Analytic Foundations of Thermal Quantum Annealing: Convergence Guarantees and Computational Advantage for Quantum Matrix Methods

Thermal Quantum Annealing (tQuA) is an open-system quantum computing architecture in which an Ising-type continuous variable array coupled non-adiabatically to an external thermodynamic reservoir evolves toward the ground state of a target Hamiltonian H by shedding energy to its environment. This paper provides a rigorous mathematical foundation for tQuA and its architectural extension to Quantum Intelligent Matrix Methods (QIMM). We enforce strict mathematical honesty by explicitly distinguishing among Theorems (proved in cited literature), Propositions (proof sketches with identified open gaps), and Conjectures (claims with open steps). This work delivers three principal contributions. First, we establish a five-pillar proof chain demonstrating that tQuA achieves global convergence to the thermodynamic Gibbs state from any initial state without adiabatic scheduling, without a priori eigenstate knowledge, and with a strictly positive escape rate from all local minima at any finite temperature T > 0. Second, we report honest numerical scaling benchmarks for the Liouvillian spectral gap Δ on gopherhole landscapes. By migrating from artificial classical Metropolis step-heuristics to a physically accurate, continuous Ohmic spectral density J(ω) satisfying the Kubo-Martin-Schwinger (KMS) detailed balance conditions, we uncover an unexpected inversion: the "Hard" scaling regime (where local minima grow proportionally to system dimension d) scales significantly better than the "Sparse" regime, exhibiting a power-law decay of Δ ~ d⁻⁰·⁷² vs. d⁻⁰·⁸⁸. Third, we formalize the adaptive annealing policy framework, positioning tQuA as a runtime-adjustable policy system that optimizes continuous variables by updating temperature paths β(t) and coupling schedules using real-time function evaluations and prior probability distributions Pr(f, W) over targeted problem classes. These results have immediate applications to nonconvex optimization, AI training loops, distributed filtering, and intensity-correlation interferometry for astronomical imaging.