A Provably One-Parameter Theory of Gravity and the Dark Sector: the de Sitter-MOND Coefficient kappa=1/2 Is Unforceable Geometry

The de Sitter-MOND framework writes the galactic acceleration scale as a0 = c² sqrt (Lambda/32pi) = kappa c sqrt (G rhoDE), and the FORM a0 ~ c² sqrt (Lambda) together with the gravitational kernel sqrt (8pi/3) (including the half-integer power of pi) are forced. This paper asks whether the single residual coefficient kappa=1/2 is forced or free, and proves a sharp result: kappa is the framework's lone free number, and it is PROVABLY UNFORCEABLE by the three standard consistency conditions -- ghost-freedom, unitarity, and a holographic / degree-of-freedom bound -- each failing for a structurally distinct, machine-verified reason. (i) Ghost-freedom is blind to kappa because kappa is the overall normalization of a phi_- -linear, degree-1-homogeneous action: the equation of motion divides cleanly by it and every no-ghost/stability inequality and sound speed is normalization-invariant. (ii) Unitarity (Schwinger-Keldysh/in-in positivity) reaches only signs, causality, and noise/dissipation ratios with a temperature -- never an absolute scale -- and for the conservative even kernel the fluctuation-dissipation relation is vacuous. (iii) Holography reaches only the scale cH (the unique hbar-free acceleration) ; the sole dimensionless group G hbar H²/c⁵ ~ 1/SdS ~ 1e-122 is hbar-laden and cancels classically, and CKN/Bousso/de Sitter-equipartition are inequalities saturated with a free O (1) whose fixing to the target IS choosing kappa (circular). The paper then identifies what kappa IS: the framework coefficient sqrt (2/Z) = (3/8pi) ^ (1/4) =0. 58779 (Z=sqrt (32pi/3) ) is PURE GEOMETRY, carrying no degree-of-freedom, thermal, or entropy content, and it equals exactly the single-degree-of-freedom (g_*=1) limit of the Cohen-Kaplan-Nelson (CKN) UV-IR bound -- the machine-exact geometric identity 3/8pi = (Schwarzschild 1/2) / (ball 4pi/3). An honest count of the Standard Model degrees of freedom (g_*=106. 75) gives a CKN coefficient of ~0. 18-0. 41 (energy version, scaling as g_*^ (-1/4) ) to ~0. 31-0. 41 (entropy version, g_*^ (-1/3) ) -- never within 5% of the framework value; a constant and a strictly-decreasing function of g_* coincide only at one tuned non-SM root (g_*~3-25), so demanding agreement is circular. kappa is therefore GEOMETRIC, not microscopic. The framework is a PROVABLY one-parameter theory of gravity and the dark sector -- one knob, irreducible and geometric -- versus LCDM's six parameters plus an undetected particle. The paper states plainly what this does and does NOT establish, both ways: it fixes the parameter economy and the irreducibility of the lone number (with anti-circularity held: kappa kept symbolic, the value 1/2 and the 8pi density route appear only in the post-hoc locating map, and the g_*=1 identity fell out of an independent derivation) ; it does NOT derive a0's value (the dS-Unruh temperature route's natural scale is 2cHLambda, ~12x too large, and the empirical a0 requires the density route plus the kappa=1/2 posit), it does NOT make the sector zero-parameter, and it does NOT confirm the framework against LCDM (a theory can be maximally economical and still wrong; that verdict belongs to the data). All identities are machine-verified and reproducible. This is a structural result for a falsifiable theory of gravity and the dark sector -- not a proven replacement for dark matter, and not a theory of everything yet, as frustrating as it may be.