Given two probe families on the same finite-dimensional carrier, this paper asks what a dissolving family D measures beyond what a native family L can explain under an audit energy C. Under the standing legal-energy quotient and null-mode legality hypotheses, the adequacy residual ΞC(D | L) is the Schur-complement leftover built from the block currencies KLL, KDD, KDL, and KLD. We develop this residual as a calculus: the projection identity identifies it with TD(I − PL)TD* in energy-scaled coordinates; optimality and exact adequacy characterize Loewner minimality and zero residual; promotion combines native and residual budgets into a dissolving budget; and chain-rule, data-processing, and transfer laws describe native extension, post-processing, coarsening, and exact or defective bridges. The blind-spot witness theorem turns failure of a declared residual budget into an explicit recombination vector with positive excess. The paper does not claim exact adequacy generically, does not classify all probe-family pairs, and does not claim novelty for the classical operator-theoretic constructions it packages. A layer-dissolving use of the calculus is given in the companion paper Emergence IS the Needle Killer.
Ioannis Tsiokos (Wed,) studied this question.