Two Barriers to the MOND Acceleration Coefficient: the de Sitter-Unruh Balance Is Orbital-Invariant at a0 = 2 c H_Lambda, and Rotation-Curve Determinations of a0 Are Definition-Limited at 30%

Abstract

The MOND acceleration scale a0 is numerically close to c H_Lambda, and several authors have proposed that it is set by the cosmological constant. The relation's form is established (Milgrom 1994, 1999, 2020); its dimensionless O(1) coefficient is not derived by any published argument. This note reports two independent barriers to fixing it and quantifies both. First, a theoretical barrier. We compute the Unruh-DeWitt response for a circular worldline in de Sitter space and extract the MOND crossover coefficient q defined by a0 = q c H_Lambda. Milgrom's (1999) balance uses a hyperbolic worldline and gives q = 2; a circular orbit is never hyperbolic, so that derivation is scope-limited to linear acceleration while galaxies are orbits. We find q = 2 exactly on orbits, for two independent reasons both forced by the identity A^2 h^2 - R^2 w^2 = 1: the short-time correlator depends only on a5^2 = a^2 + H^2, which holds for any worldline on the hyperboloid; and the full-response orbital correction is an a-independent rescaling (verified to 1e-16 over five decades in a/H) which cancels identically in the crossover ratio. The mechanism returns Milgrom's coefficient and cannot be made to yield a smaller one. Second, an observational barrier that dissolves into a theoretical one. Profiling SPARC with the mass-to-light ratio free per galaxy, the preferred a0 spans 30.6% across five admissible interpolation shapes - nearly four times the 7.87% separating the two published coefficient proposals. We identify the mechanism: the likelihood anchors on a single deep acceleration (boost nu = 3.97, y = 0.06) where the five kernels agree to 1.14%; the knee is not the anchor and correcting there over-corrects by 3.4x. The spread is already diluted 5.5x by the sample's 1.57 decades of coverage in y, so more data at one acceleration cannot reduce it. But the kernels are not degenerate - after optimal a0 rescaling they differ by 0.050 dex, 46% of the observed scatter - so the barrier is finite: it is removed by measuring the interpolation shape, needing of order 5x the effective sample. Fixing the kernel eliminates the systematic and yields a definite verdict, tabulated for five shapes: four of five favour a0 = (1/2) c sqrt(G rho_Lambda) at up to 2.69 sigma, the exponential kernel disfavours it at 0.66 sigma, and none reaches 3 sigma. The two dominant obstacles - the kernel choice (30.6%) and which cosmological density the horizon term tracks (20.9%) - both exceed the 7.87% being measured, so the present barrier is theoretical rather than observational. Nothing here derives a coefficient. The reference value kappa = 1/2 remains fitted, not derived. Prior art is conceded in section 1 before any claim is made: the circular de Sitter response is partial prior art (Hari K. and Kothawala, PRD 109, 104073, 2024; Bunney and Louko, arXiv:2406.17643); the only novelty claimed is the extraction of a MOND coefficient from an orbital detector response, which was searched for and not found. All numerical claims are reproduced by the self-checking scripts included in this record; each exits non-zero on any failed internal check. v2 (2026-08-03): two corrections, both narrowing a closure claim. Adversarial self-audit of v1 produced two successive retractions on the day of release, both in the same direction -- a door was claimed shut that was not. (a) v1 said the mechanism "cannot be made to yield a smaller" coefficient; that is a claim about ALL inertia functionals, but only functionals of the local de Sitter-Unruh TEMPERATURE were examined, and functionals of the full response F(E) -- where the paper's own KMS result forces T_eff to be gap-dependent for every nonzero orbital frequency -- were never computed (mi_orbital_q_selfaudit_2026.py). (b) The rigidity theorem that briefly replaced it is ALSO withdrawn: the Newtonian limit forces f to be asymptotically linear, but the deep limit reads f-prime AT THE FLOOR, and nothing connects two different points on f; the five functions tested were all scale-free, for which the two slopes coincide. The correct statement, derived in the new section 3.3, is the closed form q = 2 c1prime / f-prime(T_GH), so the temperature class is a ONE-PARAMETER family in r = f-prime(T_GH)/c1prime with q = 2/r. Milgrom (1999)'s f = T is the r = 1 member; Milgrom (2020)'s coefficient requires r = 4 pi EXACTLY; kappa = 1/2 requires r = 2Z = 8 sqrt(6 pi)/3 = 11.577620, and an explicit smooth, strictly increasing, asymptotically linear f delivering q = 1/Z exactly is given (mi_crossover_master_formula_2026.py, 14/14). The mechanism therefore does NOT fix the coefficient. It fixes the question: since the a0-line is identically Milgrom's balance with the floor at a0/2, the two apparent freedoms are one factor 2Z, and what is open is whether the de Sitter floor is c H_Lambda, fixed by the horizon, or (1/4) c sqrt(G rho_Lambda) -- a BARE sqrt(G rho) carrying no Friedmann 8pi/3. AGAINST THIS FRAMEWORK'S INTEREST: r is itself unfixed, so this is a reparametrisation and NOT a derivation of kappa; Milgrom (2020)'s r = 4 pi is exact and a horizon-area or solid-angle normalisation supplies 4 pi, though the converse objection does NOT hold, since 2Z = 4 sqrt(8 pi/3) makes its sqrt(pi) the FRIEDMANN factor's and an arithmetic-naturalness argument against 2Z would be spurious (mi_2Z_is_the_friedmann_root_2026.py, 8/8); the substantive objection is Deser and Levin's, that the horizon FIXES the floor at H, so H is mechanism-given and sqrt(G rho_Lambda) is a substitution for it, and nothing here defeats that; and whether an r = 2Z kernel survives the solar-system ephemeris bound and the 30.6% shape range is untested. What survives unchanged: the orbital invariance of q, the value q = 2 for Milgrom's own f = T, and the entire 30.6% shape systematic. kappa = 1/2 remains FITTED, NOT DERIVED.