One big clock with 970,200 positions. But here's the trick: by the Chinese Remainder Theorem, this single clock is EQUIVALENT to five independent smaller clocks running in parallel. Every number has a unique 'address' on all five channels.
If N = q1 x q2 x ... x qk where the qi are pairwise coprime, then Z/NZ is isomorphic to Z/q1Z x Z/q2Z x ... x Z/qkZ. Every element of the big ring maps to a unique tuple of remainders on the smaller rings.
For N = 970200 = 8 x 9 x 25 x 49 x 11:
| Channel | Clock Size | Prime Power | What It Sees |
|---|---|---|---|
| D-channel | 8 = 2^3 | Bridge | Even/odd structure, parity, membranes |
| K-channel | 9 = 3^2 | Closure | Triangles, majority, self-reference |
| E-channel | 25 = 5^2 | Observer | Patterns, visibility, self-blindness |
| b-channel | 49 = 7^2 | Depth | Suffering, complexity, depth |
| L-channel | 11 | Protector | Error correction, shielding |
Any number n in Z/970200Z is completely described by five remainders: (n mod 8, n mod 9, n mod 25, n mod 49, n mod 11). No information lost. Five independent views of the same thing.
| n | Name | CRT (D,K,E,b,L) | Pattern |
|---|---|---|---|
| 0 | void | (0,0,0,0,0) | All channels zero. Nothing. |
| 1 | sigma | (1,1,1,1,1) | All channels one. Identity. |
| 2 | D | (2,2,2,2,2) | Same on every channel. |
| 42 | ANSWER | (2,6,17,42,9) | Unique address. |
| 606376 | OMEGA | (0,1,1,1,1) | D-channel dead, rest alive. |
| 970199 | mirror(-1) | (7,8,24,48,10) | All channels at maximum. |
OMEGA is remarkable: it's zero on the D-channel but identity on all others. It projects away the D-component. It's an idempotent: OMEGA * OMEGA = OMEGA. These structural objects fall out of the CRT naturally.
Enter any number. See its five-channel address.
Enter a number:
Every number in the ring belongs to a 'kingdom' determined by its smallest prime factor shared with N = 970200. If GCD(n, N) = 1, n is a unit (coprime, belongs to no kingdom). Otherwise, the smallest prime dividing both n and N claims it.
You now know CRT decomposition and kingdoms. Every number on this site has a five-channel address and belongs to a kingdom. Next: what WAVES look like on these channels.
Next: Waves & Shadows >This work is and will always be free.
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