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 K²_BZ = 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 κ_hex = 1/(3√3) gives A_max = π/(6√3) fully closed with no free parameters. The dimensional bridge B(ℓ, A_max) = 0 has a unique solution ℓ* by the one-crossing theorem. Second (E2/ω_W — 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 ω_W 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·Δ_upd^max once ω_W 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 λ_bridge = Λ[φ; S] is bounded by 1/(3√3) ≤ λ_bridge ≤ 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 Δa_eff/a_N ≈ −α_M exp(−t_dyn/τ_M) is stated as a programme item. An observation scaffold remark clarifies that φ(x) ≠ y(r) = g_obs/g_bar until the projection Π_obs 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.