Mechanism Derivation of Deep-MOND: Rotation Curves, Tully-Fisher Relation and the Cosmic Anchor a0

Abstract

Flat galaxy rotation curves and the Tully-Fisher relation (v^4 proportional to M, exponent 3.94 +/- 0.07) are the two most robust galactic-dynamics observations. This paper derives, from the energy-field framework: (1) the precise form of the MOND characteristic acceleration a0 = GM_m/R_H^2 = cH0/6 = 1.091e-10 m/s^2 (surface gravity of the total cosmic matter mass; observed 1.2e-10, 9% deviation, same source as G's 7.5%); (2) the mechanism of the deep-MOND law: multiplicative coupling (mode locking) of the local epicyclic frequency kappa = sqrt(a_N/r) and the cosmic background oscillation omega0 = sqrt(a0/r) gives kappa_eff^2 = kappa*omega0, hence a_eff = sqrt(a0*a_N), from which the flat rotation curve v = (a0*G*M)^(1/4) and the Tully-Fisher relation v^4 = GMa0 (exponent exactly 4.0) follow simultaneously - mainstream treats deep-MOND as an assumption, this paper gives a mechanism. Additional: interpolating function a_eff = sqrt(a_N*(a0+a_N)) with transition-region prediction mu(1) = 1.41 vs 0.71 of standard MOND; structural equivalence deep-MOND = isothermal sphere + cosmic-anchor cutoff r_t = sqrt(GM/a0); dual-layer accommodation of rotation curves (collective elasticity) and the Bullet Cluster (stopband distribution). Four substantive distinctions from Machian MOND (Uruena Palomo 2024): static G, exact a0 factor 1/6, mechanism, interpolation.