Hecke-Regge Spectral Map

This note tests the next conjectural bridge step after near-diagonalization: whether the matched cotangent eigenvalues mu_i are approximately a simple scalar function of the matched centered-Hecke eigenvalues lambda_i.

For each level:

1. build the cumulative Hecke operator set
2. form B = J G J
3. build L_cot(sqrt(D))
4. match eigenspaces by maximum overlap
5. fit the matched pairs (lambda_i, mu_i) with several one-parameter or low-parameter models

Models tested:

• affine: mu = a lambda + b
• inverse_power: mu = c lambda^(-a)
• inverse_affine: mu = a/lambda + b
• rational_11: mu = (a + b lambda) / (1 + c lambda)

Local comparison: 43, 67, 163

level	best fit	rel. error	next best	next error
43	rational_11	0.000000	affine	0.127282
67	rational_11	0.046187	inverse_power	0.139191
163	rational_11	0.051635	inverse_power	0.130101

Full seven-operator cumulative levels

level	best fit	rel. error	next best	next error
163	rational_11	0.051635	inverse_power	0.130101
167	rational_11	0.033380	inverse_power	0.114459
173	rational_11	0.032869	inverse_power	0.089317
179	rational_11	0.037633	inverse_power	0.134067
181	rational_11	0.054411	inverse_power	0.111952
191	rational_11	0.022931	inverse_power	0.079568
193	rational_11	0.064548	inverse_power	0.207598
197	rational_11	0.014911	inverse_power	0.067821
199	rational_11	0.027505	inverse_power	0.064387

Read

• The affine model is never competitive in the six-dimensional full-operator cases.
• The inverse-power law is consistently much better than affine, with exponents in the range 0.364 to 0.484 across the full seven-operator levels.
• The best simple model tested is the decreasing rational map mu ≈ (a + b lambda) / (1 + c lambda).
• On the full seven-operator cumulative levels [163, 167, 173, 179, 181, 191, 193, 197, 199], its relative error ranges from 0.014911 to 0.064548.
• Over the same levels, the inverse-power residual ranges from 0.064387 to 0.207598.

Honest conclusion

The numerical bridge is stronger than mere commutator smallness: after overlap-matching eigenspaces, L_cot looks like a decreasing nonlinear spectral transform of B. But the fit is not exact, and the currently best low-complexity model is rational rather than affine.

So the plausible next conjecture is not L_cot = G, nor even L_cot ≈ a B + b I, but rather:

L_cot|_(1^⊥) ≈ f(B|_(1^⊥))

for a monotone decreasing scalar function f, with the rational (1,1) family as the current leading ansatz.