Universal Grid Mechanics (UGM):An Axiomatic, Admissibility-First Framework for Physical Reality

Universal Grid Mechanics (UGM) is an admissibility-first foundational physics framework in which physical existence is restricted to states satisfying continuity, bounded change, and local information consistency under repeated updates. Reality is modelled as a continuous persistent substrate (the grid) that admits deformation, retains history, and resists change in a bounded manner; all evolution preserves a finite admissible domain D. UGM is explicitly pre-phenomenological: no particles, fields, or spacetime geometry are assumed at the axiomatic level. Starting from five frozen axioms and the minimal local state X = (S, M), where S is a structural deformation state and M is structural memory, this paper records the current closure status of the framework. We state the primitive update loop, the coherence–selection mechanism, and the admissibility metric on deformation space; derive the admissibility-minimizing primitive operator L₆ within the class of equal-radius planar supports via coordination-normalized spectral anisotropy, whose Taylor expansion recovers the isotropic Laplacian; record the UGM gravity Hamiltonian, from which the Poisson equation and the inverse-square law follow as formal limiting theorems; and record the dimensionless Newton constant G = √3/4π² as a geometric consequence of hexagonal coordination without gravitational data. The gravity sector retains a two-regime structure: Newtonian at low memory and MOND-like screened-Poisson behaviour at high memory, with the exact interpolating function still open. Version V02. 13 consolidates three structural advances from companion papers alongside all retained results from prior versions. First (Route B spectral chain — closed): the non-degenerate second-order Brillouin-zone invariant K²BZ = 1/2 is an exact theorem (analytic proof via cos²θ decomposition and BZ inversion symmetry; numerical confirmation 0. 500000 ± 0. 000002, Track B v1. 08). The exact hexagonal geometry constant κₕex = 1/ (3√3) gives Aₘax = π/ (6√3) fully closed with no free parameters. The dimensional bridge B (ℓ, Aₘax) = 0 has a unique solution ℓ* by the one-crossing theorem. Second (E2/ωW — closed, imported): Track A v1. 05 establishes via the 2V branch-universality criterion that admissibility forces the write-law combination H*χ* + Δ*F′ (S*) to be branch-independent, making ωW a universal primitive scalar. The E2 bottleneck is thereby closed. Third (M₀ — identified, imported): the M₀ paper v1. 02 establishes that the dimensional bridge cannot introduce a second ontically independent scalar mass scale; M₀ is identified with the derived internal amplitude κ·Δᵤpdᵐax once ωW is fixed. The contraction theorem (§7. 1) is repaired with a rigorous Hilbert-space coercivity argument. The screening convention in the memory-mediated gravity sector is unified. The observational bridge factor λbridge = Λφ; S is bounded by 1/ (3√3) ≤ λbridge ≤ 1 from admissibility and hexagonal shell geometry. A dedicated M₀ paragraph in Appendix A identifies the determination programme. A Solar-System deviation scale paragraph derives ℓ ≪ 6. 9 × 10⁹ m from the Cassini radio-link bound. The Lorentz package is marked PARTIAL. The Branch III cross-system ordering expectation Δaₑff/aN ≈ −αM exp (−tdyn/τM) is stated as a programme item. An observation scaffold remark clarifies that φ (x) ≠ y (r) = gₒbs/gbar until the projection Πₒbs is formally derived. Two framework-level open items remain: (i) tensor closure beyond the scalar sector; (ii) constrained inversion of the admissible response function φ (x) from Gaia/Rubin kinematic data. This record (DOI: 10. 5281/zenodo. 19151677) consolidates the UGM axiomatic paper version chain from V02. 10a onward. The original UGM v1. 0 record is 10. 5281/zenodo. 18529620.