Track A: Branch Universality of ωW and the Derivation of g † in Universal Grid Mechanics

This paper resolves the final E2 bottleneck in Universal Grid Mechanics (UGM) by proving that admissibility and grid ontology jointly force branch-independence of the sharp Level-2 write scalar C_W(b) = H_b χ_b + Δ_b F′(S_b). The central result is a derived state-completeness theorem: using Axiom 0 (local information consistency under repeated updates) as an exclusion principle, together with Axioms 1 and 2 (grid ontology and ontological precedence), it is shown that every admissible persistent coarse distinction must be carried within the minimal local state X = (S, M). This removes the need for any closure commitment and establishes state completeness as a theorem within the frozen five-axiom canon. As a consequence, the response coefficient χ becomes uniquely determined by canonical continuation data, yielding branch-independent C_W and a universal primitive write coefficient ω_W = κ C_W / (Δτ Δupd_max). The MOND-like transition scale g† = (ω_W τ_M⁻¹ / τ_M) S̄₀ is therefore structurally determined and branch-independent, with only two remaining empirical inputs: τ_M and S̄₀. The framework yields a direct falsifiability condition: any additional admissible carrier beyond X = (S, M) would generate irreducible scatter in the radial acceleration relation (RAR). The observed low scatter across 153 galaxies (~2700 measurements, McGaugh, Lelli & Schombert 2016) is consistent with the absence of such carriers at detectable amplitude, providing empirical support for the derived closure. Finally, universality of ω_W eliminates normalization ambiguity and reduces the dimensional sector to a single scalar ℓ, completing the structural bridge to dimensional closure.