Golden Ratio as Fixed-Point Operator in Recursion Theory — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The only substantive mathematical result is the arXiv paper "On the Golden Ratio and Stable Self-Application" (2510.08934v3), which treats φ as a fixed-point operator in recursion theory, linking irrationality to computational self-reference. The YouTube results are pedagogical or irrelevant (trading, Fibonacci myths). | MATH: φ = (1+√5)/2 ≈ 1.6180339887; satisfies φ = 1 + 1/φ (fixed-point equation). The paper's core: φ is modeled as a stable local recurrence — the reciprocal map x ↦ 1/x + 1 has φ as its unique positive fixed point. In computational terms, this is a self-referential update rule where approximation error contracts by factor |φ'|⁻¹ = φ⁻² ≈ 0.381966 (the silver of the golden ratio). This is exactly the contraction constant for the map f(x) = 1 + 1/x near φ. | CONNECTION: The contraction factor 0.381966 = 2 − φ = φ⁻² is precisely the complementary golden ratio (0.382). This is the same constant appearing in: (a) the ratio of successive Fibonacci numbers' error con Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212233990
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