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^{\bullet -} \dots TrpH^{\bullet +}] 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 \hat{\Phi}_\tau 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 (\tau \to \infty), 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).