Geometric Protection of Spin-Correlated Radical Pairs: A Relational Model for Cryptochrome Magnetoreception Lifetimes

[Theoretical Research Manuscript / Quantum Biology Magnetoreception Framework]This paper presents a self-contained, classically rigorous open quantum system framework establishing an absolute geometric bound on coherence lifetimes within spin-correlated radical pairs [FAD• - TrpH• +] in avian cryptochrome photoreceptors. We translate the abstract regularized tracking properties of generalized trace-map recurrences into the peer-recognized structures of stochastic Lindblad master equations, anisotropic hyperfine tensors, and singlet-triplet quantum yield invariants. By introducing an adaptive, parameter-dependent non-local tracking potential Φ̂_τ scaled by an irrational golden-ratio projection filter, we establish a strict spectral gap over the Liouville relaxation super-operators. Under the infinite tracking depth limit (τ → ∞), we prove that random thermal dephasing is suppressed, forcing the quantum magnetoreception lifetime to remain stable and functional against ambient environmental noise. Pipeline Disclosure: Core conceptual formulation—substituting the custom trace-recurrence matrix parameters with the classical frameworks of open quantum systems, Lindblad master equations, multi-spin radical pairs, and golden-ratio projection alignments—was fully designed and authorized by the author. Initial technical layout and radical pair projection variables organized via Grok (xAI); rigorous biophysical geometry validation, Liouvillian spectral gap checking, and production-ready LaTeX typesetting finalized via Gemini (Google).