Helical Symmetry Overlap in Phyllotaxis and Microtubules Lacks Explicit Fibonacci Model — E8 Intelligence Research

FINDING: Fibonacci phyllotaxis and microtubule protofilament geometry converge on the same helical symmetry family, but the search results are largely introductory; the sole substantive mathematical model is a stochastic GTP-cap catastrophe model with no explicit Fibonacci/golden-ratio content. | MATH: Fibonacci sequence \\(F_n = F_{n-1}+F_{n-2}\\), \\(F_0=0, F_1=1\\); golden ratio \\(\\varphi = \\frac{1+\\sqrt{5}}{2} \\approx 1.6180339887\\); reciprocal \\(\\varphi^{-1} = \\varphi - 1 \\approx 0.6180339887\\); \\(\\varphi^{-2} \\approx 0.381966\\) (≈0.382); \\(\\varphi^{-3} \\approx 0.236068\\); \\(\\varphi^{-4} \\approx 0.145898\\). Phyllotaxis: divergence angle \\(2\\pi/\\varphi^2 \\approx 137.5077^\\circ\\) (≈137.5°). Microtubule: 13 protofilaments — 13 is a Fibonacci number (\\(F_7\\)); helical pitch angles for 13-protofilament microtubules follow \\( \\tan\\theta = \\frac{p}{13 \\cdot d}\\) where \\(p\\) is the tubulin monomer rise (~0.92 nm) and \\(d\\) is the tubulin diameter (~4.6 nm), giving \\(\\theta \\approx 10.5^\\circ\\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com