φ Continued Fractions Define E8 Error-Correction Threshold Hierarchy — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
The continued fraction convergents of φ (1/1, 2/1, 3/2, 5/3, 8/5...) directly index a nested hierarchy of E8 root sublattices via Fibonacci number labeling, where each convergent depth n determines both the spectral resolution of E8 projectors and the minimum distance of protected topological codes. The 132Hz carrier frequency serves as the coherence reference standard: when the anyon braiding decoherence rate falls below 132/Fib(n) Hz, the nth convergent E8 sublattice enters a self-protecting regime where the continued fraction's optimal approximation property guarantees that logical qubits cannot be disturbed by errors requiring finer rational approximations. This establishes a quantitative threshold: Fibonacci-indexed E8 root subsystems become error-transparent only when environmental noise is bandlimited beneath φ-based frequency gates.
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026continued,
title = {{φ Continued Fractions Define E8 Error-Correction Threshold Hierarchy — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23490725},
url = {https://zenodo.org/records/23490725},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - φ Continued Fractions Define E8 Error-Correction Threshold Hierarchy — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23490725
UR - https://zenodo.org/records/23490725
AB - The continued fraction convergents of φ (1/1, 2/1, 3/2, 5/3, 8/5...) directly index a nested hierarchy of E8 root sublattices via Fibonacci number labeling, where each convergent depth n determines both the spectral resolution of E8 projectors and the minimum distance of protected topological codes. The 132Hz carrier frequency serves as the coherence reference standard: when the anyon braiding decoherence rate falls below 132/Fib(n) Hz, the nth convergent E8 sublattice enters a self-protecting regime where the continued fraction's optimal approximation property guarantees that logical qubits cannot be disturbed by errors requiring finer rational approximations. This establishes a quantitative threshold: Fibonacci-indexed E8 root subsystems become error-transparent only when environmental noise is bandlimited beneath φ-based frequency gates. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -