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, every reachable response block lies in a subspace of codimension p(p-1)/2, whatever the network's size, topology and stiffnesses; at full overlap the unreachable fraction is (p-1)/2p, tending to one half. When a finite list of demands replaces the block, the charge is a list invariant that on coordinate tasks counts the terminal pairs instrumented in both directions, so a shared terminal alone is free. 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 in advance; positive definiteness adds an orthogonal term. A second-order optimiser reaches that floor within 1% in 143 of 144 runs, a contrastive rule with a bond-local update direction within 0.1% in 22 of 24; the forbidden directions are the antisymmetric ones, which no tuning reaches and an odd coupling does: we prove that a rank-revealing choice of p(p-1)/2 odd bonds spans them whenever the passive network's wedges do, and realises any small enough antisymmetric shared block exactly in projection. At a fixed number of accessed degrees of freedom, each free to be both driven and read, where the bond count does not bind, overlapping sensors onto actuators roughly doubles 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 deposited linear model.

61 pages, 7 figures, 3 tables; 22 reproducible experiments and all result tables as ancillary files. Code and data: doi:10.5281/zenodo.22765365. v3 supersedes v2: adds the Sec. 3.4 task-list deficit law, which on coordinate tasks counts terminal pairs instrumented both ways, so a shared terminal is free; with experiment 21 and five references. No conclusion is withdrawn; a forecast is sharpened