Uniqueness and reconstruction of reciprocal constitutive generators from equilibrium response maps

Abstract

We recover a shared constitutive generator from equilibrium locations on weighted graphs. For an even $C^2$ log-coordinate potential with positive curvature, the complete two-site response map determines the generator up to positive scale. A sweep over all positive boundary values at one fixed, possibly unknown, unequal weight ratio gives global identification through a Schröder-Koenigs iteration. We extend this result to a three-vertex chain with two observed free vertices under a sweep condition. For variable-ratio control, we prove finite-target recovery, local Lipschitz stability, and noisy-bisection bounds. Fixed-ratio stability and nonparametric estimation under uncertain controls remain open. In the reciprocal family $J_p$, an exact Gaussian model-selection probability benchmarks synthetic experiments under observation noise and errors in variables. These are identifiability results under controls; no measured data or physical system is analyzed.

arXiv comment36 pages, 4 figures, 3 tables. Supplemental material (19 pages) and reproducibility code included as ancillary files. v2: Substantially revised, with a new fixed-ratio identification theorem, a two-free-vertex graph extension, finite-query and stability bounds, expanded literature comparisons, and numerical validation