Crispy Fried Chicken Matching Theorem

A galaxy well today is the cosmic background at (1+z)^3 = delta: a matching theorem for dark sectors, and what it forces. Every relativistic completion of Milgromian dynamics that fits the cosmic microwave background carries a dark component, and in the modern completions that component is a field: the pressureless excitation of a scalar clock at the minimum of its potential. We prove a theorem about that entire class. Because the excitation is a conserved charge that dilutes as a^-3, and because a static well fixes the same excitation through the lapse, u = -Q_0 Psi, a galaxy well with dust overdensity delta is identically the cosmic background at redshift (1+z_match)^3 = delta, with the same sound speed, for any potential K. The Lyman-alpha forest at z ~ 3 therefore measures the dust inside galaxies today, and finds it cold. In the static limit the same field is a gamma = 2 polytrope, p_d = (2 pi G/mu^2) rho_d^2, c_s^2 = |Psi| c^2, whose hydrostatics is the Helmholtz equation of the Aether-Scalar-Tensor theory; on the background it pins that theory's free DBI amplitude, Lambda_D/Q_0 = nu_0 Omega_Lambda/Omega_dm, 18-300 times above its own power-spectrum ceiling. Two further results close the alternatives: an elliptic auxiliary that enters the lapse equation frees exactly one dust-like scalar, A = c_alpha^2 a^3/[4 H^2 (k^2/a^2 + m^2)], so a constraint-only two-degree-of-freedom metric MOND that light and matter both see does not exist at quadratic order; and any medium that holds itself up in a well has signal speed ~ v_c, so it cannot track a cluster merger, which the Bullet cluster requires. What survives is a dark component that is cold, ballistic and galaxy-clustering, and a MOND well accretes it: put into the theory's own lensing, the accreted charge multiplies the 100 kpc - 1 Mpc galaxy-galaxy lensing signal by 3-20 at realistic environments, which the KiDS-1000 isolated-lens relation excludes at Delta chi^2 >= 100. The positive result on the galactic side is that the per-galaxy acceleration scale across 147 SPARC rotation curves has zero intrinsic spread and no mass trend, where a plain halo population predicts both, and the acceleration scale a_0 = (c/2) sqrt(G rho_DE(z)) cannot rise with redshift, which the emergent scale of cold-dark-matter halos does. Every claim is a committed script with checks that can fail, a mutation control, and both footings of a_0.