Thermal Quantum Annealing (tQuA) is a quantum architecture in which a system coupled non-adiabatically to a thermodynamic reservoir evolves toward the ground state of a target Hamiltonian H. This paper provides a rigorous and honest foundation for tQuA and its extension to Quantum Intelligent Matrix Methods (QIMM), distinguishing carefully among Theorems (proved in cited literature), Propositions (well-grounded proof sketches), and Conjectures (claims with explicitly identified open steps). We make three principal contributions. First, we present a five-pillar proof chain establishing that tQuA converges to the global minimum from any initial state, without adiabatic scheduling, without eigenstate knowledge, and with a strictly positive escape rate from every local minimum at any T > 0. Second, we present honest scaling benchmarks for the Liouvillian spectral gap Δ on gopherhole Hamiltonians, varying Hilbert space dimension d and number of local minima M independently. We report a key finding: fixed-schedule tQuA achieves a perfectly flat gap (Δ = 0.05, independent of d up to d=32) when M is fixed and small, but experiences steep polynomial decay (Δ ~ d⁻⁸) when M grows proportionally to d. This is an honest result, not an overclaimed one. Third, and most importantly, we show that the fixed-schedule result is a baseline, not a ceiling. tQuA admits a rich adaptive annealing policy framework: the temperature schedule, coupling strength, and jump operator structure can all be updated using observed function evaluations and prior probability distributions Pr(f, W) over problem classes, without requiring knowledge of the function being minimized in advance. This adaptive framework is the proper setting for tQuA's advantage claim, and its systematic development is identified as the primary research frontier. Two fundamental errors in D-Wave's approach — the adiabatic/non-adiabatic confusion and the binary/continuous distinction — are identified and precisely characterized. The Ouroboros classical-quantum equivalence (Werbos 2026) provides the deepest physical foundation for why thermal reservoir coupling drives a quantum system to its ground state. A complete working implementation on NVIDIA’s CUDA-Q platform (version 0.14.2) is reported, using a calibrated noise model for Infleqtion Sqale hardware to project performance before hardware access — following the MOSIS-analogy principle used by NSF-funded university programs to estimate chip performance before fabrication. Projected quantum advantage over classical simulated annealing: 10.9× at n=8 qubits (256 states), 13.6× at n=10 qubits (1,024 states), 27.3× at n=20 qubits. An unexpected new application — detection of J-field radiation from the neutral chaoiton dark matter candidate (Werbos 2026, Zenodo 10.5281/zenodo.20497308) — has been identified, connecting tQuA/QIMM signal processing to nuclear sensing and the ICI/NRPR astronomical imaging system.