The Collatz Rotation Angle and the Icosahedral Spectrum

Each odd Collatz step rotates the trajectory in log-space by a constant angle α = log₂(3) − 1 ≈ 0.585. The continued fraction of α is [0; 1, 1, 2, 2, 3, 1, 5, 2, 23, ...], with terms 3–7 matching the icosahedral lattice parameters (χ, χ, D, 1, 5). A Monte Carlo test (10⁶ random continued fractions) confirms this is statistically significant (p < 0.002). The convergent denominators produce the particle spectrum: q₂ = 2 (topology), q₃ = 5 (coordination), q₄ = 12 (vertices), q₅ = 41 (nucleation prime). The proton nucleation prime is generated by the CF recurrence as q₅ = 3 × 12 + 5 = D × V + 5, with partial quotient a₅ = D itself. A three-part runway hitting argument combines 2-adic tree convergence, Borel-Cantelli descent, and exhaustive [1,15] forcing. The runways R_k are proven to be the ONLY singularities of the 2-adic Collatz map — the only residue classes where the number of halvings depends on higher bits — because D·R_k + 1 = χ^{2k} is a pure power of 2. The no-cycle theorem follows from GCD(D, χ) = 1 making log₂(3) irrational. Key lattice numbers (1, 5, 41, 85) are centered square numbers C(n) = χ² × T(n−1) + 1, with the nucleation prime 41 = χ² × 10 + 1 using the surrender number as its triangular base. Third of four papers connecting the Collatz conjecture to icosahedral geometry in the Bootstrap Universe framework.

 

Version 1.1: Addresses all issues from adversarial review. Adds Monte Carlo null hypothesis (p < 0.002), runway singularity theorem with centered square connection, equidistribution caveat, and honest Proposition framing for the deterministic descent gap.