Z/12,612,600 is finite -- 12,612,600 elements, 6 CRT channels -- yet its Cayley graph has a spectral gap that never closes. Across all 108 rings sharing Carmichael period 420, the smallest nonzero eigenvalue of the Laplacian is locked at 4*sin^2(pi/49). The mod-49 channel forces this floor. No combination of the other five exponents can close it.
The Spectral Plateau
108 rings share Carmichael period 420. Each is a product of prime-power rings (Z/2^a x Z/3^b x ...) with different exponents. Despite the variation, the Cayley-Laplacian gap is identical in all of them:
Gap = plateau
4*sin^2(pi/49) ~ 0.01644
Forced by the mod-49 channel across all 108 rings. The 5-dimensional exponent lattice cannot close it. (Structural analog to Yang-Mills mass gap -- NOT a Clay Millennium proof.)
Why mod-49?
49 = 7^2 is the largest prime-power factor
The spectral gap of Z/n is 4*sin^2(pi/n). Among channels {8, 9, 25, 49, 11, 13}, the 49-channel has the smallest gap and therefore dominates.
Gabriel-Horn cap
Finite volume, unbounded surface
Z/N truncates Fourier modes at k <= N-1. Bounded L^2-energy fills a Gabriel-Horn-shaped spectral cap. (Structural analog to Navier-Stokes regularity -- NOT a Clay Millennium proof.)
CRT orthogonality
7 independent Laplacians
On Z/8 x Z/9 x Z/25 x Z/49 x Z/11 x Z/13 x Z/17, simultaneous singularity across orthogonal channels is forbidden. The plateau is structurally protected.
Finite Energy, Infinite Iteration
The Carmichael period 420 means every unit's orbit cycles within 420 steps: a^420 = 1 for all elements coprime to the ring. Finite elements, infinite repetition. Even non-units eventually reach a fixed point -- 2^420 mod 12,612,600 = 1,576,576, an idempotent that kills the mod-8 channel and preserves the rest.
Dyson (1979) observed the same structure in physics: if each cycle of thought costs less energy than the last by a fixed ratio f, the total energy is a geometric series E_0/(1-f). It converges. Finite energy buys infinite experience.
f = 0.5
Total = 2 * E_0
Double the first cycle. Infinite iterations.
f = 0.9
Total = 10 * E_0
Ten times the first cycle. Generous.
f = 0.99
Total = 100 * E_0
Barely cooling. Still converges.
f -> 1
Total -> infinity
No cooling = infinite cost. Must cool to survive.
Explore: Geometric Series
Enter a cooling percentage (1-99). The geometric series 1 + f + f^2 + ... = 1/(1-f). At 50%: 2x. At 90%: 10x. Finite sum, infinite terms.