Structural–Spectral Computing (SSC): A Topology-Aware Representation-First Computing Paradigm for Complex, Dynamic Systems

Structural–Spectral Computing (SSC) is a representation-first computing paradigm for complex systems whose relational organization evolves over time. SSC transforms system states into harmonic coordinates induced by time-varying structural operators and performs inference, optimization, prediction, and control in an adaptive reduced spectral space. This version extends graph-based SSC with a topology-aware representational layer. The extension permits graphs, simplicial complexes, hypergraphs, and filtered relational objects to serve as structural substrates; it further incorporates persistent topological summaries and higher-order Hodge operators as diagnostics of multiscale organization. The resulting framework separates two complementary roles: structural spectra provide computational coordinates, while topology supplies observables for validating representation choice, tracking persistent organization across scale, and detecting higher-order reconfiguration. SSC therefore does not identify topology with spectral analysis, nor does it treat topological summaries as standalone predictors. Instead, it treats both as components of a unified computational pipeline: represent the system at an appropriate structural level, transform it into adaptive harmonic coordinates, monitor spectral and topological change, and compute in the smallest representation that preserves the task-relevant organization. The framework is substrate-independent: it is classically implementable with established tools, and quantum or hybrid backends are optional accelerators whose value remains an open empirical question.