The a0 Galaxy Law and its Screened Khronon Action: Field Equations, Lean Certificates, and the Closure Record (Working Note)

Working note, version 1.0 (2026-11-07). AI-assisted research programme; not peer reviewed; not reproduced by any outside group. kappa = 1/2 is fitted; the theory is not closed; nothing here says the data favour this framework over any other. Gravity carries one extra scale, a0 = kappa c sqrt(G rho_DE), kappa = 1/2 fitted (proved not derivable from the framework's premises), realised by the v3s screened khronon action, with AC2q as the candidate successor. This note states the action in Legendre form, derives the covariant field equations of both actions and certifies the algebra (sympy Euler-Lagrange residuals, Lean 4 theorems with mutation twins, finite-difference solvers on known solutions), and gives every gate the programme set itself its frozen verdict in one table. PASS: Cassini, PPN, tensor speed, galaxy lensing = dynamics, SPARC on the action's exact field, PN sign, well-posedness of the Pi^6-blended switch in 1D, 2D and 3D. FAIL: AGC 114905, Malin 1, the canonical FG1 tilt, light-cone causality on anisotropic backgrounds (physical, unobservable). After the joint systematic budget explains the X-ray ellipticals, the KiDS colour split and the bright M31 dwarfs, the residue is X-COP clusters, groups and Coma UDGs, whose needed extra mass varies about a hundredfold between classes, and separately the CMB-lensing shape, which needs cold extra gravity at z 3400. Four walls close every sector class tried: cold fluid, fixed-temperature sea, hot collisionless fermion, shock-sourced entropy sector; primordial black holes give no shared window; every switch key between discs and hot systems is dead. Three frozen kill tests remain: Gaia DR4 wide binaries (the action predicts Newton), a0(z) flat to z 2.33, and lensing = dynamics. The supplement archive carries every Lean source and build output, the closure tables, the frozen criteria and the prediction file at the cited commits. Companion to PAPER67 (10.5281/zenodo.23968171), PAPER68 (10.5281/zenodo.23990944) and PAPER69 (10.5281/zenodo.24025170).