One Free Number: An Exhaustive Closure on the Normalization of the de Sitter-MOND Acceleration Scale a0 = c² sqrt (Lambda/32pi)

The MOND acceleration scale a0 ~ 1. 2e-10 m/s² can be written exactly as a de Sitter horizon surface gravity, a0 = c² sqrt (Lambda/32pi) = (c/2) sqrt (G rhoDE) = c HLambda / Z with Z = sqrt (32pi/3) ~ 5. 789. The dimensionless content factorizes: the group sqrt (8pi/3) is FORCED (the Einstein coupling 8pi times the Friedmann 3, a sympy-exact identity), and the ONLY undetermined number is the outside coefficient kappa = 1/2. Whether a0's value is derived or merely fitted reduces to one binary question: can kappa = 1/2 be forced? This note reports an exhaustive negative answer. Across ~24 structurally-distinct routes -- ghost-freedom, unitarity, holography, a microscopic degree-of-freedom count, the Atiyah-Patodi-Singer eta-invariant on the de Sitter horizon, five symmetry-breaking mechanisms (Hopf-bundle flux, the Witten SU (2) anomaly, the gravitational Chern-Simons framing anomaly, a cosmic-string conical defect, a zeta-regularized Casimir energy), eight further families (on-shell dS-Unruh, 4-form flux quantization, action-level quantization, non-Dirac index operators, horizon entropy, Goldstone normalization, limit-matching, the dS-CFT central charge), and a gated brute-force over six classes of modified-inertia action -- no construction forces kappa = 1/2 non-circularly. The failures share one structural reason, made precise here: kappa is the absolute classical normalization OUTSIDE the gravitational square-root, carrying the Einstein 8piG (how the dark-energy density gravitates), while every number-fixing probe in physics (signs, ratios, scales, quantized levels, spectral asymmetries, vacuum energies) is a mod-Z or dimensionless object type-mismatched to a bare multiplicative coupling. The mismatch is formal: a0 (2 kappa) = 2 a0 (kappa) exactly. A genuine caveat closes the loop: the framework DOES contain a rigorously forced 1/2 -- the binomial 1/2 in the de Sitter-Unruh response sqrt (a²+ (cH) ²) - cH = a²/ (2cH) -- but it sets the TEMPERATURE root a0 = 2 c HLambda, separated from the density-root kappa by EXACTLY Z = 5. 789. It is a real 1/2 of the framework's own making, provably not kappa. Conclusion (strengthened from, and consistent with, the author's prior work): a0 ~ c sqrt (Lambda) is a forced SCALE, while its normalization is ONE free posit -- the framework is a PROVABLY one-parameter effective theory, not zero-parameter and not a theory of everything. The result is a strong negative (no non-circular construction across ~24 routes), explicitly NOT a formal no-go against every conceivable exotic operator. The live content is entirely empirical: a declining a0 (z) proportional to sqrt (rhoDE (z) ) and a Lorentz-violating sTX dipole, both falsifiable this decade. All symbolic checks are reproducible from the public repository. This note documents a closure and claims LESS, not more.