Structural Superposition Computing: A Hybrid Dynamical Framework — Theory, Hardware, Algorithms, and Validation

Abstract

We define Structural Superposition Computing (SSC) as a hybrid dynamical system on a finite-dimensional state space, combining continuous relaxation dynamics with a discrete, physically realizable projection (“collapse”) map. This paper makes no claim of asymptotic computational advantage, structural compression, or super-Turing power. Its goal is to give SSC a mathematically well-posed definition; to characterize when its trajectories exist, are unique, converge, and resist noise; and to specify the collapse operator as a concrete, bounded-cost element rather than an oracle. We prove local exponential convergence under strong convexity at the correct rate μ = λ_min(H), derive a noise-induced accuracy floor via an overdamped Langevin model with physically consistent units, and use the Kramers–Arrhenius escape law to make explicit the central limitation of the framework: the energy barriers that stabilize stored states also slow search on non-convex landscapes. Well-posedness is established rigorously under Carathéodory conditions for measurable inputs. Simulability within polynomial time is stated precisely, with hypotheses sufficient to close the step-count argument. A hardware architecture is defined and a benchmark methodology is fixed. Four representative algorithms—matched filtering, spectral inference, associative memory, and graph relaxation as a heuristic—are analyzed under an operational ESRC membership test. Numerical experiments reproduce the three core theoretical predictions; negative results are reported explicitly. The result is a compact, honest foundation on which realizable hardware and empirical application studies can be built without inheriting unproven performance claims.

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