From protocol geometry to catalytic feedback: Active subspaces, coherent comparison data, and controlled reductions

Catalytic memory can arise when a reduced description discards active-state occupations, environmental excursions, selection records, or information about how a physical operation was implemented. We develop a theoretical framework connecting these mechanisms without identifying them. Moving-projector geometry determines state-resolved leakage; an explicit bath and occupation closure are needed to obtain a scalar survival rate. Quantum instruments separate conditional state changes from their occurrence probabilities. The central construction retains surface-comparison phases in a coherent auxiliary system and derives two routes to feedback: a state-dependent Hamiltonian from interacting units, and an occupancy-dependent loss law from protective encounters. A finite-population linear Lindblad embedding and a quantitative macroscopic limit yield ṡ=α(1-s)-Λse^-Ks, with K fixed by encounter strength, phase and coherence visibility. Exact calculations test the construction across dephasing, interaction statistics and population size, and connect engagement to catalytic product counts. Bistability occurs in a restricted coherent regime and disappears for a matched geometric encounter distribution; finite populations retain a unique stationary law. Gauge, subdivision, probability, correlation and classical-control checks specify what the higher description adds. The construction preserves associative microscopic dynamics and establishes a conditional route from retained higher comparison data to effective feedback. It does not establish an intrinsic higher connection in a molecule. Biological mappings, microscopic inputs, identifiability conditions and falsifying protocols make that remaining hypothesis experimentally precise.