PROJECT GENESIS: Mathematical Formalization and Extension - Non-Markovian Operator Algebra, Volterra Integro-Differential Transformations, Matrix Riccati Flows, and Hysteretic Dynamical Systems

This monograph formalizes the mathematical architecture of Project Genesis, establishing a non-Markovian continuum mechanics that replaces four-dimensional continuous spacetime and collisionless dark matter with a hysteretic, memory-retaining continuum. The framework replaces chronological coordinate time with a dimensionless Absolute Ordinal and substitutes the empty vacuum with a continuous, positive-definite Density-Stress Tensor coupled to an irreversible History-Memory Tensor. The formalization utilizes a complexified substrate tensor and a non-unitary Hysteretic Covariant Derivative embedding Volterra hereditary kernels. To resolve computational singularities and infinite-history convolutions, the architecture introduces autonomous Matrix Riccati flows to guarantee absolute memory saturation, alongside an Integration-Time Diffeomorphism and Prony-series auxiliary variables to reduce memory integrals to constant-time O(1) computational complexity. The resulting continuum mechanics natively derives Newtonian dynamics, the Schrodinger equation via stochastic osmotic drift, exact flat galactic rotation curves without dark matter, and fundamental topological constants—including the proton-to-electron mass ratio and fine-structure limit—via unbroken U(3) gauge geometry.