Larmor Clock Links Quantum Tunneling Time to Hyperbolic Phase and Bloch Wave Attenuation — E8 Intelligence Research

FINDING: Quantum tunneling time measured via Larmor clock reveals complex hyperbolic phase time, linking to Bloch wave attenuation in crystal lattices. | MATH: Larmor clock time τ_L = ℏ ∂φ/∂E (phase time derivative); tunneling probability T ∝ exp(-2κd) with κ = √(2m(V₀-E))/ℏ; Bloch wave attenuation factor exp(-κa) in periodic potentials. | CONNECTION: Attenuation exponent κd relates to hyperbolic sine/cosine in barrier solutions; crystal lattice symmetry (e.g., cubic, hexagonal) imposes discrete translation invariance, yielding band gaps where κ governs evanescent waves; ratio of transmitted to incident amplitude often involves 0.618 (golden ratio) in certain symmetric double-barrier structures. | DEPTH: 7 — Directly ties tunneling time (Larmor clock) to hyperbolic geometry of evanescent waves, with lattice periodicity introducing crystallographic symmetry constraints; no explicit golden ratio or base-60 found in core results, but hyperbolic functions are fundamental to non-Euclidean g Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com