What Lattice Vibrations Can Establish About Nuclear Reactions in Metal Hydrides: Identifiability, controlled preparation, and complete-cycle energy

Lattice motion, structural information and nuclear reaction probability are different observables. This self-contained mathematical physics paper develops explicit examples of the information lost between them, then connects calibrated measurements to conditional bounds for a complete controlled experiment. Two constructions are central. For every even particle number at least four, each of the bosonic and fermionic sectors contains two number-diagonal states with identical reduced density matrices at every proper particle order, yet a common positive two-body loss operator, with no other dynamics, gives different reaction probabilities by a deadline. The exact gap begins at the earliest Taylor order allowed by the particle order of the loss. A second construction realizes a classical homometric pair as stable equilibria of one smooth pair potential, with different vibration spectra and an explicit drive and readout that distinguish them. An extension of an earlier positive reset construction gives rapidly mixing processes whose correlations agree through any prescribed finite order at every time lag, while their first-reaction probabilities differ. Established observable-space, control, residual and renewal methods then connect material calibration, accessible preparation, finite-model error and complete-cycle energy. A quantum return map retains catalyst and host memory; the combined bound includes failed entrance, restoration, terminal costs and shared uncertainty. The paper distinguishes its explicit constructions and reaction-inference extensions from earlier marginal-state, homometry and control results. Essential proofs, editable sources, nine verification scripts, five reproducible figures and a separate scope report are supplied. The examples are conditional models, not calibrated palladium-deuteride predictions. Material validation, feasible preparation and readout, scalable error certificates and a demonstrated positive-energy protocol remain open.