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

Abstract

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 \(1\times n\) board with half-squares and \((\tfrac12,\tfrac12)\)-fence tiles (arXiv:1907.06517). - Duplex numbers: \(a + b\varepsilon\), \(\varepsilon^2 = +1\) (split-complex) — Schrödinger ↔ diffusion via Wick rotation \(t \to it\), but algebraic structure prevents full equivalence (arXiv:1201.3069). - Rational-index inverses: \(1/F_{p/q}\), \(1/L_{p/q}\) via codenominator functions (arXiv:2607.20754). - Heilbronn: \(\nexists\) guarantee \(\Vert h\alpha\Vert < \varepsilon\) for all \(\alpha\) with \(h \in \{F_n\}\) (arXiv:250 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com