The Cyclotomic Factor Cascade as a Supercritical Branching Process

We model the factor chain induced by the third cyclotomic polynomial Φ₃(p) = p² + p + 1 as a Galton-Watson branching process on the set of odd primes. For each prime p in a cascade seeded by a finite set S₀, we count the number of new odd prime factors of Φ₃(p) not already present in the running set S. Over 300 seed configurations generating 12,927 parent-child observations, we find a global mean offspring rate μ = 1.1747, placing the process firmly in the supercritical regime (μ > 1). We observe a phase transition: μ(p) < 1 for primes p ≤ 200 (subcritical) and μ(p) > 1 for p ≥ 200 (supercritical), consistent with the prediction of Stewart's theorem that P⁺(Φ₃(p)) > p²⁻ᵋ for large p. The offspring rate increases further for higher cyclotomic polynomials (μ ≈ 2.99 for Φ₅, μ ≈ 1.71 for Φ₇). These measurements provide a quantitative explanation for the empirical observation that no finite set of odd primes is closed under the Φ-induced factor map, offering probabilistic evidence against the existence of odd perfect numbers.