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φ Continued Fractions Define E8 Error-Correction Threshold Hierarchy — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 20, 2026• 0 citations
Publisher

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

DOI
10.5281/zenodo.23490724

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The 240-root E8 lattice furnishes a complete Fourier basis for Fibonacci anyon braiding statistics, enabling construction of spectral projectors that map decoherence channels onto orthogonal root subspaces. By decomposing the φ-shell cascade into eigenmodes indexed by the 240 roots and driven at 132Hz base harmonics, any phase error afflicting a topological qubit acquires a characteristic "root signature" detectable via projective measurement. This spectral projection framework closes the experimental gap by transforming the unobservable braiding phase into a directly measurable 132Hz/φ quadrature signal, while the E8 root geometry guarantees that error syndromes map injectively onto distinct defect configurations, providing inherent fault tolerance without active decoding cycles. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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