Exact Tangent-Mean Coordinates and Equi-Affine Orbit Rigidity: A Defect–Wronskian Principle

Let nu be a compactly supported Borel probability measure on R, let theta_1, ..., theta_n be distinct real nodes, and let w be the interpolatory quadrature vector. We prove a defect-Wronskian-orbit principle for identities integral alpha'(s+ht) dnu(t) = sum_j lambda_j(h) alpha'(s+h theta_j) on normalized nondegenerate curves. There is exactly one non-rigid configuration: if nu = sum_j w_j deltatheta_j, the identity is tautological for every curve. Otherwise let p >= n be the first missed moment. For every Cᵖ⁺¹ curve in equi-affine arclength, constant curvatures, the special-affine one-parameter-orbit property, constancy of the high frame jet E⁻¹(alpha')⁽ᵖ⁾, differentiation invariance of the tangent-coordinate space, a base-point-independent exact quadrature on one set of nonzero scales accumulating at zero, and a locally uniform o(|h|^p) residual are all equivalent. No a priori boundedness or regularity of the supplied coefficients is assumed. Exactness forces lambda(h)-w = O(|h|ᵖ⁻ⁿ⁺¹) and makes the high frame jet E⁻¹(alpha')⁽ᵖ⁾ constant. The analytic input is a classification of finite-dimensional exponential-polynomial spaces with pure-exponential Wronskian Wr(X) = C etau s != 0: X = direct sum_a egamma_a s C[s]<r_a, with sum_a r_a gamma_a = tau. In particular X is differentiation invariant; the normalized case has tau = 0. This collapses any constant high-jet relation to the order-n constant Frenet equation. For positive measures, either the rule is tautological or n <= p <= 2n; equality p = 2n is exactly the orthogonal-nodal endpoint, hence the Gaussian configuration when the moment form is positive definite below degree n, and the residual threshold is sharp. The theorem extends from point evaluations to every unisolvent family of compactly supported distributions; Hermite derivative sampling is a direct corollary. In the plane, conic branches acquire exact Lagrange mean-value coordinates. In dimension three, a three-tangent reconstruction recovers both spatial equi-affine curvatures from its first coefficient corrections.