Reciprocity can halve what a mechanical network can learn

Abstract

Tunable mechanical networks are being built as materials that learn in place. Capacity is estimated by counting parameters against target constraints, ignoring Maxwell-Betti reciprocity, the symmetry every passive, linear elastic network obeys. When p degrees of freedom are both driven and read out, every reachable response block lies in a subspace of codimension p(p-1)/2, whatever the size, topology and stiffnesses; at full overlap the unreachable fraction is (p-1)/2p, approaching one half. The consequence for training is a number: any learning rule leaves an error at least the norm of the target's antisymmetric part on the shared block, computable before training, and positive definiteness adds an orthogonal second term. A second-order optimiser with the exact Jacobian reaches that floor within 1% in 143 of 144 runs across 36 configurations, a contrastive rule with a bond-local update direction within 0.1% in 22 of 24 runs at a fixed step budget; the forbidden directions are the antisymmetric ones, which no tuning reaches and an odd coupling does: we prove that a rank-revealing selection of p(p-1)/2 odd bonds spans them whenever the passive network's wedge vectors span the relevant exterior square, and then realises every sufficiently small antisymmetric shared block exactly in projection. At a fixed number of accessed degrees of freedom, each of which may be both driven and read, the layout rule inverts: where the bond count does not bind, overlapping sensors onto actuators buys about twice the ceiling on reachable dimension; at a fixed count of sensors plus actuators it does not. Prescribed-displacement drives obey a companion law. The symmetry is classical; its fixed-graph consequence at partial overlap is new. For a published robotic metamaterial, no symmetric positive-definite stiffness matrix meets both targets in the linear model they deposit.