Extrapolation, Inverse Scattering, and the Mathematical Necessity of Rigidity Projections in Formal Machine Reasoning

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Large language models struggle with deep, multi-step mathematics because autoregressive generation acts as an ill-posed extrapolation problem and a shallow approximation of inverse scattering. Without global constraints, small token-level perturbations compound exponentially across reasoning chains, triggering hallucinations, compositional collapse, the reversal curse, and circuit complexity bottlenecks. The DSM architecture resolves this breakdown by replacing unconstrained continuous drift with layer-by-layer rigidity projections, transforming an unstable statistical predictor into a stable, error-correcting dynamical system: Spectral Filtering: Restricts candidate states to a low-frequency harmonic subspace across an invariant discrete network gap, suppressing high-frequency prompt noise and stabilizing the extrapolation trajectory. Algebraic Integer Quantization: Snaps continuous representations onto the Voronoi cells of a rigid algebraic integer lattice, guaranteeing that quantization error cannot compound across arbitrary network depth and bypassing standard circuit depth ceilings via exact arithmetic. Topological Conservation: Enforces discrete exterior calculus nilpotency at the hardware level, ensuring that information flows remain strictly solenoidal and conservative, which mechanically blocks logical self-contradiction. Modular Clocking: Employs an intrinsic packing symmetry defect that injects a persistent phase bias, preventing reasoning trajectories from stalling in repetitive loops or degenerate limit cycles. Thermodynamic Gating: Evaluates a structural uncertainty metric in real time to shunt off-manifold candidates before switching digital registers, confining energy expenditure strictly to geometrically admissible deductions near the fundamental Landauer erasure limit. The resulting system establishes provable error non-accumulation, bounds transversal drift to zero, and provides the rigorous mathematical foundation required for deterministic, hallucination-free reasoning.