Dirichlet Form and Reconstruction of Macroscopic Geometry

Abstract

Can macroscopic geometry be derived from a finite stochastic mechanism rather than assumed in advance? This article develops a rigorous route from reversible microscopic jump dynamics to an intrinsic metric, measure, differential operator, and curvature. The starting point is deliberately pregeometric: a symmetric jump measure defines a Dirichlet form and a self-adjoint Markov generator, while dimension, topology, and spatial scale are not built into the kinetic model. Geometry appears only after a transport carrier has been selected and the microscopic dynamics admits an appropriate local limit. For an explicitly defined locally periodic class, the paper proves joint convergence of the Dirichlet form and the absolute measure. The conductances and masses of one slowly modulated finite cell therefore determine, in a single limit, both the principal symbol (A) and the weight (w), and hence the transport metric (g=A^{-1}), the limiting measure, and the associated diffusion operator. Ordinary and affinely shifted Mosco limits are obtained from the same microscopic sequence. Three results give the paper its central structure. First, the direct microscopic-to-geometric passage is complemented by a global inverse theorem: for every prescribed (C^{2,\alpha}) metric on a fixed toroidal carrier, a single finite one-vertex bouquet cell—kept fixed throughout the carrier—can realize that metric through a smooth positive conductance field. Arbitrary positive densities can be incorporated independently. Second, marked microscopic data provide a quantitative reconstruction of the second jet and therefore of Riemannian, weighted, and connection curvature. An explicit counterexample shows why this additional information is essential: Mosco convergence, even together with uniform convergence of the principal symbols, does not by itself imply convergence of curvature. Third, the finite-cell description yields an exact differential calculus with respect to edge conductances. It gives computable sensitivity formulas, concavity and optimization results, identifiability criteria, degeneration laws, and a precise distinction between geometric information and microscopic degrees of freedom invisible to the metric. The inverse-selection problem exhibits a genuine barrier bifurcation: the Karush–Kuhn–Tucker discriminant and the Maxwell set describe two different mechanisms by which a canonical microscopic representative may fail to be unique. The article also derives a minimal metric-compatible connection on the transport bundle, clarifies its distinction from the Levi–Civita connection, and establishes a three-dimensional variational obstruction: under the stated full-submersion and fixed-volume hypotheses, every critical point of the spatial total scalar-curvature functional is necessarily flat. The principal value of the work is not another analogy between diffusion and geometry, but a closed “synthesis–limit–verification” framework. A finite cell generates a macroscopic geometry; the same cell admits inverse design and optimization; and marked observations determine when that geometry, including its curvature, can actually be recovered. The general passage beyond the locally periodic class and any extension from spatial geometry to spacetime are isolated as separate research problems rather than used as hidden assumptions. The paper is intended for readers working on Dirichlet forms, stochastic and periodic homogenization, metric-measure geometry, finite-network analysis, inverse problems, and the microscopic foundations of emergent geometry.