A reciprocal graph-deviation functional: structural identifiability from controlled equilibrium maps

We constructed a reciprocal graph-deviation functional for positive ratios on a finite weighted graph. Its scalar generator $J(x)=\frac{1}{2}(x+x^{-1})-1$ was characterized previously by ratio-composition and unit log-curvature. Our principal new result was structural identifiability: already on a two-site, one-edge system, the complete minimizer-location map obtained by sweeping the boundary and site-to-edge weight ratio determines any reciprocal generator whose log-coordinate representation is $C^2$ with everywhere-positive curvature, up to a positive scale; unit-curvature calibration fixes that scale. For the calibrated reciprocal family~$J_p$ we obtained the equilibrium response in closed form. A synthetic $15$-condition sweep with seven simulated measurements per condition and log-scale noise $\sigma =0.05$ selected $J$ over~$J_{1.1}$ with probability~$0.997$ under exact boundary and weight parameters; a declared representation-uncertainty screen lowered that value to approximately~$0.88$. These probabilities characterize the simulated design under the declared data-generating model. The finite sweep discriminated prespecified generators, whereas nonparametric recovery required the complete response map. We also recorded the associated convex gradient-flow and Langevin dynamics.