Hecke-Regge Bridge at Level 163

Exact statement

Let G be the 7x7 Hecke Gram matrix on the Heegner set and let D(d,d') = G(d,d) + G(d',d') - 2 G(d,d'). Let J = I - (1/7) 11^T.

Then the centered Hecke geometry is given exactly by

B := -1/2 J D J = J G J.

Proof

Write D = diag(G) 1^T + 1 diag(G)^T - 2 G. Multiplying on the left and right by J, the first two terms vanish because J 1 = 0. Hence

J D J = -2 J G J,

so

-1/2 J D J = J G J.

This is the canonical centered Gram matrix recovered from the Hecke distance geometry.

Obstruction to literal equality with the cotangent Laplacian

Let L_cot be the cotangent Laplacian built from the Regge edge lengths ell(d,d') = sqrt(D(d,d')).

Then L_cot != G for a structural reason:

• L_cot 1 = 0
• G 1 = [-72.0, -40.0, 114.0, 118.0, 168.0, 458.0, -4.0]

So any exact theorem has to compare L_cot to the centered Hecke operator B, not to G itself.

Numerical bridge on the level-163 data

• ||B + (1/2) J D J||_F = 2.908e-13
• nonzero spectral correlation between B and L_cot: 0.982148
• relative commutator norm ||[B,L_cot]|| / (||B|| ||L_cot||) = 0.030399
• in a B-eigenbasis, the relative off-diagonal size of L_cot is 0.071352
• optimal eigenvector matching overlap: min 0.930724, mean 0.973626

B eigenvalue	matched L_cot eigenvalue	eigenvector overlap
10.632739	26.837106	0.996905
60.562594	17.814893	0.930724
96.738276	15.303978	0.937400
251.545366	10.591287	0.990881
455.849771	7.378571	0.989801
810.671253	5.250892	0.996042

Honest conclusion

• The exact theorem is the double-centering identity B = -1/2 J D J = J G J.
• The cotangent Laplacian is not literally equal to G, and not even well-described by a simple affine rescaling of B.
• But L_cot and B are numerically close to simultaneously diagonalizable on 1^perp.

So the credible mathematical lead is:

Hecke distance geometry -> exact centered Gram operator B

together with the conjectural refinement

L_cot(sqrt(D)) is a geometric near-diagonalization of B.