Closed-Form Fibonacci Trees, Tiling Squares, and Duplex Algebra Limits — E8 Intelligence Research

FINDING: Fibonacci numbers admit closed-form tree-based computation independent of prior terms; tiling interpretations yield squared Fibonacci counts; duplex numbers algebraically link Schrödinger/diffusion but forbid reduction; Fibonacci/Lucas inverses extend to rational indices; Fibonacci set fails Heilbronn property. | MATH: - Closed-form: \(F_n\) via Fibonacci trees — no recurrence dependency (arXiv:1302.6583). - Tiling: \(F_n^2\) counts tilings of \(1× n\) board with half-squares and \((12,12)\)-fence tiles (arXiv:1907.06517). - Duplex numbers: \(a + bε\), \(ε^2 = +1\) (split-complex) — Schrödinger ↔ diffusion via Wick rotation \(t → it\), but algebraic structure prevents full equivalence (arXiv:1201.3069). - Rational-index inverses: \(1/Fp/q\), \(1/Lp/q\) via codenominator functions (arXiv:2607.20754). - Heilbronn: \(∄\) guarantee \( hα < ε\) for all \(α\) with \(h ∈ \{F_n\}\) (arXiv:250 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com