CRT Evolution

Evolutionary algorithms search by mutating, crossing, and selecting solutions. The genome structure determines how efficiently evolution can explore the space. CRT decomposition splits a genome into 7 independent genes -- one per channel. Mutation in one channel cannot disrupt another. Crossover swaps entire channels: meaningful genetic units, not random segments.

Test: evolve lookup tables for bigram text prediction. CRT genome = 7 per-channel tables (132 entries, avg search space 19 per gene). Monolithic genome = one direct table (128 entries, search space 128 per gene). Same population (16), same generations (500), same selection (tournament), same corpus (471 chars).

Convergence: CRT Climbs, Mono Stalls

CRT per-channel accuracy rises from 10% to 27% over 500 generations, steadily improving throughout. Monolithic accuracy jumps from 3% to 3.8% by generation 50, then completely stalls. The monolithic search space (128 possible values per gene) is too large for random mutation to explore. CRT's smaller per-gene spaces (8 to 49) allow rapid optimization.

Accuracy: 4.6x Advantage

Final bigram prediction accuracy (full CRT decode via majority vote): CRT 96/470 (20.4%). Best monolithic across 3 random seeds: 21/470 (4.5%). CRT wins by 4.6x. The tally-based analytical optimum (from the HDC experiment) is 172/470 (36.6%) -- the evolutionary approach reaches 56% of the analytical ceiling with zero domain knowledge.

Why CRT Wins: Three Structural Advantages

1. SMALLER SEARCH SPACES. Each CRT gene searches a space of size q_i (8 to 49, average 19). Each monolithic gene searches 128 values. A random mutation in CRT has 1/19 chance of finding the optimal value vs 1/128 for monolithic -- 6.7x better odds per mutation.

2. DENSE FITNESS SIGNAL. CRT fitness counts correct per-channel predictions (max 3290). Monolithic fitness counts exact character matches (max 470). CRT gets credit for partially-correct predictions: 5 of 7 channels right is rewarded, even if the decoded character is wrong. This gradient-like signal guides evolution.

3. MEANINGFUL CROSSOVER. CRT channel-swap exchanges a complete, independent genetic unit -- like swapping a chromosome. Monolithic single-point crossover cuts at an arbitrary position, mixing unrelated genes. CRT crossover preserves what works in each channel.

Search Space Decomposition

The CRT search space factorizes: 7 independent tables with a total of 132 entries. The monolithic space is 128 entries but each entry ranges over 128 values (vs average 19 for CRT). Effective search volume: CRT = product of q_i^q_i, but only one gene mutates per step, so the per-step search is bounded by max(q_i) = 49. Monolithic per-step search: 128. The same algebraic independence that enables efficient learning enables efficient evolution -- channel independence decomposes a hard search into easy ones.