Broad Hecke-Regge Bridge Scan

This scan tests the centered-Hecke-vs-cotangent bridge at every prime level N <= 199 for which the chosen operator family has at least 4 elements.

Families:

• cumulative_h1: S_N = {p in {3,7,11,19,43,67,163} : p <= N}
• split_plus_vacuum: S_N = {p in H1 : p != N and (-p/N)=1} union {N}

Metrics:

• off-diagonal Frobenius fraction of L_cot in a B-eigenbasis
• relative commutator norm ||[B,L_cot]|| / (||B|| ||L_cot||)
• mean matched eigenvector overlap

Headline

• The raw global minima are misleading: several low-rank 4-operator cases are nearly trivial and drive the off-diagonal fraction essentially to zero.
• After controlling for family size/rank, level 163 is not the global minimizer in this scan.
• In the cumulative family with the full 7-operator set, the best levels up to 199 are:
  • 199 with off-diagonal fraction 0.048721
  • 197 with 0.049438
  • 173 with 0.049383
  • 163 with 0.071352
• So the earlier three-level observation was real but local: 163 beats 43 and 67, but not the broader prime-level field up to 199.

Family: cumulative_h1

• best off-diagonal fraction: level 29 with 0.000000
• best commutator norm: level 41 with 0.000000
• best mean overlap: level 41 with 1.000000

Skipped cases: - skipped level 11 with operators [3, 7, 11] because set_too_small - skipped level 13 with operators [3, 7, 11] because set_too_small - skipped level 17 with operators [3, 7, 11] because set_too_small - skipped level 19 with operators [3, 7, 11, 19] because degenerate_distance_geometry - skipped level 37 with operators [3, 7, 11, 19] because degenerate_distance_geometry

level	operators	dim	rank(B)	offdiag	commutator	mean overlap
23	[3, 7, 11, 19]	2	2	0.000000	0.057684	0.983308
29	[3, 7, 11, 19]	2	2	0.000000	0.031872	0.999452
31	[3, 7, 11, 19]	2	2	0.000000	0.086066	0.994509
41	[3, 7, 11, 19]	3	2	0.000000	0.000000	1.000000
43	[3, 7, 11, 19, 43]	3	3	0.100321	0.047024	0.979476
47	[3, 7, 11, 19, 43]	4	4	0.081928	0.065847	0.997255
53	[3, 7, 11, 19, 43]	4	4	0.103446	0.021655	0.944693
59	[3, 7, 11, 19, 43]	5	4	0.039610	0.029771	0.996788
61	[3, 7, 11, 19, 43]	4	4	0.115407	0.056997	0.979804
67	[3, 7, 11, 19, 43, 67]	5	5	0.138548	0.059168	0.952021
71	[3, 7, 11, 19, 43, 67]	6	5	0.065749	0.035575	0.989000
73	[3, 7, 11, 19, 43, 67]	5	5	0.228265	0.043880	0.884264
79	[3, 7, 11, 19, 43, 67]	6	5	0.097381	0.041809	0.967171
83	[3, 7, 11, 19, 43, 67]	7	5	0.092448	0.028778	0.978706
89	[3, 7, 11, 19, 43, 67]	7	5	0.078808	0.036298	0.819503
97	[3, 7, 11, 19, 43, 67]	7	5	0.110470	0.044281	0.938880
101	[3, 7, 11, 19, 43, 67]	8	5	0.098805	0.031636	0.961722
103	[3, 7, 11, 19, 43, 67]	8	5	0.103286	0.042696	0.980386
107	[3, 7, 11, 19, 43, 67]	9	5	0.059764	0.032580	0.983812
109	[3, 7, 11, 19, 43, 67]	8	5	0.056793	0.028618	0.990813
113	[3, 7, 11, 19, 43, 67]	9	5	0.127661	0.049976	0.961210
127	[3, 7, 11, 19, 43, 67]	10	5	0.070436	0.019053	0.873404
131	[3, 7, 11, 19, 43, 67]	11	5	0.097602	0.029750	0.972172
137	[3, 7, 11, 19, 43, 67]	11	5	0.115552	0.046420	0.966397
139	[3, 7, 11, 19, 43, 67]	11	5	0.039167	0.014495	0.996626
149	[3, 7, 11, 19, 43, 67]	12	5	0.050662	0.018779	0.989794
151	[3, 7, 11, 19, 43, 67]	12	5	0.069072	0.027280	0.990866
157	[3, 7, 11, 19, 43, 67]	12	5	0.110957	0.031255	0.954166
163	[3, 7, 11, 19, 43, 67, 163]	13	6	0.071352	0.030399	0.973626
167	[3, 7, 11, 19, 43, 67, 163]	14	6	0.072143	0.042024	0.992382
173	[3, 7, 11, 19, 43, 67, 163]	14	6	0.049383	0.015832	0.947038
179	[3, 7, 11, 19, 43, 67, 163]	15	6	0.092065	0.032817	0.974553
181	[3, 7, 11, 19, 43, 67, 163]	14	6	0.118878	0.018822	0.915619
191	[3, 7, 11, 19, 43, 67, 163]	16	6	0.081827	0.034996	0.943459
193	[3, 7, 11, 19, 43, 67, 163]	15	6	0.102643	0.029737	0.935196
197	[3, 7, 11, 19, 43, 67, 163]	16	6	0.049438	0.015448	0.989348
199	[3, 7, 11, 19, 43, 67, 163]	16	6	0.048721	0.017578	0.980381

Meaningful same-size comparison:

• for the full 7-operator family [3,7,11,19,43,67,163], the scanned levels are 163,167,173,179,181,191,193,197,199

• among those, level 163 is not minimal; 199 is best and 173,197 are very close ## Family: split_plus_vacuum

• best off-diagonal fraction: level 31 with 0.000000

• best commutator norm: level 191 with 0.002313

• best mean overlap: level 73 with 0.999829

Skipped cases: - skipped level 11 with operators [7, 11, 19, 43] because degenerate_distance_geometry - skipped level 13 with operators [3, 13, 43] because set_too_small - skipped level 17 with operators [17, 19, 43, 67] because degenerate_distance_geometry - skipped level 19 with operators [3, 19, 67] because set_too_small - skipped level 23 with operators [7, 11, 19, 23, 43, 67] because degenerate_distance_geometry - skipped level 29 with operators [7, 29, 67] because set_too_small - skipped level 37 with operators [3, 7, 11, 37, 67] because degenerate_distance_geometry - skipped level 41 with operators [41, 43, 163] because set_too_small - skipped level 89 with operators [11, 67, 89] because set_too_small - skipped level 101 with operators [19, 43, 101] because set_too_small

level	operators	dim	rank(B)	offdiag	commutator	mean overlap
31	[3, 11, 31, 43]	2	2	0.000000	0.047887	0.998603
43	[3, 7, 19, 43, 163]	3	3	0.047859	0.013215	0.990520
47	[11, 19, 43, 47, 67, 163]	4	4	0.138280	0.052892	0.946374
53	[7, 11, 43, 53, 163]	4	4	0.112203	0.013262	0.991078
59	[11, 43, 59, 67]	5	3	0.062732	0.019059	0.990787
61	[3, 19, 61, 163]	4	3	0.098771	0.003501	0.992994
67	[3, 7, 11, 43, 67]	5	4	0.061132	0.035939	0.994674
71	[7, 11, 67, 71, 163]	6	4	0.035473	0.024297	0.998995
73	[3, 19, 67, 73]	5	3	0.018141	0.007377	0.999829
79	[3, 7, 43, 79]	6	3	0.035122	0.029685	0.999398
83	[19, 43, 67, 83, 163]	7	4	0.043772	0.022094	0.996817
97	[3, 11, 43, 97, 163]	7	4	0.033744	0.009833	0.943729
103	[3, 11, 43, 67, 103]	8	4	0.068022	0.023304	0.993411
107	[7, 43, 67, 107]	9	3	0.033387	0.028389	0.999648
109	[3, 7, 43, 109]	8	3	0.063350	0.023891	0.998492
113	[7, 11, 113, 163]	9	3	0.031944	0.012755	0.998915
127	[3, 7, 43, 67, 127]	10	4	0.074598	0.033363	0.995495
131	[19, 67, 131, 163]	11	3	0.027122	0.016250	0.999527
137	[7, 11, 19, 137]	11	3	0.034129	0.012723	0.999250
139	[3, 19, 43, 139]	11	3	0.039105	0.025654	0.999366
149	[7, 19, 67, 149]	12	3	0.052472	0.032643	0.998773
151	[3, 7, 67, 151, 163]	12	4	0.046589	0.009608	0.996505
157	[3, 11, 19, 67, 157]	12	4	0.098744	0.018411	0.981976
163	[3, 7, 11, 19, 67, 163]	13	5	0.064190	0.033603	0.983322
167	[43, 67, 163, 167]	14	3	0.030796	0.020323	0.998118
173	[43, 67, 163, 173]	14	3	0.011075	0.006754	0.999654
179	[7, 11, 163, 179]	15	3	0.016483	0.011141	0.999548
181	[3, 11, 43, 67, 181]	14	4	0.086232	0.021393	0.993849
191	[7, 11, 19, 191]	16	3	0.005137	0.002313	0.999816
193	[3, 7, 43, 67, 193]	15	4	0.050675	0.037848	0.997788
197	[7, 19, 43, 163, 197]	16	4	0.042223	0.022480	0.998954
199	[3, 11, 19, 67, 163, 199]	16	5	0.039501	0.014647	0.997140

Meaningful same-rank comparison:

• the split-plus-vacuum family is much more heterogeneous, so raw minima are dominated by small 4-operator rank-3 cases
• in this family too, level 163 is good but not exceptional in the broad scan