, well above 0.027984.
• The interesting objects are therefore low wall-dominant resonances inside the continuum, not isolated bound states in a clean gap.
• The probe frequencies are still meaningful: 0.196 comes directly from the low-end solve, and 0.392, 0.746, 0.864 are confirmed by shift-invert on the exact operator.
• The particle-spectrum interpretation is therefore weaker than the original sub-band picture, but stronger than “mere ringing”: these are bona fide interior wall modes of the frozen Hecke background.