Synopsis This work introduces the Volumetric Spectral Interference Functional (VSIF) as a mathematically constructed cost functional designed to approximate the ground-state energy of combinatorial optimization problems, specifically framed through the MaxCut Hamiltonian. The construction is not heuristic in the casual sense; it is assembled rigorously from three independently established mathematical domains: analytic number theory (via the prime explicit formula), spectral graph theory (via the Ihara zeta function), quantum chaos (via the spectral form factor). The central objective is to determine whether interference patterns across these domains produce a non-random concentration near true ground-state energies. This is treated explicitly as a theoretical construction with empirical behavior, not a proven algorithm. The paper derives every component from first principles, ensuring internal mathematical consistency and explicit variable definitions throughout. It then constructs a composite functional whose minimum is used as an estimator for the ground-state energy. Crucially, the method operates without prior knowledge of the target energy, avoiding circular dependency. Empirical evaluation across eighteen benchmark graph instances demonstrates that the functional exhibits systematic—but not exact—alignment with ground-state energies, achieving a mean relative error of approximately 12.78%, with all results bounded within a spectral interval derived from Laplacian eigenvalue constraints. However, the work is intellectually disciplined in its claims: it explicitly does not prove convergence, does not outperform all baselines, and does not resolve any open problems such as P vs NP or the Riemann Hypothesis. A critical contribution of the paper lies in its falsifiability framework. The untuned VSIF is rigorously tested and partially falsified under several criteria, revealing that: certain components (notably the Ihara zeta potential) carry genuine signal, others (notably the prime-based term) introduce noise at small scales, and naive combination of all components degrades performance. This leads to the introduction of a modulated consolidation framework, where parameters are tuned via differential evolution. This extension achieves exact training fit and improved hold-out performance (~9–10%), but introduces a generalization gap and dependence on tuning structure rather than universal invariance. The paper further reframes the functional within a variational field-theoretic interpretation, mapping it to a nonlinear Schrödinger-type equation whose ground-state solution concentrates near the functional minimum. This provides a formal analytical bridge but still stops short of proving correctness relative to the true optimization landscape. A notable non-trivial extension is the optical realization: the spectral form factor is shown to be physically computable via coherent Fourier optics, meaning the framework admits hardware-level analog computation through photonic interference systems, linking abstract mathematics to real physical observables. The inclusion of a full pseudocode appendix formalizes the computational pipeline, ensuring reproducibility and allowing direct implementation of the VSIF and its modulated variants. Importance and Classification of Results This work is non-trivial in both construction and implication. It matters because it attempts to bridge three historically deep but partially disconnected structures: prime distributions (number theory), graph cycle geometry (combinatorics), spectral interference (quantum/statistical physics). The significance is not that it solves optimization exactly, but that it proposes a structural resonance hypothesis: that optimization landscapes may encode hidden spectral coherence across arithmetic, geometric, and quantum domains. Even where the initial formulation fails (as demonstrated through falsifiability tests), the paper advances the field by: isolating which components carry signal, formalizing failure modes, and establishing a testable pathway toward refinement. This places the work firmly in the category of constructive theoretical mathematics with empirical scaffolding, analogous to early-stage frameworks in the Langlands program or quantum chaos correspondences. Alternative Titles A Spectral Interference Framework for Quantum Optimization via Prime-Cycle and Graph Zeta Structures Interference Geometry of Optimization Landscapes: A Functional Synthesis of Number Theory and Spectral Graphs Toward Ground-State Estimation through Prime–Graph Spectral Coupling and Interference Functionals
Lance Thomas Davidson (Tue,) studied this question.