Reciprocity can halve what a mechanical network can learn

Abstract: Tunable mechanical networks are being developed as materials that learn in place. Capacity is estimated by counting tunable parameters against target constraints, ignoring Maxwell-Betti reciprocity, the symmetry every passive, linear elastic network obeys by construction. 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 nearly half the target space is unreachable. 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 degrees of freedom, 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, a bond-local contrastive rule within 0.1% in 22 of 24, and odd couplings restore the lost directions at the price of an external torque source: the network becomes active. Prescribed-displacement drives obey a companion law we prove. The symmetry is classical; its consequence on a fixed graph with finitely many tunable stiffnesses at partial overlap is new. For a published robotic metamaterial it shows that no symmetric positive-definite stiffness matrix meets both targets in the linear model their deposited data integrate.