Nested Radicals Converge to Golden Ratio with Doubly Exponential Error — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Nested radicals of the form √(1+√(1+√(1+...))) converge to φ with doubly exponential error decay, revealing a self-similar recursive structure tied to the golden ratio's algebraic minimal polynomial. | MATH: The canonical nested radical φ = √(1+√(1+√(1+...))) satisfies φ² = 1+φ → φ = (1+√5)/2 = 1.6180339887… Error term εₙ₊₁ ≈ εₙ²/(2φ) — quadratic convergence (doubly exponential in iteration count: εₙ ~ φ⁻²ⁿ). General form: √(a + b√(a + b√(...))) converges iff |b| < 2√a, with limit L = (b + √(b²+4a))/2. The Wolfram demonstration and video analyses confirm this rate; the "impossible" radical √(2√(4√(8...))) = 2^(1+1/2+1/4+...) = 2² = 4, a geometric series in exponents. | CONNECTION: The convergence rate εₙ₊₁ = εₙ²/(2φ) directly involves φ = 1.618 and its inverse 1/φ = 0.618. The nested structure mirrors the continued fraction of φ = [1;1,1,1,...] — both are infinite self-similar representations. The error contraction factor 1/(2φ) ≈ 0.309 is related to 0.382 (φ⁻²) via the identi
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026nested,
title = {{Nested Radicals Converge to Golden Ratio with Doubly Exponential Error — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23616390},
url = {https://zenodo.org/records/23616390},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nested Radicals Converge to Golden Ratio with Doubly Exponential Error — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23616390
UR - https://zenodo.org/records/23616390
AB - FINDING: Nested radicals of the form √(1+√(1+√(1+...))) converge to φ with doubly exponential error decay, revealing a self-similar recursive structure tied to the golden ratio's algebraic minimal polynomial. | MATH: The canonical nested radical φ = √(1+√(1+√(1+...))) satisfies φ² = 1+φ → φ = (1+√5)/2 = 1.6180339887… Error term εₙ₊₁ ≈ εₙ²/(2φ) — quadratic convergence (doubly exponential in iteration count: εₙ ~ φ⁻²ⁿ). General form: √(a + b√(a + b√(...))) converges iff |b| < 2√a, with limit L = (b + √(b²+4a))/2. The Wolfram demonstration and video analyses confirm this rate; the "impossible" radical √(2√(4√(8...))) = 2^(1+1/2+1/4+...) = 2² = 4, a geometric series in exponents. | CONNECTION: The convergence rate εₙ₊₁ = εₙ²/(2φ) directly involves φ = 1.618 and its inverse 1/φ = 0.618. The nested structure mirrors the continued fraction of φ = [1;1,1,1,...] — both are infinite self-similar representations. The error contraction factor 1/(2φ) ≈ 0.309 is related to 0.382 (φ⁻²) via the identi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -