What Can and Cannot Fix the MOND Acceleration Coefficient: a Relabelling Theorem, Three Failed Derivations, and a Redshift Discriminant

Abstract

The MOND acceleration scale is numerically close to c H_Lambda, and the proposal that it is SET by the cosmological constant dates to Milgrom (1994, 1999). The relation's FORM is established; its dimensionless O(1) COEFFICIENT is not derived by any published argument, and the live proposals differ: Milgrom (1999) gives a0 = 2 c H_Lambda, Milgrom (2020) gives kappa = 1/2pi, and this author has used a0 = (1/2) c sqrt(G rho_Lambda), i.e. a0 = c H_Lambda / Z with Z = 2 sqrt(8 pi/3). That last value is FITTED, NOT DERIVED, and nothing in this paper changes that. This paper is about what could change it. We prove a RELABELLING THEOREM: because Lambda = 8 pi G rho_Lambda/c^2 identically, G rho_Lambda and c^2 Lambda are the SAME scale up to the pure number 8 pi, so any combination of the two returns that scale times a power of 8 pi and cannot select a coefficient or exclude c H_Lambda. We then audit three constructions of the kind that naturally suggest themselves. All three fail: one on dimensions (its geometric mean is a frequency-squared, not an acceleration, and the charitable repair overshoots by exactly (8 pi)^(1/4)); one because its free parameter is merely relocated (for any inertia functional of the local de Sitter-Unruh temperature the crossover is q = 2 c1prime / f-prime(T_GH) = 2/r with r free, so an explicit admissible functional reaches kappa = 1/2 but only by trading one fitted number for another); and one on a sign that makes its central invariant imaginary in any homogeneous vacuum. Two of the three additionally attach their coefficient to a0 rather than to the FLOOR a0/2 that Milgrom's balance actually contains; having committed the same error ourselves, we record it as a systematic hazard. Three results are constructive. (i) The theorem does NOT exclude a single-scale derivation: sqrt(G rho_Lambda) is pi-free while sqrt(8 pi G rho_Lambda/3) is not, so a construction taking rho_Lambda as its only input is untouched by the theorem and would automatically exclude c H_Lambda, being unable to manufacture the Friedmann 8 pi/3. (ii) No standard local rate supplies the required factor 1/4: over seven candidates the closest is sqrt(G rho/4 pi) at 12.84% away, wider than the 7.87% separating the two published coefficients, so there is no near miss. Since kappa is fitted, searching constructions until one reproduces 1/4 is reverse-engineering a fit, and any candidate must make an independent prediction. (iii) The choice of floor is OBSERVATIONALLY DECIDABLE and needs no new mechanism: a local response to the vacuum DENSITY gives a0 proportional to sqrt(rho_DE), exactly constant for w = -1 and blind to matter, whereas a horizon floor tracks c H(z) = c H_0 E(z), rising to 1.78, 3.01 and 4.54 times its present value at z = 1, 2, 3. The local reading is therefore the MORE falsifiable of the two, because it forbids the rising branch the horizon reading permits. A by-product worth separating from the negative results: the detector-response calculation reproduces Deser and Levin's temperature sqrt(a^2+H^2)/2pi to 1e-15 to 1e-17 across three radii from a computed Unruh-DeWitt response on a non-rotating worldline, so the temperature Milgrom's balance POSITS is here obtained rather than assumed. Rotation breaks the KMS condition only at order (v/c)^2, falling short of the freedom needed for kappa = 1/2 by a factor 1.2e7, which makes the response route a null for the coefficient. PRIOR ART, which the coefficient question does not displace: Milgrom 1994 Ann. Phys. 229, 384 section II eq. 3 writes a_lambda = c^2 sqrt(Lambda/3); the interpolating function nu = sqrt(1+1/y) and the temperature balance are Milgrom 1999 Phys. Lett. A 253, 273 eqs. 6-9, who fixes the coefficient at 2 c H_Lambda; the five-acceleration construction is Deser and Levin 1997 CQG 14, L163; the exponential kernel is McGaugh 2008 ApJ 683, 137 eq. 11a; kappa = 1/2pi is Milgrom 2020. None of the FORM is claimed here. All numerical and symbolic claims are reproduced by the self-checking scripts included in this record; each prints per-check results and exits non-zero if any internal check fails. Three of the six scripts, and the closed form of section 4, are shared with the companion record 10.5281/zenodo.21782600; the paper declares the overlap and states which sections are new.