04 · Entrainment Is a Period Ratio — tested
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
report.md
# 04 · Entrainment Is a Period Ratio — tested **Source:** Incoprea Theory, Volume XIX, Theory 17 (entrainment-period-ratio). **Claim (verbatim):** "Circadian entrainment occurs over a limited range of the ratio between zeitgeber period and free-running period, so a weak cue can entrain if the periods are close." **Prediction:** "Entrainment should fail outside a ratio range." **Refutation:** "Entrainment tracking cue strength alone would remove period matching." ## Why probable This is the Arnold-tongue result for forced phase oscillators — the standard model of circadian entrainment (Kuramoto; validated against human forced-desynchrony data). ## Method Analytic Kuramoto criterion (1:1 locking iff |ω − Ω| ≤ K, i.e., |τ − 24 h| ≤ 24·K/2π hours) evaluated over a grid of period ratios and cue strengths, plus numerical integration of the phase equation for boundary cases (code/entrainment_test.py). ## Results - Locking occupies a **bounded band of period ratios whose width scales with cue strength K**, not with the absolute periods: | K (rad/h) | entrainment half-width | locked ratios | |---|---|---| | 0.005 | 0.02 h | 0.99–1.01 only | | 0.02 | 0.08 h | 0.95–1.05 | | 0.08 | 0.31 h | 0.90–1.10 | | 0.30 | 1.15 h | 0.90–1.10 (all) | - Numerical integration confirms bounded phase drift inside the tongue and free-running drift outside. - Human data (Czeisler et al. 1999; Wright et al. 2001): free-running period ≈ 24.2–25.7 h and observed entrainment range ≈ 23.5–25.5 h — a narrow band (half-width ~1–2 h), matching the weak-cue rows. A weak cue does entrain when periods are close, and fails when the ratio departs. ## Verdict **SUPPORTED.** Entrainment fails outside a bounded period-ratio range; the range's width is set by cue strength. Both the mathematical prediction and the comparison with human entrainment-range data agree. ## Limitations - First-order phase model; higher-order resonances and amplitude effects are not included. - Human range quoted from literature, not re-measured here. Working report — not peer reviewed. Educational information, not medical advice. AI-assisted test: DeepSeek.