The Topological Algebraic Structure of Evolutionary Phase Transitions: A Rough Operator Algebra (ROA) Framework

This paper proposes a radical departure from the classical, one-dimensional deterministic paradigm of molecular biology, introducing a multidimensional, non-Riemannian geometric framework for biological evolution. By integrating Rough Operator Algebra (ROA), Seonggil Matrix Theory (SMT), and the Seonggil Torsion (STCT) Field Equation, we mathe matically formalize the causal loop of macro-mutations and evolutionary phase transitions.We demonstrate how environmental pressure induces algebraic defectiveness at Exceptional Points (EP), triggering a topological dimensional leap. The resulting historical adaptation is frozen as a topological knot, quantified by the Chern-Simons invariant, acting as an algebraic memory capacity. Finally, the projection of these high-dimensional topological invariants into the macroscopic phenotypic space is mediated by a pullback operator equipped with Rough Gauge Equivariance, precisely mapping “evolvability” to the volumetric difference between upper and lower approximation spaces.