Constrained Inversion of the Admissible Response Function ϕ(x) from the Radial Acceleration Relation

Universal Grid Mechanics (UGM) is an admissibility-first framework in which physical existence is restricted to states satisfying continuity, bounded change, and local information consistency under repeated updates. The frozen UGM_02 canon defines the minimal local state X=(S,M), with S a finite admissibility/deformation state and M bounded structural memory evolving inside a forward-invariant admissible domain.

At the Route B level, the spectral chain is treated as fixed in the scoped sense required by the revision packet: the second-order Brillouin-zone invariant K²_BZ = 1/2, the exact hexagonal normalization κ_hex = 1/(3√3), and the one-crossing dimensional bridge fix the spectral/dimensional structure at that interface, while broader framework-level items remain logically distinct. The present paper formulates and executes a pilot present-epoch inversion protocol for the remaining Route B response sector, namely the admissible response law ϕ(x), with the broader open items E2/ω_W, M₀, and tensor closure kept explicit.

The admissible response law is bounded by 0 ≤ ϕ(x) ≤ 1, with continuity, one-crossing compatibility, and finite closure. The stronger exclusion of a purely radial total response under persistent anisotropic morphology is imported from Track B Proposition 13.1. The shell-sum forward form y(r) = Σ βₙ ϕ(rₙ(r)/ℓ) is structurally anchored by the Track B boundary factorization. The remaining open interface is the derivation that y(r) = g_obs/g_bar is the radial projection of that boundary response through the observation scaffold Π_obs.

Pilot data products and outcome: This version advances from formulation to pilot execution. The pilot uses public RAR point and bin tables (2,693 point rows; 14 binned rows from McGaugh–Lelli–Schombert) and a statistically representative reduced Gaia DR3 present-epoch working table (71,343 rows; 27 columns) derived from a fixed 1/250 extraction protocol with quality filtering. A sector summary table (3,600 rows: 10 radial bins × 360 angular centers) is used for the angular requirement test.

On this dataset, the pilot recovers a nontrivial monotone pilot radial observational response proxy ϕ^obs_{r,pilot}(x) with x = log₁₀(g_bar), rising from ≈1.02 at x ≈ −8.9 to ≈8.71 at x ≈ −11.7 over 14 RAR bins. The reduced-Gaia present-epoch D_r values in the inner/mid disk (4.5–8.5 kpc) range from 0.012 to 0.028, consistent with the quasi-steady kinematic assumption. A nonzero present-state angular correction δϕ(x,θ) is not decisively required by the reduced-Gaia pilot: independent-sector angular constancy is not rejected at 5% (χ² = 22.76, dof = 14, p = 0.064); first- and second-harmonic terms each give Δχ² ≈ 5.8 (p ≈ 0.055), which is suggestive but not decisive.

Because the observation-scaffold bridge from B_S(x) = F(S)ϕ(x) to y(r) = g_obs/g_bar remains open, the recovered pilot curve is interpreted as an empirical response proxy rather than a final bridged recovery of the canonical bounded response law. All uniqueness claims are conditional on the Hessian positive-definiteness hypothesis of Proposition 9.3.