Three universality theorems for ensemble-derived MOND interpolating functions
Year: 2026
Type: preprint
Abstract: 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.
Author: Rusishvili, Mikheil
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7170301472
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