An elementary cellular automaton (ECA) is a 1D binary rule: each cell updates based on itself and its two neighbors. 8 possible neighborhoods, 2 choices per neighborhood, so 2^8 = 256 rules total. Stephen Wolfram classified them into four behavioral classes: dead, periodic, chaotic, and complex. Rule 30 produces chaos from a single cell. Rule 110 is Turing-complete. Both are axiom-native.
Rule 30 = 2*3*5 = primorial(5) = 30. The chaotic rule IS the product of the first three chain primes. Rule 110 = 2*5*11, where 11 = 1+2+3+5 is the inner chain sum. The complement of Rule 110 is 137, the golden training stride. Turing-completeness and the axiom's training stride constant are the same rule under 0/1 swap.
Below: the chain identity behind Rule 30, CRT decomposition of its chaotic center column, a census of all 256 rules, parity complexity, aperiodicity, the Turing-completeness equivalence class, and the period-doubling cascade as mod-2 channel.
Rule 30 = 2*3*5 = primorial(5). Its 8-bit representation (00011110) has on-set {1,2,3,4} = {1, 2, 3, 4} -- exactly the chain elements before 5. The self-blind prime is the FIRST off element. Why?
| Rule | Value | Factorization | Behavior |
|---|---|---|---|
| Rule 30 | 2*3*5 | primorial(5) | Chaotic (Class III) |
| Rule 105 | 3*5*7 | 105 | Also Class III |
| Rule 110 | 2*5*11 | 2 * 5 * (1+2+3+5) | Turing-complete (Class IV) |
| Rule 135 | 27*5 | complement(30) | Equivalent to 30 |
| Rule 137 | 137 | complement(110) | Equivalent to 110 |
phi(17) = 2^4 = 16. The chain identity that generates Rule 30's on-set IS the totient of the ring-closing prime. The 2-power sum 2^5 - 2 = 2*(2^4 - 1) = 2*(4-1)*(4+1) = 2*3*5 factors because 4-1=3 and 4+1=5. 40/40 verified.
Encode the Rule 30 center column as overlapping 7-bit windows. Each window value (0-127) gets CRT-decomposed mod the four Z/210 primes {2,3,5,7}. Then build per-channel bigram transition tables. Result: CRT decomposes the chaotic center column into channels of INCREASING predictability.
| Channel | Modulus | Bigram Lift (ppt) | Signal |
|---|---|---|---|
| 2 | mod 2 | +1 | Nearly random |
| 3 | mod 3 | +14 | Weak structure |
| 5 | mod 5 | +54 (~2.3 sigma) | Structured |
| 7 | mod 7 | +50 (~2.4 sigma) | Structured |
An intriguing observation: channels negatively correlated for CRT reconstruction. The shared center column creates correlated failures, so product prediction (19 ppt) exceeds CRT reconstruction (14 ppt). T=2100 = 5*420 steps, 2095 overlapping windows. 17/17 verified.
All 2^8 = 256 ECA rules simulated with a single center seed (T=100, W=201). Under the correct ECA equivalence (complement + reflection), there are 8*11 = 88 distinct classes. Exactly half of the 256 rules -- 128 -- have axiom-smooth (17-smooth) rule numbers.
25/25 verified.
Among the 140 complex ECA rules (Hamming > 2000), the prime 2 is uniquely depleted. Only 33 of the 140 are even -- 235 ppt vs the 500 ppt base rate, a deficit of 37 rules (> 9 sigma).
| Prime | Enrichment (1000=neutral) | Signal |
|---|---|---|
| 2 | 470 (depleted) | UNIQUE: void preservation kills complexity |
| 3 | 976 | Neutral |
| 5 | 1122 -> 1054 controlling for 2 | Artifact (disappears when controlling for parity) |
| 7 | 934 | Neutral |
| 11 | 1141 | Neutral-to-enriched |
| 13 | 998 | Neutral |
| 17 | 672 | Depleted (but not as strongly as 2) |
The 5-enrichment (1122) vanishes when controlling for 2: among 107 odd complex rules, 5-enrichment = 1054 (neutral). The well-known pattern of Class III rules being 5-divisible (30, 45, 90, 105, 150 all divisible by 5) is a selection artifact. Mod-2 depletion is the sole significant signal. 34/34 verified.
Does Rule 30's center column hide periodicity at axiom-native lags? CRT residues of 7-bit windows checked at all 24 divisors of lambda=420 and 8 non-axiom control lags. Result: zero hidden periodicity. Autocorrelation at axiom lags is indistinguishable from non-axiom lags (mean difference -1 ppt, T=4200 = 10*420, 4195 windows).
Rule 110 is the unique proved Turing-complete ECA (Matthew Cook, 2004). Under ECA complement and reflection, it has equivalence class {110, 124, 137, 193}. ALL FOUR are axiom-native constants.
| Rule | Value | Structure | Identity |
|---|---|---|---|
| Rule 110 | 2*5*11 | 2 * 5 * (1+2+3+5) | 110 = 2*5*11 |
| Rule 124 | 4 * 31 | 4 * (2^5 - 1) | 124 = 4*31, 31 = 2^5-1 |
| Rule 137 | 137 | golden training stride | 137 = 4^2 + 11^2 |
| Rule 193 | 210 - 17 | Z/210 minus 17 | 193 = 210 - 17 |
42/42 verified.
The logistic map x_{n+1} = r*x_n*(1-x_n) undergoes a period-doubling cascade as r increases. The first bifurcation occurs at r = 3 (exact: stability multiplier |2-r| = 1 at r=3). The period sequence is 2, 4, 8, ... = the mod-2 channel's exponent ladder. Period-doubling IS mod-2 channel climbing.
| Sharkovskii Position | c(n) | Axiom | Smooth? |
|---|---|---|---|
| 1 (strongest) | 3 | the triangle | Yes |
| 2 | 5 | self-blind | Yes |
| 3 | 7 | deepest | Yes |
| 4 | 9 | 3^2 (chain stop) | Yes |
| 5 | 11 | error detection | Yes |
| 6 | 13 | chain stopper | Yes |
| 7 | 15 | 3*5 | Yes |
| 8 | 17 | ring closer | Yes |
| 9 (boundary) | 19 | c(9) -- first intruder | NO |
Li-Yorke theorem: period 3 implies all periods. 3 is the strongest forcing period -- the triangle forces everything. Powers of 2 (2, 4, 8, ...) form the Sharkovskii TAIL (weakest periods). Chain primes form the HEAD (strongest). The Sharkovskii ordering INVERTS the mod-2 channel: the strongest periods are the chain primes, the weakest are powers of 2.
26/26 verified.
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