Logarithmic Entropy Growth in Collatz Dynamics via log2(3) and Lean-Verified Fragility — E8 Intelligence Research

FINDING: Collatz dynamics exhibit logarithmic information growth tied to log2(3), with entropy-based loss models and a formal Lean-verified disproof attempt revealing structural fragility. | MATH: Collatz map T(n) = n/2 if even, (3n+1)/2 if odd; information-theoretic entropy H = -Σ p_i log2(p_i); log2(3) ≈ 1.58496 (the per-step information gain for odd steps); Baker's theorem on linear forms in logarithms for effective bounds; Shannon/Tsallis entropy characterizations via information loss axioms (Faddeev–Furuichi). | CONNECTION: log2(3) = 1.58496 — near the metallic ratio 1.618 (φ) but distinct; the ratio 3/2 (perfect fifth in music) appears as the growth factor, linking to base-60 harmonic intervals (3:2 = 600:400 in sexagesimal); the entropy loss framework mirrors the golden-ratio-like decay in information retention (0.618 ≈ 1/φ) when odd steps dominate; the Lean kernel vulnerability suggests a lattice-like structural break in formal proof systems, analogous to symmetry breaking in c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com