Golden Ratio as Stable Fixed Point of Self-Application Recurrence — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The arXiv paper "On the Golden Ratio and Stable Self-Application" (2510.08934v3) is the sole substantive mathematical result; the rest are pedagogical videos or tangential graph theory (Tree Forest Ratio). The paper operationalizes the golden ratio as a stable fixed point of local recurrence under self-application, linking irrational approximation to computational self-reference. | MATH: Let Φ = (1+√5)/2 ≈ 1.6180339887. The paper treats Φ as a limit of a recursive update rule: x_{n+1} = 1 + 1/x_n, which converges to Φ from any positive start. Key identity: Φ = 1 + 1/Φ, equivalently Φ² − Φ − 1 = 0. The "stable self-application" boundary likely involves a map f(x) = 1 + 1/x whose derivative at Φ is |f'(Φ)| = 1/Φ² ≈ 0.381966 (which is 1 − 1/Φ = 0.381966, the complementary golden ratio). This derivative magnitude < 1 guarantees local stability. The paper's core claim: Φ is the unique real number where the reciprocal map and its self-application (iteration) remain locally stable wh
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026golden,
title = {{Golden Ratio as Stable Fixed Point of Self-Application Recurrence — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23431281},
url = {https://zenodo.org/records/23431281},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Golden Ratio as Stable Fixed Point of Self-Application Recurrence — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23431281
UR - https://zenodo.org/records/23431281
AB - FINDING: The arXiv paper "On the Golden Ratio and Stable Self-Application" (2510.08934v3) is the sole substantive mathematical result; the rest are pedagogical videos or tangential graph theory (Tree Forest Ratio). The paper operationalizes the golden ratio as a stable fixed point of local recurrence under self-application, linking irrational approximation to computational self-reference. | MATH: Let Φ = (1+√5)/2 ≈ 1.6180339887. The paper treats Φ as a limit of a recursive update rule: x_{n+1} = 1 + 1/x_n, which converges to Φ from any positive start. Key identity: Φ = 1 + 1/Φ, equivalently Φ² − Φ − 1 = 0. The "stable self-application" boundary likely involves a map f(x) = 1 + 1/x whose derivative at Φ is |f'(Φ)| = 1/Φ² ≈ 0.381966 (which is 1 − 1/Φ = 0.381966, the complementary golden ratio). This derivative magnitude < 1 guarantees local stability. The paper's core claim: Φ is the unique real number where the reciprocal map and its self-application (iteration) remain locally stable wh Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -