The Coefficient of the a0-Lambda Relation: a Zero-Mode Theorem for Local MOND Actions, Two Failed Repairs, a Four-Form Reframing, and an H0 Degeneracy

The empirical relation a0 = κ c √(G ρΛ) with κ ≈ 1/2 (measured 0.465 ± 0.076 from the baryonic Tully-Fisher intercept and 0.551 ± 0.043 by a distance-free estimator) has never been derived. We ask whether the covariant candidate action of the de Sitter-MOND closure programme (a khronometric clock, a dynamical MOND scalar with a bounded-boost kernel, and a coherence operator) can relate a0 to Λ at all, and answer in the negative with a theorem and three computed dead ends. (i) The MOND primitive enters the static field equations only through its derivative and the FLRW background only through Λ + (2−KB)J(0)/2 + K(Q0)/2: with Λ explicit, no equation of the action relates the two scales (a normalisation zero mode). (ii) Deleting Λ and fixing the primitive's zero by an empty Newtonian vacuum yields a negative vacuum energy of 0.4-0.7% of ρΛ; the constant 4π4/15 of the RAR primitive that would give κ ≈ 1 belongs to the unsaturated branch the bounded-boost theorem forbids. (iii) A sequestering-type global constraint tying Λ to the spacetime average of the scalar's Lagrangian misses by five orders of magnitude today and tends to zero in the de Sitter future. (iv) Promoting a0 to a conserved four-form flux reverses the sign, makes a0 ∝ √(GρΛ) structural with the flux amplitude cancelling, and leaves the coefficient as the coupling ratio Z/β2 = 7.96; the induced environmental a0 is invisible in galaxies, switches the scalar off above 155 a0, and shifts the Gaia DR4 wide-binary boost at 2 kAU by −0.019, about one sigma. On the observable side the only coefficient-free alternative left standing, the horizon form a0 = c2/(2π LdS) with κ = 0.461, is separated from 1/2 by 8.5%, below the 9.5% mass-budget floor of the Tully-Fisher route, and is exactly degenerate with the H0 tension: κ = 1/2 on the Planck H0 and κ = 0.461 on the SH0ES H0 predict the same a0 to 0.2%. We conclude that κ = 1/2 is an empirical boundary condition that this class of local actions cannot derive, and that its determination is a precision problem gated by the stellar mass-to-light zero point, the absolute gas scale and H0. Every number is produced by a committed script with checks that can fail (kappa_closure/k01-k04 in the public repository). AI-assisted research draft; not peer reviewed.