Three universality theorems for ensemble-derived MOND interpolating functions

MOND interpolating functions are ordinarily chosen rather than derived, and the several derivations that exist each supply one function from one mechanism. This note takes the complementary route: it asks what is forced on any such function if it arises as an ensemble average, so that the results constrain a whole class of constructions rather than propose a new member of it. The construction considered is a population of independent couplings, each with its own critical acceleration a_c distributed as P(a_c), each responding with the classical depinning mobility. Three properties turn out to be determined by P alone and to be independent of everything else in the construction. (i) If P(a_c) ∝ a_c^β as a_c → 0, the deep-MOND slope is β+1; the square-root force law, and with it flat rotation curves and a baryonic Tully–Fisher slope of four, therefore holds if and only if P(0) is finite and non-zero. (ii) The deep-limit amplitude is exactly (π/4)P(0), for every admissible P. (iii) Because every term of the complement decomposition is non-negative, there is a universal floor on the Newtonian approach: 1 − µ(y) ≥ ⟨s²⟩/2y² ≥ 1/2y² holds exactly, at every finite y, for every admissible P — no ensemble can approach Newtonian gravity faster than y⁻², and the floor is saturated only by a single threshold — while an approach slower than y⁻² requires a divergent second moment. The class is therefore pinned from both sides, and a corollary follows: the empirical radial-acceleration-relation function is unreachable by any ensemble at all, and the "simple" function µ = y/(1+y) requires a coupling population with logarithmically divergent mean critical acceleration — so it cannot arise from any ensemble for which a₀ is a mean. Working the exponential case gives a closed form, µ_B(y) = (π/2)[I₁(y) − L₁(y)], with both MOND limits and no shape freedom. All statements are verified numerically to four or more digits, and a practical warning is recorded: the Bessel–Struve representation loses every significant digit by y ≈ 40 and must be replaced by quadrature. Because the results are class-level, they apply to existing derivations as well as to future ones, and §7 makes that application explicit.