The Constitutive Gravitational Framework VIII: A Finite Vacuum-Mode Origin for the Shape of the Weak-Field Response, Tested Against SPARC Rotation Curves

The constitutive framework for weak-field gravity models the observedgravitational field as the response of a vacuum medium, with the weak-fieldequation ∇·(κ_g g) = −4πG ρ_eff. In Paper VII of this series it was establishedthat the infrared normalization coefficient F₀ of the constitutive potentialcannot be supplied by the spherical-Casimir mechanism: the obstruction isdimensional, and any attempt to fix F₀ from a finite vacuum mode count fails byroughly ninety-five orders of magnitude. The present paper takes that no-go asits starting point and asks a different, dimensionally admissible question:whether the finite-mode structure of the vacuum response governs the SHAPE ofthe response — specifically the acceleration scale at which the weak-to-strong(deep-MOND-to-Newtonian) transition occurs — rather than its absolutenormalization. In a spherical harmonic basis the number of angular modes up todegree ℓ_max is N₀ = (ℓ_max + 1)², and a heuristic acceleration-cutoff argumentpredicts a dimensionless transition scale g_resp/a₀ = π/ℓ_max, independent of anycoefficient. The value ℓ_max = 15 gives N₀ = 256 and predictsg_resp/a₀ ≈ 0.209. Using SPARC-style rotation-curve data for fourteen diskgalaxies, a narrow-window diagnostic and a clean κ-only response-accelerationscan are performed. With SPARC-standard stellar mass-to-light valuesΥ_disk = 0.5 and Υ_bulge = 0.7, the median preferred response acceleration isg_resp/a₀ ≈ 0.217, corresponding to ℓ ≈ 14.5 and N ≈ 240. For the representativesmoothing width σ_lnx = 0.18 the best-fit value is g_resp/a₀ ≈ 0.204, withinabout 2.5% of π/15. These results are interpreted as preliminary observationalconsistency with a finite spherical vacuum-mode SHAPE for the response, fullycompatible with the empirical status of F₀ established in Paper VII, and not as aderivation of the normalization.