CRT Cellular Automata

A cellular automaton applies the same local rule everywhere, iteratively. Complex patterns emerge from simple rules. CRT decomposes the CA into 7 independent layers -- each channel runs its own CA rule at its own modular scale. After 3 iterated steps, each cell absorbs a 4-gram context window. The channels evolve independently, coupled only at sync points.

Test: evolve CA rules for bigram text prediction. CRT CA = 7 per-channel lookup tables (132 entries total), each applied 3 times with left-neighbor context. Monolithic CA = one 128-entry table applied 3 times. Same population (16), generations (500), corpus (471 chars). Both get 4-gram context via iteration -- the question is whether CRT decomposition helps evolution find better rules.

How the CA Predicts Text

Initialize: each position in the text gets its CRT residue (char mod q per channel). Update: each cell combines its left neighbor and itself (combined = (left + self*2) mod q), looks up the result in the channel's rule table. After 3 steps, position t has information from positions t-3 through t. The CA's prediction at position t should match the next character's residue at t+1. The rule table has only q entries but captures 4-gram patterns through iteration.

This is genuinely different from direct lookup (prev -> next). Direct lookup uses 1 step with 2-gram context. The CA uses 3 iterated lookups with 4-gram context. Same parameter count (132 entries). The depth gives more context at the cost of a harder optimization landscape.

Results: 2.4x Advantage

CRT CA bigram accuracy: 85/470 (18.1%). Best monolithic CA across 3 seeds: 35/470 (7.4%). CRT wins by 2.4x. Both use the same CA iteration depth (3 steps) and the same evolutionary framework. The difference is purely structural: CRT decomposes the rule space.

Convergence: Steady Climb

CRT per-channel accuracy rises steadily: 9% (gen 0) to 15% (gen 50) to 21% (gen 500). The curve shows no plateau after 500 generations -- more evolution would improve further. The trajectory matches the direct-lookup experiment (10% to 27% over same generations), confirming that per-channel credit assignment drives efficient search regardless of the prediction mechanism.

Cross-Channel Sync

After 3 independent CA steps, channels may have drifted apart. A sync step couples them: decode all 7 channels to a character via majority vote, then re-encode back to channel residues. This corrects channels that disagree with the consensus. Run 3 more CA steps from the synced state.

Result: sync improves accuracy from 85 to 87 (+2.4%). Small but positive -- the independent channels are already quite consistent (they evolved under the same corpus), so correction has limited room. The key insight: minimal sync at specific points is all that's needed. Continuous coupling would destroy channel independence.

CA vs Direct Lookup: Depth vs Width

Direct lookup: CRT 96/470, mono 21/470, ratio 4.6x. CA iteration: CRT 85/470, mono 35/470, ratio 2.4x. The CA helps both methods by providing 4-gram context instead of 2-gram. But it helps mono MORE (21 to 35, +67%) than CRT (96 to 85, -11%). Why?

The CRT advantage scales with the gap between search space complexity and available information. Direct lookup has a huge gap: mono searches 128-dim space with only bigram signal, while CRT searches 7 small spaces. With CA's 4-gram context, mono gets richer signal, partially closing the gap. CRT's structural advantages (channel independence, small spaces, meaningful crossover) persist but are less decisive when the baseline has more information to work with.