Conservation capacity of local learning rules in physical networks

Abstract

How many memories can a local learning rule protect, and what fixes the number? Physical learning rules train resistive networks through local measurements, and the standard rules conserve a mass-like function of the conductances, a law whose general form was posed as an open problem with the expectation that no useful solution theory exists. We answer it, in the direction the expectation ran against, and the answer is a capacity theory. The Tellegen identity behind the conserved mass localizes: the feedback state is pinned to zero at the inputs, so every sector of the circuit that the input electrodes separate carries its own private conserved mass, by an argument consuming only Kirchhoff's law and that boundary condition, hence valid for arbitrary nonlinear branch laws. The number of independent sector masses is a topological property of the circuit, bounded by the number of output electrodes. An untrainable element can destroy the mass of its own sector and of no other, which says where fixed nonlinear components may be placed; per-edge learning rates select which functionals are conserved and never how many; and adjoint coupled learning, which clamps its outputs before measuring, drains every output-carrying sector at a rate set by the squared output multipliers. In linear circuits we then classify the identities: exact rational computation over all 502 circuits on up to five vertices with two inputs and one or two outputs, and hundreds of larger ones, yields a closed-form count, proved on that family and conjectured in general, with the sector statement a theorem at every circuit size. The anticipated series-parallel polynomial laws appear as exactly the series-class cubic differences, and the number the opening question asks for is a budget: one designed mass per sector, the differences the topology donates, and one broadcast scalar for each protected functional beyond.