Regge-Hecke Family Trend

This note tests the centered-Hecke-vs-cotangent comparison on the natural class-number-1 prime-level families:

• level 43: operators [3, 7, 11, 19, 43]
• level 67: operators [3, 7, 11, 19, 43, 67]
• level 163: operators [3, 7, 11, 19, 43, 67, 163]

For each level:

1. compute the Hecke kernel G
2. form the centered operator B = J G J
3. build the cotangent Laplacian L_cot from edge lengths sqrt(D) with D = diag(G) 1^T + 1 diag(G)^T - 2 G
4. restrict to the positive-eigenvalue Hecke subspace of B
5. measure the relative off-diagonal size of L_cot in a B-eigenbasis

level	operators	centered rank	offdiag fraction	min overlap	mean overlap
43	[3, 7, 11, 19, 43]	3	0.100321	0.967153	0.979476
67	[3, 7, 11, 19, 43, 67]	5	0.138548	0.914631	0.952021
163	[3, 7, 11, 19, 43, 67, 163]	6	0.071352	0.930724	0.973626

Read

• from 67 to 163, the off-diagonal fraction drops from 0.138548 to 0.071352
• so the near-diagonalization conjecture does show the trend you asked for on the two nontrivial higher levels
• level 43 does not sit on that monotone line; its rank-corrected off-diagonal fraction is 0.100321
• the natural reading is therefore “improves from 67 to 163” rather than “monotone across all class-number-1 levels”