A Covariant Screened Theory for a Dark-Energy-Tied Galaxy Law: the Khronon Form Fails Linear Growth, an AeST Form Survives at Linear Order, and Both Fail Cluster Cores (Working Note)

Working note, version 1.0 (2026-10-22). AI-assisted research programme; not peer reviewed. kappa = 1/2 is fitted; the theory is not closed. The owner's programme studies a gravity framework whose galactic law is g_N = g^2/(g + a0), a0 = kappa c sqrt(G rho_DE), kappa = 1/2 fitted; its extra gravitating mass is a collisionless 'cold energy' medium, not a particle. The companion note PAPER58 found that only a screened version, with a stiffness length L*, passes both Cassini and the galactic law. This note writes that screened theory covariantly in two forms and tests both; criteria frozen before computing, mutation controls run, both a0 footings reported and never pooled. (1) Clusters fail in both forms: rebuilt from raw eRASS1 and KiDS data, cluster cores sit 1.68-1.85 (canonical) / 1.54-1.70 (alternative) times above the law on their baryons (Z about +5 to +6); any cold profile that fixes the cores overshoots the outskirts, and AeST's static dust has the wrong shape. (2) v3, the Blanchet-Skordis Khronon in its literal form plus an independent spatial covector and one stiffness term, has an exact AQUAL mu = x/(1+x) static limit (free function postulated), is healthy at linear order, well posed in the weak field, and passes Cassini (24/24 cells, by 4.8e7 to 5.1e9; frozen rule UNDECIDED on a float-cancellation check), the planets and the outer globulars (L* >= 21.68 / 22.85 pc), but fails linear growth: J'(0) = -1 and the dust mass has no allowed window (gap 24-172x). (3) v4, AeST with the same stiffness term and a capped free function, keeps the exact law, Cassini and BBN and removes the growth failure at linear order; its flagged health defect was a mis-specified criterion and the mode is ghost-free at linear order only; open are a contact-term binding up to 2.3e4 times Newton, the causal cone of the stiffness term, nonlinear behaviour and strong fields. v4 is the better-surviving candidate, not established. (4) Outer galaxy lensing is undecidable; dwarf satellites remain in tension (+0.89 dex); binary pulsars and GW170817 do not constrain the theory; ALMA is expected unscorable. kappa, L* and the dust abundance are fitted, not derived. (5) Pending: the native CMB (validated code, no spectrum run; tests only the dust sector and the cosmological G ratio) and the native cosmic-shear run (running, no verdict). Gaia DR4 is the sealed decisive test of screening: gamma_v - 1 = 8.1e-6 screened against about 1.17 unscreened. Companion notes: PAPER58 v1.1 (doi:10.5281/zenodo.23579379), PAPER57 (doi:10.5281/zenodo.23532885), PAPER49 v1.2 (doi:10.5281/zenodo.23533323). AI assistance: Anthropic's Claude (analysis lanes, review, drafting).