O0-CRP-028 — Scientific Record

**Title:** NK-Model Scaling Toward Biological Realism

**Program:** Track 3 followup R3

**Version:** 1.0.0

**Status:** COMPLETED · Executed 2026-07-28

**Preregistration:** v1.0.0 frozen before execution

Claim status

CLAIM STATUS: **SUFFICIENCY_SCALES · Order-emergence sufficiency holds at N=50, G=2000**

EVIDENCE TYPE: COMPUTATIONAL SIMULATION

PHYSICAL VALIDATION: NONE

Verdict: all five sufficiency criteria pass at 25× larger genome (N=50 vs 20) and 4× longer evolution (G=2000 vs 500). Mean |Δ vs theory| = 0.087 (below 0.10 tolerance).

Key finding

**The O0-CRP-023 sufficiency-of-known-forces finding scales.** At biological-toy scale (N=50, K up to 10), standard evolutionary dynamics (mutation + tournament selection + finite population) produces order-increase at rates consistent with adaptive-walk theory. Random search does not. Frozen dynamics do not. High-mutation dynamics regress relative to tournament (Muller-ratchet-like). Selection beats random by > 5%.

Results by condition

| config | K=0 | K=2 | K=5 | K=10 |

|---|---|---|---|---|

| **tournament3** (mean fit_max) | 0.678 ± 0.039 | 0.740 ± 0.022 | 0.766 ± 0.019 | 0.753 ± 0.016 |

| random | 0.566 ± 0.034 | 0.586 ± 0.025 | 0.595 ± 0.016 | 0.595 ± 0.019 |

| frozen | 0.579 ± 0.037 | 0.593 ± 0.022 | 0.596 ± 0.014 | 0.606 ± 0.019 |

| high_mutation | 0.595 ± 0.036 | 0.614 ± 0.022 | 0.615 ± 0.017 | 0.596 ± 0.012 |

| theoretical | 0.808 | 0.770 | 0.713 | 0.618 |

| Δ = observed − theory | −0.130 | −0.030 | +0.054 | +0.136 |

Interesting: at K=10, tournament3 (0.753) *exceeds* the theoretical adaptive-walk prediction (0.618) by 0.136. The theoretical prediction formula assumes a single-locus adaptive walk without population-level effects. At high K, population-based tournament search finds better local optima than a solo random walk, so the "theoretical prediction" is a pessimistic bound. This is not a violation — it's information that the theory used here is a bound rather than a precise expectation at high K.

At K=0, the reverse occurs: tournament3 (0.678) *underperforms* theory (0.808) by 0.130. K=0 has a single global optimum discoverable by any hill-climber; the shortfall likely reflects finite-time convergence and the fact that we're taking the population-max as a fitness measure, but the population is not homogeneous at N=50 in G=2000 generations at μ = 1/50.

Figures





Substantive reading

1. **Hatcher's residual-force argument does not gain force at biological scale.** The CRP-023 finding that measurable order-increase matches theoretical predictions from known forces alone holds — and in fact holds *better* at intermediate K where the theory is most reliable.

2. **The tournament3 - random gap widens at higher K.** At N=20 (CRP-023) the gap was ~0.15. At N=50 (CRP-028) the gap is ~0.15-0.17. Scaling does not close the gap; if anything, selection's advantage over random becomes more pronounced.

3. **High-mutation regression persists.** At all K, high_mutation < tournament3. Excessive mutation drives populations off local optima faster than selection can consolidate them. This is Muller-ratchet-adjacent behavior and consistent with population-genetics theory.

Adversarial interpretation

• **The theoretical prediction formula is calibrated on the K=0..N-1 continuum but is a rough approximation.** The K=0 mismatch (−0.130) and the K=10 overshoot (+0.136) both suggest the formula is loose at the extremes. A more careful theoretical calculation would improve tolerance.
• **N=50 is still far from biological scale (N > 1000).** These results probe the scaling *trend* but do not directly test biological genomes.
• **Population size stayed at P=100.** In real biology, effective population sizes vary from 10^3 (mammals) to 10^9 (bacteria). Scaling in P is a separate axis (registered followup).
• **Sample size reduced from 30 to 20 seeds** to fit compute budget. This is a minor reduction; std errors above are per-seed variance and standard errors of the mean are smaller.

Not established

• Sufficiency at N > 100.
• Sufficiency with realistic biological complications (diploidy, recombination, environmental change).
• Any inference about *specific* biological systems.
• Refutation of Hatcher's argument in the strong form. This study only shows the empirical claim (that evolution *observably* fails to close the theoretical gap) does not scale.

Retrospective note on O0-CRP-023

CRP-023 reported theory-observation agreement within 0.10 tolerance at N=20. This study confirms that agreement extends to N=50 with the same tolerance (mean |Δ| = 0.087). Both studies used the same rough analytical theory; the theory is loose but the empirical picture is consistent.

Replication procedure

cd research/studies/O0-CRP-028/src
python run_study.py    # ~17.5 min
python analyze.py

Reuses O0-CRP-023 NKLandscape, run_evolution, and apply_verdict.

Registered followups

• R1: N=100, G=5000 — full biological-adjacent scaling.
• R2: Recombination — add crossover operator and re-test.
• R3: Environmental change — non-stationary landscape.
• R4: Rigorous Wright-Fisher analytical prediction (replace rough adaptive-walk bound).

Revision history

• v1.0.0 (2026-07-28): initial record.