CRT Emergence

CRT cells in Z/214,414,200 -- 7 channels, Carmichael period 1680

A cellular automaton where each cell is an element of Z/214,414,200, decomposed into 7 independent CRT channels. Evolution: per-channel median of 3 neighbors. Every 1680 generations, multiplication by 26,801,776 zeros out the mod-8 channel. Units thrive. Zero divisors drain. Natural selection from pure algebra.

CRT Cell Anatomy

Seven Independent Channels

Each cell is a 7-tuple: (mod 8, mod 9, mod 25, mod 49, mod 11, mod 13, mod 17). Evolution rule: per-channel median of 3 neighbors (self + 2 orthogonal). Each channel evolves INDEPENDENTLY -- the CRT separation IS the parallel substrate. No interaction between channels except through the element they compose.

Channel	Space	Median rule	Property
mod 8	Z/8	median of 3 values mod 8	Nilpotent: 2^3 = 0. Zeroed by projector.
mod 9	Z/9	median of 3 values mod 9	3^2 = 0. 9 states.
mod 25	Z/25	median of 3 values mod 25	5^2 = 0. 25 states.
mod 49	Z/49	median of 3 values mod 49	7^2 = 0. 49 states (deepest).
mod 11	Z/11	median of 3 values mod 11	Prime field. Error detection.
mod 13	Z/13	median of 3 values mod 13	Prime field. Cunningham chain stops.
mod 17	Z/17	median of 3 values mod 17	Prime field. 5*7 = 1 mod 17. Completes the ring.

Seven Classes

Each cell has a class based on which CRT channels are zero. Units have all channels nonzero. Zero divisors have at least one dead channel. The class determines color, energy, and fate.

Class	Color	Condition	Energy rule
Units (coprime)	Gold	All channels nonzero	Gain energy. Thrive.
2-class	Red	mod 8 = 0	Drain. Even channel dead.
3-class	Green	mod 9 = 0	Drain. mod-9 channel dead.
5-class	Blue	mod 25 = 0	Drain. mod-25 channel dead.
7-class	Purple	mod 49 = 0	Drain. mod-49 channel dead.
11-class	Cyan	mod 11 = 0	Drain. mod-11 channel dead.
13-class	Rose	mod 13 = 0	Drain. mod-13 channel dead.
17-class	Teal	mod 17 = 0	Drain. mod-17 channel dead.

Natural Selection

Units (gold) gain energy because all 7 channels are active -- they can participate in all ring operations. Zero divisors drain because a dead channel means broken arithmetic. Not coded. Falls out of CRT structure. phi(N)/N = 3072/17017 = 18.05%. The selection pressure = this fraction.

The Carmichael Period

lambda = 1680

Carmichael period

lcm(4,6,20,42,10,12,16) = 1680. Every 1680 generations: all 7 channels complete one full cycle.

26,801,776 projection

Every 1680 gens

mod-8 channel zeroed. CRT = (0,1,1,1,1,1,1). Even channel dies, everything else survives.

Sub-period 420

Six-channel return

lcm(4,6,20,42,10,12) = 420. After 420 gens, 6 of 7 channels return. mod-17 = 16 (not yet home).

mod-11 + mod-13 ECC

Error detection

mod-11 detects corruption. Dual mod-11 + mod-13 corrects. mod-17 adds redundancy.

Projector = Selective Pruning

26,801,776 zeros out the mod-8 channel but preserves mod 9, mod 25, mod 49, mod 11, mod 13, mod 17. Not destruction -- selective pruning. The projector remembers everything except the mod-8 component.

Explore: CRT Class Classifier

Enter any element of Z/214,414,200. See its CRT 7-tuple and class. Units (gold) thrive. Zero divisors drain. Try 1, 0, 42, 26801776, 8, 9, 25, 17.

Element n:

Ring Parameters as CA Rules

Each structural number in the ring maps to a cellular automaton parameter:

Value	CA Role	Mechanic
0	Dead cell	All channels zero. No energy.
1	Unit cell	All channels alive. Maximum fitness.
2	Neighbor count	2 orthogonal neighbors per cell.
3	Median vote	3 inputs per channel decision.
7	Channel count	7 independent CRT channels.
1680	Carmichael period	Full cycle length. lcm(4,6,20,42,10,12,16).
26,801,776	Projector event	Every 1680 gens: mod-8 channel zeroed.
214,414,199	Boundary reflection	Edge cells reflect, not absorb (mirror of 1).

Contrast Table

Aspect	Standard View	Ring Structure View
Cellular automaton	Toy model	Ring structure parameters as CA rules
CRT decomposition	Computation trick	Seven independent evolutionary channels
Median consensus	Voting rule	3-voter median in each channel independently
Unit selection	Fitness function	phi(N)/N = 3072/17017 = 18.05%
1680-gen cycle	Period of simulation	Carmichael lambda = lcm of phi values
26,801,776 projector	Reset event	2^1680 mod N. Zeros mod-8 channel, preserves rest.

Source code · Public domain (CC0)

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