A Topological Framework for Cortical Visual Prostheses: The Ψ-Ω-H Homological Perception Calculus and Concrete Substrate Isomorphism (Phase I)

Abstract

I present a unified, mathematically closed, and non-circular calculus for cortical visual prostheses that models visual perception as a functorial mapping between the category of visual scenes and the category of cortical stimulation patterns. By treating perception through algebraic topology rather than classical pixel-to-electrode discretization, the signal pipeline preserves persistent homology and cohomology invariants under continuous deformations, patient-specific cortical geometry shifts, and material electrode degradation. The framework is governed by three foundational operators: 1. (Perception Homology Functor): Extracts persistent homology features () across cubical complexes derived from optical input and maps them to canonical neural activation manifolds. 2. (Stimulation Cohomology Functor): Solves the ill-posed non-linear inverse problem of deriving optimal 4096-channel current vectors using a Preconditioned Conjugate Gradient (PCG) solver accelerated by forward- and reverse-mode automatic differentiation (Jacobian-Vector Products), bypassing explicit Hessian allocation while avoiding subnormal float singularities. 3. (Hardware Abstraction Functor): Decouples mathematical representations via a platform-agnostic Neuromorphic Intermediate Representation (NIR), stabilized by Parameter-Agnostic Adaptive Compensation () via regularized Tikhonov pseudo-inversion. I validate this system against a high-fidelity laminar spiking neural network (SNN) model of primate primary visual cortex ( leaky integrate-and-fire neurons across 6 cortical layers) reproducing experimental tuning curves with correlation . A topological security suite (-Operator) utilizes Takens delay embeddings, persistent landscape norms, and chaotic Lorenz phase-decorrelation to neutralize exogenous neural injection attacks (JAM, FLO, and Trojan bit-flips). Finally, I derive the operating system invariants that guarantee execution stability: Copy-on-Write (CoW) memory consolidation reducing peak memory consumption by , zero-touch autonomous cryptographic keypair synthesis, and asynchronous socket isolation under Unix Discretionary Access Control (DAC).