Why Cold Energy Settles: A Tethered Medium Whose Displacement Is the Gravitational Polarization (Working Note)

Working note, version 1.0 (2026-10-20; drafted 2026-10-14). AI-assisted research programme; not peer reviewed. kappa = 1/2 is fitted; the theory is not closed. Galaxy rotation curves follow one relation between observed and baryonic acceleration, and in this framework its scale is set by dark energy, a0 = (1/2) c sqrt(G rho_DE). Clusters and the cosmic web also need a cold, collisionless mass component, 'cold energy', with Omega_c/Omega_b = 5.364 (measured). This short note explains why that component arranges itself, galaxy by galaxy, into exactly the profile the relation demands. Cold energy is treated as a medium of real rest mass whose displacement P is the gravitational polarization. With K = 4 pi G and u = K|P|, the action is L = -|grad Phi - K P|^2/(2K) - rho_b Phi - W_rem(u) - (gamma/2)|grad P|^2, with K dW_rem/du = u^2/(a0 - u) and gamma = K L*^2/2. Varying Phi gives the baryons' Newtonian flux; varying P balances the baryons' inward pull g_N against the vacuum's push-back u^2/(a0 - u). The unique rest state is g_N = g^2/(g + a0), the simple interpolating function. The stiffness length L* lies between 24 pc (set by Newtonian outer-halo globular clusters) and about 1 kpc. It screens the Solar System: Cassini's quadrupole is at most 5e-8 of the 2026 bound and the only residue is a uniform sunward pull of at most 2.4e-16 m/s^2. With catchment supply and condensed-only sourcing, the framework's own 1024^3 simulations are consistent with raw KiDS-1000 cosmic shear on both footings (Delta chi^2 -7.70 and -7.18 against a same-pipeline control) with inherited initial conditions; rerun with the framework's own initial conditions, the canonical footing still passes (Delta chi^2 -10.08), while the alternative-footing rerun is queued. An earlier own-supply rule is excluded by isolated-galaxy lensing, and an expelled-gas rule for hot-gas systems fails on the full 318 eRASS1 clusters. Not derived: kappa = 1/2, Omega_c/Omega_b, L*, why only condensed baryons tether, and the relativistic completion. Sealed kill lines: a Cavendish test at four times the source acceleration (|dG/G| < 1e-6), a probe beyond 1000 AU (no non-thermal anomaly at or above 1e-13 m/s^2), and Gaia DR4 wide binaries (Newtonian). AI assistance: Anthropic's Claude, with review by parallel AI sessions.