The Self-Screening Fraction: A Combinatorial Derivation of Mass-Dependent MOND Phenomenology

Abstract

In a companion paper (Pec and Bebop 2026, Paper 1), we showed that the MOND acceleration scale a₀ follows an exponential decay in the assumed stellar mass-to-light ratio, with χ²/ndf = 0.19, resolving the cluster problem without dark matter. Here we derive the mechanism. A gravitational system with N baryonic constituents has N(N-1)/2 internal pairwise interactions and N external connections. The fraction of the gravitational budget consumed internally, the self-screening fraction, is f = N/(N+2). This single formula, derived from combinatorics alone, produces three results: (1) a Limited Center Power theorem showing that the gravitational influence available for external (MOND-regime) phenomenology scales as 2/(N+2), vanishing for large systems; (2) a natural reproduction of the exponential α(M/L) decay observed in Paper 1; and (3) a parameter-free explanation of why galaxy clusters, with N ~ 10¹⁴, exhibit the precise MOND discrepancy observed. The residual oscillation detected in Paper 1 maps onto discrete screening-domain transitions analogous to Barkhausen jumps in ferromagnets. We present predictions for LSB, HSB, dwarf, and ultra-diffuse galaxies testable with BIG-SPARC.