Scale Without Law: Why the de Sitter-Unruh Temperature Forces the MOND Acceleration but Not the Interpolation

A foundations/obstruction note on Milgrom's 1999 'modified dynamics as a vacuum effect'. The MOND acceleration scale satisfies a0 ~ c² sqrt (Lambda/32pi) = cHLambda/Z; Milgrom proposed reading this via the Deser-Levin de Sitter-Unruh temperature T (a) ~ sqrt (a² + aLambda²), aLambda = cHLambda, with inertia responding to the excess over the de Sitter vacuum. This note asks, with explicit and adversarially-checked sympy/numpy calculation, how far that mechanism actually goes, and reports an honest two-sided result. WHAT WORKS: the de Sitter-Unruh temperature FORCES the scale (a0 ~ c sqrt (Lambda) recovered, not assumed) and yields the deep-MOND square-root law and the baryonic Tully-Fisher relation (both exact). WHAT DOES NOT WORK without inserting more: the full interpolation function, the precise coefficient Z, and a covariant equation of motion -- the standing open problem since 1999. THREE RESULTS make the obstruction precise: (i) the candidate interpolation mufw= (sqrt (1+4x²) -1) /2x and the bare dS-Unruh interpolation mu= (sqrt (1+x²) -1) /x are the SAME one-parameter family, separated by a single rescaling, so the whole gap collapses to one undetermined coefficient Z; a principled free-energy postulate does give a valid (simple-nu) MOND interpolation with correct limits and sign, but not mufw and not forced, and a decoy-family test proves the coefficient is reverse-engineered, not derived. (ii) a NUMBER-FIELD obstruction: thermodynamic normalizations (Unruh 2pi, Bekenstein-Hawking 1/4, surface-gravity 1/2, d (d-1) =6) live in the field Q (pi), while Z=sqrt (32pi/3) lives in Q (sqrt (pi) ) ; the two fields do not meet, so no horizon-entropy or temperature factor can ever OUTPUT the coefficient -- only an intrinsic sqrt (rho) route lands on Z, with one free O (1) (the Z²=4* (8pi/3) factorization is killed: the '4' is 1/kappa² at the posited kappa=1/2, not the area-law 1/4). (iii) a STRENGTHENED sign theorem: the MOND-signed response requires breaking ghost-freedom or supplying an ACTIVE drive (a passive de Sitter bath gives deltaₘ = 2 integral rho/omega² >= 0, the anti-MOND sign), needing ONLY ghost-freedom not passivity; combined with the Ostrogradsky (local) and Cassini (field-theoretic) horns this blocks a passive-bath covariant completion. The one honest crack is external: a non-de-Sitter, in-band, phase-coherent galactic drive could supply the active kernel, but none is known and it would owe an explanation of a0's universality. OBSERVATIONAL HANDLE: a0 (z) ~ sqrt (rhoDE (z) ) declines on the DESI dynamical-dark-energy branch (~25-40% by z=3) and is constant for w=-1, the one place the de Sitter-Unruh reading is separable from a static coincidence. CONCLUSION (deliberately modest): the de Sitter-Unruh route owns the SCALE and the deep-MOND LIMIT but not the LAW; the residual freedom is reduced to a single provably-unforced number and a clean sign obstruction, both stated as theorems. No derivation of MOND, no fixing of the coefficient, nothing about the Standard Model. All numbers reproducible from the public repository. The author retracted all earlier 'theory of everything' claims (2026-06-23) ; this note makes none. Supplements the author's de Sitter-Unruh papers (DOI 10. 5281/zenodo. 20973740, 10. 5281/zenodo. 20965016).