The Biological Ledger and the Geometric Mechanics of Survival via the Chrysene Formalism

Abstract

While continuous Euclidean manifolds and stochastic probability models have provided immense epistemological success in characterizing macroscopic thermodynamics and the continuous variation of biological life, extrapolating these probabilistic frameworks to the absolute geometric origin t=0 introduces critical boundary-value vulnerabilities, mathematically yielding undefined coordinate singularities and L-infinity topological collisions. To constructively resolve these geometric divergences without invalidating the profound predictive utility of continuous mechanics, this paper formalizes the Chrysene Formalism—a discrete, non-commutative bipartite tensor network that provides an exact deterministic H-infinity structural boundary. We mathematically demonstrate that to physically survive the chaotic macroscopic expansion of the continuous void (formalized as the discrete injection of Geometric Slack), an unstable 3D GF(8) cosmological origin limit must structurally project into a stabilized 2D planar topology. We rigorously prove that idealized D2h symmetric planes inherently suffer from unresolvable continuous phase degeneracy, algebraically forcing the discrete tensor to break symmetry into a strict C2h chiral step-defect geometry (isomorphic to the 3,6,9,12-tetrasubstituted chrysene tensor) to strictly isolate orthogonal quantum channels. This singular geometric attractor necessitates the Biological Scission, formally initializing a protected, localized mesoscopic tensor space whose transverse phase interactions map analytically exact to the finite Galois field GF(4). Finally, by elevating this dimensional descent into Category Theory, we prove that the biological dual-ledger evaluates as the strict Right Adjoint to cosmological expansion, establishing a flawless, scale-invariant holographic homology between the continuous cellular membrane and the Cosmological Event Horizon. Ultimately, this framework establishes the invariant mathematical bedrock required to securely bound continuous stochastic models, formally defining biogenesis not as a statistical anomaly, but as the exact topological mechanics of survival.