Operator-Based Reconstruction of Riemannian Metrics and Lorentzian Conformal Geometry from Distance and Timing Data

We formulate a two-track inverse-geometric framework for reconstructing spatial metric information and spacetime causal geometry from finite, noisy distance and timing data. Track A treats positive-definite Riemannian geometry. In a normal neighborhood, the mixed Hessian of the squared distance determines the metric, while the leading non-Euclidean term of the two-point distance expansion determines the Riemann tensor. We recast the latter as a symmetric curvature operator on the bivector space Λ2TpM and propose a constrained, weighted regression whose design-matrix rank explicitly diagnoses local identifiability. Track B treats Lorentzian geometry. Causal order, null-arrival sets, and the principal symbols of hyperbolic wave operators determine light cones and therefore, under standard causality and regularity assumptions, the conformal class rather than an absolute metric. Proper-time, volume-density, or dynamical information is then required to calibrate the conformal factor. The computational pipeline combines local embeddings, graph and connection Laplacians, frame synchronization, regularization, bootstrap uncertainty, and three independent curvaturechecks: tensor contraction, small-ball volume growth, and heat-kernel asymptotics. We correcta common conceptual error by separating chart-transition cocycle consistency from physical holonomy: the former should close, whereas the latter carries curvature and must not be forced to the identity. A synthetic FLRW consistency example illustrates nonlinear uncertaintypropagation and demonstrates that a nonzero central curvature estimate can remain statistically inconclusive. The paper distinguishes exact identities, theorem-level results underassumptions, proposed algorithms, synthetic demonstrations, and speculative extensions. It does not claim unrestricted recovery from arbitrary sparse data or a derivation of quantum gravity.