The constitutive framework for weak-field gravity models the observedgravitational field as the response of a vacuum medium, with the weak-fieldequation ∇· (κg g) = −4πG ρₑff. 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 ℓₘax is N₀ = (ℓₘax + 1) ², and a heuristic acceleration-cutoff argumentpredicts a dimensionless transition scale gᵣesp/a₀ = π/ℓₘax, independent of anycoefficient. The value ℓₘax = 15 gives N₀ = 256 and predictsgᵣesp/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 valuesΥdisk = 0. 5 and Υbulge = 0. 7, the median preferred response acceleration isgᵣesp/a₀ ≈ 0. 217, corresponding to ℓ ≈ 14. 5 and N ≈ 240. For the representativesmoothing width σₗnx = 0. 18 the best-fit value is gᵣesp/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.
Ralph C DeMartino (Thu,) studied this question.