01 · Prime Cycles Dodge Predators — tested
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
report.md
# 01 · Prime Cycles Dodge Predators — tested
**Source:** Incoprea Theory, Volume XXIII, Theory 08 (cicada-prime-predator). Posted at incoprea.com/research.
**Claim (verbatim):** "Periodical cicada brood intervals are set by the ratio of predator cycle length to developmental time, favouring prime numbers because they minimise overlap with shorter cycles."
**Prediction:** "Brood intervals should be prime or rarely composite."
**Refutation:** "Intervals tracking climate would remove the number-theoretic argument."
## Why probable
This is the standard Goles–Yoshimura / Lloyd–Dybos number-theoretic account of Magicicada periodicity; specialists consider the prime-interval argument quantitatively plausible, though not the only factor.
## Method
Two independent computations (code/bicie... see `code/cicada_test.py`):
1. **Divisor count:** for every brood interval N in 2–50, count predator cycles k (2 ≤ k < N) that divide N — a predator cycling every k years meets the brood at every emergence exactly when k | N.
2. **Encounter model:** expected predator-peak coincidences over 10^5 years for brood intervals N = 12–20 against predator cycles {2,3,4,5,6,7,8,9,10,12,15} using lcm(N,k).
3. **Observed nature:** compare with the real Magicicada periods.
## Results
- Mean number of divisors: **0.00 for prime N** vs **3.18 for composite N** (N = 2–50). The primes 13 and 17 have no divisors in 2…N−1.
- Expected predator coincidences per 10^5 years: N=13 → 15,992 and N=17 → 12,229, versus neighbouring composites N=12 → 54,802, N=16 → 29,504, N=18 → 33,294, N=20 → 30,437. **Prime intervals cut predator overlap by roughly 2.5–3.4× relative to adjacent composites.**
- Observed periodical cicadas use exactly **13-year and 17-year broods — both prime** (M. tredecim group; M. septendecim group).
## Verdict
**SUPPORTED.** The number-theoretic prediction is verified exactly, the simulated overlap is minimised at 13 and 17 among nearby intervals, and both observed Magicicada periods are prime. The test confirms the mechanism's arithmetic; it does not by itself prove selection history (developmental/life-history constraints on period evolution are not modelled).
## Limitations
- The encounter model assumes predators cycle with the chosen periods; real predator dynamics are more complex.
- Why cicadas landed on 13/17 specifically (rather than any prime) involves emergence synchrony and developmental constraints not tested here.
- This is a model + natural-history check, not a field experiment.
Working report — not peer reviewed. Educational information, not medical advice. AI-assisted test: DeepSeek.