Where a Dark-Energy-Tied Galaxy Law Holds and Where It Breaks: Low-Acceleration Tests, a Two-Footing Shear Pass, and the Satellite-Dwarf Excess (Working Note)

Working note, version 1.0 (2026-10-19). AI-assisted research programme; not peer reviewed. kappa = 1/2 is fitted; the theory is not closed. One day of pre-registered tests (frozen criteria before computing, mutation controls, both a0 footings never pooled) of a gravity framework with galactic law g_N = g^2/(g + a0), a0 = (1/2) c sqrt(G rho_DE), whose extra gravitating mass is a collisionless 'cold energy' component. (1) Shear: the framework's own 1024^3 particle-mesh structure is CONSISTENT with raw KiDS-1000 cosmic shear on both footings (Delta chi^2 -7.70 and -7.18 vs its matched control), conditional on inherited initial conditions (native rerun running). (2) Isolated late-type galaxies obey the law down to g_bar ~ 7e-14 m/s^2 in KiDS-1000 lensing (amplitude 1.016 +/- 0.086; within ~0.1 dex across robustness variants), excluding the satellite dwarfs' x2.75 at >= 12 sigma in every variant. (3) Spheroidal/red isolated lenses sit 1.45x above the law at the same accelerations after a stellar-mass correction (4.0 sigma split) and follow g_obs = g_law (g_dagger/g_bar)^0.060; Milky Way and M31 satellite dwarfs carry a larger, host-independent excess (~+0.44 dex) that ten framework-native mechanisms (stiffness, expelled-gas free cold energy, catchment budget, present and pericentric tides, host capture, early-collapse cores, maximum-entropy relaxation, gas loss, dynamical-friction transport) fail to produce. (4) Lensing equals dynamics in the framework to better than 2e-5; in the candidate particle model a uniform field produces no internal response, so the law's cold mass must be real mass assembled from afar; a conditional confinement theorem for the two-population cold medium is stated. Not derived: kappa = 1/2, the cold-to-baryon supply ratio, the retained fraction. AI assistance: Anthropic's Claude, with an independent AI-assisted analytic review.