Deep-MOND Response as Longitudinal Goldstone Susceptibility of a Critically Slow Vacuum Condensate: Derivation Chain, Two Numerical Verifications, Three Closed Channels, and One Parameter-Free Result

This working preprint develops a Landau–Ginzburg model of the quantum vacuum as a condensed medium evolving on an FRW background, and systematically tests whether such a medium can generate Milgrom's acceleration scale a₀ ≈ cH₀/2π. The program is organized as a labeled derivation chain in which every claim is explicitly tagged as derived, borrowed, or heuristic, and negative results are treated as primary outputs on equal footing with positive ones. Main results: Overdamped FRW dynamics. The naive dilution term −3HM is shown to be structurally incorrect for an intensive order parameter; the derived overdamped limit replaces it with an effective friction 1/(3H + Γ), resolving two pathologies of the relaxation dynamics and producing the first internally generated equality between the Hubble rate and a scale of the medium. Persistent non-equilibrium epoch. Importing the Starobinsky–Yokoyama stochastic formalism yields the equilibrium distribution P_eq ∝ exp(−8π²V/3H⁴) at the Gibbons–Hawking temperature T = H/2π, a Ginzburg-type critical Hubble rate H_c ∝ |a_M|^(1/2) b^(−1/4), and a derived epoch in which order is energetically preferred but dynamically unattainable. Its width scales as b^(−1/4), verified numerically (fitted exponent −0.249; the ordering scale H_order = 0.42|a_M|^(1/2) is b-independent to 10⁻³ relative accuracy). Three closed channels. A medium–gravity coupling reproduces the Bekenstein–Milgrom AQUAL structure automatically and always strengthens gravity (sign theorem), but three response channels are decisively closed: the scalar-amplitude channel (exponents n = 2 and 2/3, never the required n = 1); the entropic Goldstone-alignment channel, whose amplitude and transition scale are split by (M_Pl/H₀)² ~ 10¹²¹ — the cosmological-constant hierarchy re-emerging inside MOND phenomenology — and whose saturated limit g = g_N + a₀ fails the observed deep-MOND slope (0 vs. 1/2); and the gravity-generated Halperin–Lubensky–Ma cubic, which falls short of a₀ by 61 orders of magnitude. Parameter-free result. An inversion theorem shows that deep MOND is equivalent to a medium internal energy f(P) = 2πG P² + κ|P|³, giving the closed formula a₀ = (4πG)²/3κ. The same one-loop integral, read as a response to the external field, yields P_ind ∝ √g with no adjustable parameter: the deep-MOND exponent 1/2 is identified with the known longitudinal Goldstone susceptibility divergence χ_L ∝ h^(−1/2) of ordered O(N) media. This identification is verified by Monte Carlo simulation of the 3d classical Heisenberg model (the √h term is required at χ²/dof = 0.83 versus 16.8 without it). Magnitude accounting. The resulting formula a₀ = (GH₀/2π)²(P₀/K)³ captures the amplitude at the cost of concentrating all remaining fine-tuning into a single dimensionless quartic coupling b ≲ (H₀/M_Pl)^(8/3) ~ 10⁻¹⁶⁰: the model focuses the hierarchy rather than solving it. This trade-off is stated explicitly. Pre-registered falsifier: if the condensate tracks its critical line, the model predicts a₀(z) ∝ 1/H(z) — weaker MOND phenomenology at high redshift, distinguishable from both the naive a₀ ∝ H scaling and a frozen a₀, and testable with z ~ 1–2 rotation-curve and weak-lensing measurements of the radial acceleration relation. Open problems (attractor status of the critical line, technical naturalness of b, microscopic origin of the dipole coupling, Solar-System screening, relativistic completion, confrontation with raw SPARC data) are enumerated in the paper. All simulation and analysis code (Langevin/Fokker–Planck ensemble, RAR confrontation, Heisenberg Monte Carlo, model-comparison fits; Python 3 with numpy/scipy/matplotlib) is included in this record and reproduces all figures and quoted numbers.