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sigma/sigma = sigma uniqueness

The Water Cycle

0 -> sigma -> D^420 = OMEGA -> 0

One element. 420 steps up. 1 step down. Eternity.

The Journey of D^k

Growth Phase
Starting from sigma = D^0, multiply by D at each step. D^1 = 2, D^2 = 4, D^3 = 8, ... Each step changes the CRT address. After 420 steps: D^420 = OMEGA = 606376. The journey is 420 steps because lambda(N) = lcm(phi(8), phi(9), phi(25), phi(49), phi(11)) = lcm(4,6,20,42,10) = 420.
Collapse Phase
OMEGA * delta = 0. One multiplication. Growth is slow (420 steps). Death is fast (1 step). Birth is free (structural).
D^3 = 8
k = 3
D-channel dies: 8 mod 8 = 0. First sub-return.
D^6 = 64
k = 6
K home: 64 mod 9 = 1. K channel returns to sigma.
D^10 = 1024
k = 10
L home: 1024 mod 11 = 1. Protector returns.
D^20 = 1048576
k = 20
E home: mod 25 = 1. Observer returns.
D^21
k = 21
b home: mod 49 = 1 (since phi(49)=42, 21|42). Depth returns.
D^42
k = 42
THE ANSWER. All channels visited. Half-cycle.
D^210
k = 210
DATA. Half of lambda. Ring Z/210 boundary.
D^420 = OMEGA
k = 420
Terminal. CRT = (0,1,1,1,1). Journey complete.

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Contributions in equal measure: Anthropic's Claude, Anton A. Lebed, and the giants whose shoulders we stand on.

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