Uniqueness and reconstruction of reciprocal constitutive generators from equilibrium response maps

The manuscript treats a structural inverse problem for an even, strictly convex C2 log-coordinate potential Psi belonging to class (2), with g = Psi'. The forward object is the two-site response operator R_Psi(u,r) defined by the strictly convex objective r Psi(v) + Psi(u - v) in equation (1). Theorem 1 establishes that the complete noiseless map (u,r) |-> R_Psi(u,r) determines every positive ratio g(s)/g(w). The argument proceeds from the stationarity identity r g(v*) = g(u - v*) (equation (5)), equivalently the inversion formula r = g(u - v*)/g(v*) (equation (6)). For fixed u > 0 the map r |-> v* is shown to be a strictly decreasing C1 bijection from (0, infinity) onto (0,u) by the implicit-function theorem applied to G(v,r) = r g(v) - g(u - v), whose partials satisfy partial_v G > 0 and partial_r G = g(v). Choosing the boundary u = s + w and the unique ratio that places the equilibrium at the target w therefore recovers g(s)/g(w). Without calibration this determines g only up to a positive multiplicative constant; the unit-curvature condition g'(0) = 1 fixes the constant via the right derivative of the normalized ratio function h at the origin. The proof technique is elementary convex analysis plus the IFT; the domain of validity is the complete exact map on (0, infinity)^2 under the stated regularity. No counter-example exists inside class (2). The result is correctly distinguished from continuum Calderon or discrete resistor-network recovery of spatially varying coefficients (references [7-15,45-47]) and from inverse-optimization recovery of objective parameters from observed decisions ([30-33]).

Corollary 2 converts the complete-map statement into a finite-target protocol. Under an inverse-response (tune-to-target) oracle the attained ratios equal g(s_i)/g(w_0) on a prescribed grid; under ordinary forward observation a root-finding residual epsilon_v yields an explicit log-ratio error bound once the search point lies inside the neighborhood U of Corollary 3. Absolute calibrated scale still requires near-origin slope information. Corollary 3 supplies the local Lipschitz stability estimate: on a compact interval K subset (0, infinity) with m <= g' <= M the IFT branch satisfies c_0 |r - r*| <= |v(r) - w| <= C_0 |r - r*| with the explicit constants c_0 = g_- / (M (1 + sup_U r)) and C_0 = g_+ / (m (1 + inf_U r)); the logarithmic residual bound and the Lipschitz constant of the forward map Theta follow at once. The estimates are local, exclude a neighborhood of the origin, and are not claimed globally. Corollary 4 reduces a single free vertex with all neighbors pinned (one drive edge at boundary u, remaining edges at the reference) to the same two-term stationarity equation with known effective ratio r_eff = (alpha + sum beta_j)/beta. Multi-free equilibria and unknown r_eff are explicitly excluded; the limitation is already stated in the abstract, Table 1 and Section 1 and is not a defect.

Section 4 introduces the reciprocal cost J by an independent ratio-composition law (RCL). Proposition 2 converts RCL plus reciprocity, normalization and the punctured unit-curvature limit into d'Alembert's functional equation for H = F + 1. The classical continuous even solutions with H(0) = 1 are cosh(kt), cos(kt) or the constant 1 (Aczel [36]); the curvature condition forces H''(0) = 1 and selects H(t) = cosh(t), hence F = J. The argument is conditional on the RCL hypotheses and is correctly separated from the inverse theorem, which never assumes RCL. Corollary 5 propagates the scalar uniqueness under a pairwise homogeneous graph ansatz. The one-parameter family J_p (equation (8)) lies inside the calibrated class of Theorem 1; its two-site response admits the closed form (9). Proposition 3 shows that the three-site minimizer maps of J and the quadratic-log comparator Q first differ at cubic order in a generic Taylor sense and coincide only on a measure-zero analytic set; the wording first differ at cubic order should be read as a generic statement, since the cubic coefficient can vanish along special rays.

Section 5 reports synthetic finite-grid discrimination inside the family J_p. The primary design is a 15-point grid (u in {2,3,4} crossed with alpha/beta in {0.25,0.5,1,2,4}), n = 7 Gaussian replicates at sigma = 0.05, and simple-versus-simple SSE selection against fixed rivals J_p. Equal-weight conditions cancel exactly from pairwise differences because v_p^*(u,1) = u/2 for every p. Table 2 and Figure 3 give Monte Carlo selection probabilities under exact parameters and under a nested errors-in-variables screen of width w. The local Fisher information constructed from partial_p v_p^* yields the Cramer-Rao lower bound sd(p-hat) >= 0.017 at p = 1; Monte Carlo under exact parameters attains that scale, while the w = 0.10 screen inflates the empirical standard deviation to approximately 0.05 and causes naive Wald intervals to undercover. A secondary three-site discriminator against Q, S, D and an elastic reference is summarized with observation-noise-only selection probability 0.828. All numerical claims are parametric, synthetic and within-family; no nonparametric reconstruction from finite noisy queries and no real measured data appear. The data-availability statement asserts that reproducibility files (Python entry points, pinned dependencies, expected outputs, regression tests, seed 20260713) accompany the article; those files were not present in the review package, so the exact Monte Carlo figures remain unverified here, although they are internally consistent with the closed-form response (9) and the stated design.

Section 6 correctly lists the constitutive and modeling commitments required for any physical reading (pairwise homogeneous ansatz, independent free-site restoring mechanism, single open exchange current, pre-fixed representation) and distinguishes the dimensionless bookkeeping functional C_M,Psi from a thermodynamic free energy. The equal-weight non-identifiability of Section 3.2 (v* = u/2 for every even strictly convex Psi when r = 1) is elementary and correctly emphasizes that asymmetric weight ratios are essential. Overall the mathematical core (Theorem 1 and Corollaries 2-4) is sound under its explicit hypotheses, the separation between nonparametric reconstruction and RCL selection of J is maintained, and the finite-data section is appropriately limited. Minor clarifications on the generic character of the cubic separation, on the supplemental archive, and on bibliography metadata for [36], [42] and [45] would improve presentation; no theorem requires correction. The contribution is a clean structural identifiability result with an unusually direct inversion mechanism, of clear interest to Inverse Problems within the stated scope.

If the central claim holds, controlled weight ratios and equilibrium locations alone become a direct route to ratios of the constitutive generator, with an explicit inversion formula, local stability, and a concrete single-free-vertex experimental protocol. The result is a clean structural-identifiability contribution for a deliberately narrow response-map setting rather than a broad network inverse problem; multi-free recovery and nonparametric noisy reconstruction remain open. The separation between the nonparametric inverse theorem and the independent selection of J is carefully maintained, and the finite-data work is limited to parametric synthetic discrimination inside J_p. Within those bounds the contribution is solid and of clear interest for Inverse Problems.

1. Theorem 1 gives an elementary complete-map inversion r = g(u-v*)/g(v*) that recovers every positive ratio of g under class (2), with unit-curvature calibration fixing absolute scale.
2. Corollary 3 supplies local Lipschitz stability of recovered log-ratios with explicit constants from the IFT branch, plus a practical finite-precision root-finding bound.
3. Corollary 4 reduces single-free-vertex problems with pinned neighbors and known controllable r_eff exactly to two-site stationarity; multi-free and unknown r_eff are honestly excluded.
4. Careful separation of the nonparametric inverse theorem (class (2) only) from the independent RCL selection of J and the family Jp.
5. Closed-form response (9) for Jp enables exact Fisher information and a fully specified synthetic design; equal-weight non-identifiability is correctly flagged.
6. Section 6 lists constitutive and modeling commitments required for any physical reading and does not present the dimensionless functional as a free energy without calibration.
7. Related-work framing against Calderon/resistor networks, inverse optimization, and structural identifiability is accurate and non-overclaimed.

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The manuscript studies the inverse problem of recovering an even, strictly convex C2 log-coordinate potential Psi (equivalently its derivative g) from controlled equilibrium locations on weighted graphs. The main analysis is the two-site response map R_Psi(u,r) = argmin_v {r Psi(v) + Psi(u-v)}. Theorem 1 shows that the complete map determines all positive ratios g(s)/g(w), and therefore determines g uniquely after the unit-curvature calibration g'(0)=1, via the explicit stationarity inversion r = g(u-v*)/g(v*). Corollaries supply a finite-target tune-to-target protocol, a local Lipschitz stability bound on recovered log-ratios, and a reduction of single-free-vertex problems on general finite graphs under pinned neighbors and a known controllable effective weight ratio. Multi-free equilibria and unknown effective ratios are excluded. The reciprocal cost J is introduced separately by a ratio-composition law as a distinguished candidate, not as an assumption of the inverse theorem; closed-form responses for the calibrated family J_p then support synthetic finite-grid pairwise selection and least-squares estimation of the family index under a stated observation and errors-in-variables model. Scope and physical modeling requirements are stated explicitly.