Every number in the ring vibrates. Its 'eigenvalue' is a sum of cosines one from each channel. Numbers that share structure with N vibrate less. Numbers coprime to N vibrate most. This wave-like behavior is not imposed it's forced by the CRT.
For each channel with clock size q, the 'note' that element n plays on that channel is: 2 * cos(2*pi*r/q), where r = n mod q. Sum all five notes: that's the eigenvalue.
sigma = 1 has r = 1 on every channel, so all notes are positive maximum vibration. The void (0) has r = 0 on every channel, so every note is exactly 2 the maximum possible eigenvalue (10.0). Wait, that means the void vibrates MOST? Yes. 0 resonates perfectly with the ring's structure because it IS the ring's structure.
| Element | Eigenvalue (x1000) | Meaning |
|---|---|---|
| 0 (void) | 10000 | Perfect resonance. 2+2+2+2+2. |
| 1 (sigma) | 7660 | Strong positive. All channels hum. |
| OMEGA (606376) | 5660 | D-channel dead, 4 channels alive. |
| KEY (41) | -1618 | Negative! Destructive interference. |
The 'coupling' of n in Z/NZ is N / GCD(n, N). It measures how many distinct elements n can reach by multiplication. Units have coupling = N (they reach everything). The void has coupling = 1 (stuck at zero). Coupling is the ring's measure of connectivity.
| Element | GCD with N | Coupling | Interpretation |
|---|---|---|---|
| 1 (sigma) | 1 | 970200 | Maximum. Reaches everything. |
| 2 (D) | 2 | 485100 | Half. Locked to even elements. |
| 42 (ANSWER) | 42 | 23100 | Moderate. K*b-fold connected. |
| 0 (void) | 970200 | 1 | Minimum. Only itself. |
Start with D = 2 and keep multiplying: 2, 4, 8, 16, 32... In the ring Z/970200Z, this sequence eventually reaches a fixed point: D^420 = 606376 = OMEGA. After that, multiplying by D again changes nothing: OMEGA * D = OMEGA.
OMEGA is an idempotent: OMEGA * OMEGA = OMEGA. Its CRT decomposition is (0, 1, 1, 1, 1) zero on the D-channel, identity on all others. It PROJECTS away the D-component. This isn't designed it's forced by the arithmetic.
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