A Regularized Variational Framework for Metric-Type Geometry from Discrete Anchors

We formulate and analyze a regularized variational model in which finitely many discrete anchors constrain a scalar field on a bounded Lipschitz domain and, through second-order response, determine a tensorial object of metric type. The analytic point of departure is that singular pointwise anchoring is incompatible with the natural Sobolev setting of the Dirichlet energy. To overcome this difficulty, each anchor is represented by a mollified averaging functional, so that the full anchor mechanism becomes continuous on the admissible class and remains compatible with weak convergence. The resulting action consists of a Dirichlet term, a weighted anchor-fidelity term, and an auxiliary regularization term. Within this framework we derive the first variation, the weak Euler--Lagrange equation, and the second variation in complete form. We then prove existence of minimizers under standard coercivity and weak lower-semicontinuity hypotheses, establish uniqueness under strict convexity, and show that the second-order response is symmetric and positive semidefinite when the regularization is convex. In the quadratic regularization case we obtain a linear elliptic field equation with smooth localized forcing and record the corresponding interior regularity consequences. The geometric conclusion of the paper is stated at its natural level of generality: whenever the second-order response admits a local tensor representation and satisfies an explicit nondegeneracy condition relative to a positive definite reference tensor, it induces a continuous metric-type tensor on the region under consideration. A finite-difference discretization of the quadratic model is also constructed, validated by a manufactured-solution experiment, and used to study the dependence of the reconstructed response tensor on the anchor-fidelity and anchor-width parameters. The paper therefore provides a mathematically controlled Euclidean variational framework in which localized discrete data determine a field and, under explicit hypotheses, a geometric response of metric type.