UDM acceleration

The radial–acceleration relation of disc galaxies is governed by a single scale a₀ ≃ 1.1×10⁻¹⁰ m s⁻², whose ratio to cH₀— w ≡ a₀/(cH₀) = 0.156–0.165 — has no accepted explanation. We propose a spectral–geometric hypothesis for this dimensionless ratio. Treating gravity in the dense-medium picture as the gradient response of a compression field, and allowing the effective spatial dimension to be emergent, the lowest eigenmode of a hypothesised Dirichlet cavity at the Hubble radius fixes a geometric coefficient w_geom = 1/j²_ν,1 with ν = d_H/2 − 1; identifying the third-dimension activation with the lowest-mode susceptibility, ε = 1/π², gives d_H = 2 + 1/π² = 2.1013 and

with a dimensionally self-consistent variant w_geom = 0.1569; both values lie in the observed band, and neither uses the measured ratio as input. The eigenvalue problem, however, fixes only the wavenumber of the mode, not its amplitude: the physical claim therefore has the form a₀ = C_A·w_geom·cH₀, where C_A is a dimensionless amplitude normalisation that the construction does not derive and sets to unity by hypothesis. The geometric factor is parameter-free; the physical identification is conditional on C_A = 1, and this distinction — geometry derived, dynamics hypothesised — is maintained throughout. The hypothesis carries a falsifiable consequence, a₀(z) = C_A·w_geom·cH(z); applied to all 175 SPARC galaxies with fixed mass-to-light ratios and no per-galaxy tuning it reproduces rotation curves with 0.098 dex scatter in velocity space, a figure stable at 0.098–0.102 dex as the global disk mass-to-light ratio varies across 0.3–0.7. An appendix develops a candidate microscopic action from which the linear activation law follows at leading order, with all residual corrections quantified. The paper is offered as a testable hypothesis for the ratio a₀/(cH₀), not a derivation of a₀.