Quantum Twin of Golden Ratio Emerges in Quasicrystal Stability Operators — E8 Intelligence Research

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Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-17
DOI
https://doi.org/10.5281/zenodo.22805487
Primary Topic
Quasicrystal Structures and Properties
Type
preprint
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preprint

Quantum Twin of Golden Ratio Emerges in Quasicrystal Stability Operators — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Quasicrystal Structures and Properties
preprint

Quantum Twin of Golden Ratio Emerges in Quasicrystal Stability Operators — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: The golden ratio's "twin" (likely the silver ratio or reciprocal pair) appears in quantum systems and self-application stability, with quasicrystal inflation/deflation operators as eigen-structures. | MATH: Twin of φ = 1/φ = φ−1 ≈ 0.618 (if "twin" = reciprocal); or silver ratio δ_S = 1+√2 ≈ 2.414 (if "twin" = metallic mean). Quantum system: φ appears as eigenvalue of a Hamiltonian or transfer matrix — exact value φ = (1+√5)/2, satisfying φ² = φ+1. Inflation/deflation operator T on quasicrystal tilings: T acts on substitution rules, eigenvalues λ = φ, φ−1, −φ−1 (for Penrose/Pisot tilings). Stable self-application paper (arXiv:2510.08934): treats φ as a fixed point of the map x ↦ 1+1/x, with convergence rate governed by φ−1. | CONNECTION: φ and φ−1 are the two roots of x²−x−1=0 — their sum = 1, product = −1. This pair generates the golden ratio's self-similarity: φ−1 = 1/φ = 0.618, and φ−1 + φ = √5 ≈ 2.236. Quasicrystal inflation: 5-fold symmetry (crystallographically forbidden Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Decent work and economic growth
Quasicrystal Structures and Properties
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