The Topological Signal Processor (TSP): Deterministic Spectral Manifold Architectures, Arithmetic Rigidity, and Algebraic Error Recovery

The Topological Signal Processor replaces conventional floating-point approximations with structural admissibility on an invariant discrete spectral manifold. Standard numerical methods, such as tensor networks and low-rank singular value truncation, fail in non-stationary and turbulent environments due to cumulative round-off drift and runaway variance inflation. The processor eliminates these instabilities by confining state trajectories to an eight-hundred-sixty-one-node Johnson graph whose discrete Laplacian spectrum is locked against continuous deformation by class-number-one arithmetic rigidity. Signal admissibility is enforced using an orthogonal projection operator that completely eliminates off-manifold noise components, guaranteeing that non-physical perturbations vanish identically. Rapid regime changes, acoustic shocks, and sharp transitions are captured by a third-order difference stencil that diverges strongly across discontinuities while remaining stable on smooth trajectories. When kinematic acceleration and geometric off-manifold deviations simultaneously exceed calibrated statistical thresholds, a syndrome gate initiates deterministic recovery by re-aligning states across the manifold's finite permutation automorphism group. Operating on a fixed, sparse graph topology allows the architecture to maintain bounded computational complexity per step, providing an algebraically stable alternative to continuous floating-point filtering