Hecke Spherical Spectrum

Construction

icosphere subdivision levels analyzed: 162 and 642 vertices
seven Heegner seed sites chosen by greedy maximin on the 162-vertex sphere
Heegner labels assigned to those seven geometric sites by minimizing the normalized pairwise-distance mismatch against the exact 7x7 Hecke distance matrix
label assignment mismatch (Frobenius norm on normalized distances): 2.152094
assigned site labels in geometric site order: [7, 19, 3, 67, 43, 11, 163]

first nontrivial bulk-like mode: 0.16311747088501743
wall-gap-state count under the wall-vs-bulk classifier: 0
potential stats: {'min': 0.0, 'max': 0.5173436492585429, 'mean': 0.08673841116709517}
c^2 stats: {'min_c2': 0.0203125, 'max_c2': 1.0}
wall-mode counts: {'(3, 19)': 1, '(3, 163)': 1, '(7, 43)': 1, '(7, 163)': 1, '(11, 67)': 3, '(19, 43)': 6, '(19, 67)': 4, '(43, 67)': 3, '(43, 163)': 2}

mode	omega	kind	dominant wall	wall overlap	dominant basin	basin overlap	participation volume
1	0.132637	bulk	(11, 67)	0.005	67	0.999	9.0
2	0.163117	bulk	(19, 43)	0.005	43	0.982	7.2
3	0.179592	bulk	(3, 19)	0.024	19	0.962	11.8
4	0.191110	bulk	(11, 67)	0.016	67	0.997	8.3
5	0.211948	bulk	(3, 11)	0.028	67	0.775	12.4
6	0.214739	bulk	(3, 11)	0.101	67	0.227	68.4
7	0.218935	bulk	(19, 43)	0.011	43	0.980	6.6
8	0.240486	bulk	(11, 67)	0.018	67	0.997	6.2
9	0.247731	bulk	(3, 11)	0.008	43	0.944	5.8
10	0.251598	bulk	(11, 67)	0.014	67	0.976	5.5
11	0.267077	bulk	(19, 43)	0.008	43	0.992	5.0
12	0.268869	bulk	(19, 67)	0.008	67	0.966	6.2
13	0.270861	bulk	(7, 67)	0.008	67	0.886	7.7
14	0.272948	bulk	(11, 19)	0.062	19	0.464	32.8
15	0.274743	bulk	(11, 67)	0.057	19	0.408	30.2

first nontrivial bulk-like mode: 0.07133511251500721
wall-gap-state count under the wall-vs-bulk classifier: 0
potential stats: {'min': 0.0, 'max': 0.5173436492585429, 'mean': 0.045047211024542504}
c^2 stats: {'min_c2': 0.0203125, 'max_c2': 1.0}
wall-mode counts: {'(3, 163)': 1, '(7, 43)': 2, '(7, 67)': 1, '(11, 67)': 1, '(19, 43)': 15, '(19, 67)': 11, '(43, 67)': 12, '(43, 163)': 5}

mode	omega	kind	dominant wall	wall overlap	dominant basin	basin overlap	participation volume
1	0.062686	bulk	(11, 67)	0.001	67	1.000	42.9
2	0.071335	bulk	(19, 43)	0.000	43	0.999	37.8
3	0.086550	bulk	(3, 19)	0.003	19	0.996	48.5
4	0.097054	bulk	(11, 67)	0.002	67	0.999	37.7
5	0.102058	bulk	(11, 67)	0.001	67	0.999	37.1
6	0.105083	bulk	(7, 43)	0.001	43	0.999	34.7
7	0.117986	bulk	(7, 163)	0.002	43	0.975	33.5
8	0.127003	bulk	(3, 11)	0.047	11	0.513	157.5
9	0.128169	bulk	(11, 67)	0.003	67	0.974	41.7
10	0.135499	bulk	(11, 67)	0.005	67	0.986	38.3
11	0.138814	bulk	(19, 43)	0.002	43	0.898	38.1
12	0.139421	bulk	(3, 19)	0.009	19	0.746	74.0
13	0.140259	bulk	(19, 67)	0.002	67	0.885	44.6
14	0.141785	bulk	(7, 163)	0.014	19	0.601	99.7
15	0.143314	bulk	(7, 163)	0.015	19	0.453	131.7

This spherical construction is deterministic once the maximin site set is fixed, but the embedding of Heegner labels onto the seven geometric sites is still a modeling choice resolved here by best-fit distance matching rather than a theorem.
The 162- and 642-vertex spectra can be computed in full, so there is no sparse-solver ambiguity here.
The useful test is whether low modes become wall-dominant before bulk-dominant. The note reports that classifier directly rather than forcing a continuum interpretation that the discrete sphere does not justify on its own.