Cold Fusion in Metal Hydrides: Mechanism Identification, Finite-Control Bounds, and Energy Closure

A detector counts fusion products, but it does not measure the fuel sampled by the beam. In a metal hydride, a rising yield may reflect more deuterium along the stopping path, a greater probability of reaction, or both. This paper develops a mathematical framework for distinguishing these explanations and testing the mechanisms they suggest. For fuel profiles of known total mass, the paper determines the exact worst-case error in estimating the reaction-weighted inventory from noisy diagnostic measurements. It also finds the greatest probability of a first reaction by a prescribed time for an initially inactive quantum system with bounded access, independently adjustable absorption and no dissipative feeding into the reactive sector. A two-level system attains this optimum through a simple protocol: prepare with absorption off, switch it fully on once, and stop transfer when the surviving state becomes fully reactive. A complementary analysis constrains what unresolved resonances can contribute to a measured yield. The two-channel resonance-area ceiling is the scattering form of the steady-current bound of Walschaers, Buchleitner and Fannes (2017). The paper treats arbitrary fixed channel widths without additional loss and quantifies the shortfall when coherent mixing is limited. Incorporating the full incident spectrum sharpens the experimental test: for a triangular beam, an area budget equal to its half-width permits a mean reaction probability of at most three quarters. Maximizing integrated resonance strength therefore need not maximize the yield seen by an experiment. Structural contrasts, compatible many-particle states and renewal laws connect these bounds to reaction counts and complete-cycle energy, including catalyst loss in muonic molecules. The proofs and worked models are self-contained. Their physical application still requires independently calibrated nuclear reaction strengths, accessible configurations, populations and lifetimes. The measurements examined establish neither a changed microscopic reaction probability in palladium nor positive net energy; the analysis identifies the information needed to decide those questions.