Geometry of Events in Deformed Cellular Spacetimes

We develop the geometry of events in a deformable cellular spacetime, extending our earlier cellular-spaces framework from cellular complexes to cellular events complexes. The framework operates within the conformal class of Minkowski space; in four dimensions, this is the vanishing-Weyl-tensor sector, which excludes Schwarzschild, Kerr, and gravitational-wave spacetimes. The framework treats integer counts of cell crossings as the primitive geometric data: spatial separation between events is the shortest count of face-adjacent cells; temporal separation is the cell-crossing count of a reference light pulse. Newton’s universal clock is replaced by an operational one: the temporal count distance is the ratio of cell length to the speed of light through a cell, and because both quantities are invariants of the co-deformation, the temporal count is itself an invariant: temporal separation is operationally measured via light-pulse counts rather than posited as an external coordinate. Under the co-deformation principle, a single positive scalar field ρ (cell density) controls both the rod length and the clock period. We prove six results, all expressed in terms of counts on the cellular events complex, with a smooth conformally flat metric g˜=e2φη (φ=−13lnρ) appearing only as the comparison/calibration object for convergence statements. First, the scalar curvature of the smooth comparison metric is the closed-form differential operator R˜=2□ρ/ρ1/3−(8/3)(∂ρ)2/ρ4/3. Second, the volume of a small Alexandrov interval admits an explicit asymptotic expansion in the interval height T, with leading correction Q(m,u)T2 involving an anisotropic invariant at the midpoint m. Third, Q is irreducible to scalar and Ricci-directional invariants alone; the explicit decomposition Q=145R˜+15R˜uu+12J exhibits a third independent invariant J(m,u)=(u·∂)2φ(m) as new structural content of the Lorentzian diagnostic. Fourth, the discrete-to-continuum convergence of counts on the cellular events complex yields a counts-only curvature estimator with rate O(a) at the joint scaling T≍a. Fifth, the smooth comparison metric itself is reconstructible from counts on the discrete complex at rate O(a): the conformally flat Lorentzian geometry is uniquely determined, up to background Minkowski calibration, by the cellular events complex. Sixth, a finite collection of Alexandrov-interval volume measurements at a fixed midpoint suffices to recover the full local curvature data {R˜(m),R˜μν(m),J(m,u)} at rate O(a) (curvature spectroscopy); and the temporal light-tick count λ is essential in a precise sense—there exist conformally flat Lorentzian geometries indistinguishable on every spatial slice by the earlier spatial-only diagnostic but distinguished at the origin by the events-space directional invariant. The framework’s scope is the conformal class of Minkowski: flat FLRW in conformal time, leading-order weak-field gravity, and 2D gravity. This paper is a mathematical contribution to discrete-to-continuum geometry on cellular events complexes; it is not a physical theory of gravity.