Topological Spectral Inversion: Resolving Isomorphic Divergence in Inverse Generative Therapeutics via the Discrete Dirac Operator within the Chrysene Formalism

Abstract

The emergence of Inverse Generative Therapeutics (IGT) has shifted the pharmacological paradigm from heuristic ligand discovery to the systematic reconstruction of disease state-spaces. However, a persistent barrier to this paradigm is isomorphic divergence—a phenomenon wherein pathologically distinct cellular states present identical classical network topologies (isomorphic graphs), yet exhibit radically divergent dynamical trajectories and drug-response profiles. This paper introduces a novel mathematical framework to resolve this ambiguity by applying the Chrysene Formalism, which bridges non-commutative geometry and discrete topologies. By lifting classical biological networks into non-commutative spectral triples over the Chrysene manifold Ic, we demonstrate that isomorphic divergence is not an intrinsic dynamical anomaly, but rather a projection artifact of omitting non-commutative metric relations. Crucially, we ground the complex phase factors of our spectral triple in the classical non-equilibrium thermodynamics of biochemical cycles, where phases represent localized chemical potential gradients and holonomies correspond to thermodynamic cycle affinities.

At the core of this methodology is the Discrete Dirac Operator, which acts on a graded Hilbert space of network states. Unlike the classical graph Laplacian, which is blind to directional phase relations and non-local topological coupling, the discrete Dirac operator captures the underlying discrete spin structure and holonomy of the biological state-space. We establish a rigorous theoretical foundation, correct previous algebraic inconsistencies regarding the adjoint relation and the Laplacian definitions, and outline a computational validation pipeline to demonstrate how the spectral flow of the Discrete Dirac Operator can be inverted. By calculating the precise spectral perturbations required to drive a pathological network back to a healthy basin of attraction, we establish a mathematically rigorous, topologically invariant foundation for generative de novo therapeutic design.