Two Kinds of Cold Energy: A Capacity-Limited Tethered Component and a Free Core Component Under a Vacuum-Tied MOND Law (Working Note)

Working note, version 1.0 (2026-10-13). AI-assisted research programme; not peer reviewed. We work with the acceleration law a0 = kappa c sqrt(G rho_DE) (kappa = 1/2, fitted), equivalently a0 = 0.173 c H_inf, where H_inf is the asymptotic Hubble rate of a cosmological-constant universe. The law is written as the static limit of a single action for a polarization field P with a saturating vacuum term W'(u) = u/(1 - u/a0) (no new constant), giving g_N = g^2/(g + a0); it fits raw SPARC rotation curves (0.1009/0.0994 dex) and, with free-fall basins and a stiffness rule, passes wide-binary, cluster-satellite and star-node tests. Reading the data first, we find cold energy plausibly comes in two kinds. (i) The field's bound charge has zero net mass and no inertia, so it cannot be the collisionless mass clusters, the Bullet Cluster, Local Group timing and the cosmic web require. (ii) Only a carrier with real rest mass meets the evidence; the viable carrier is a cold collisionless medium whose displacement is P. (iii) That coupling yields a conservative tether with the law as its unique static minimum and a capacity Sigma <= a0/(pi G) (214/258 Msun/pc^2). (iv) In raw SPARC data galaxies sit below the capacity (median ~0.3 of it), while the Bullet Cluster's cores hold 3.6-4.8 times it. Cold energy is therefore plausibly tethered (law-shaped, capped) in galaxies, with a free collisionless component pooling in massive cores, a candidate for the excess core mass measured in early-type galaxies, groups and 318 eRASS1 clusters. If the medium is a condensate, the data still allow its self-interaction (sigma/m <= 0.71 cm^2/g vs a required 0.11-0.82). We list six framework-specific predictions, including a decisive one (the free central density in relaxed clusters), and the limitations: the carrier is particle-like; kappa is fitted and Omega_c/Omega_b measured; the capacity rests on stated conditions; hot vs condensed baryons is unresolved; massive early types still fail (Z ~3.7); the programme's full raw-KiDS shear test is pending; Bullet lensing masses are provisional. Nothing here closes the theory. AI assistance: Anthropic's Claude.