From Galaxies to Clusters with the Derived Interpolation Function

The MOND interpolation function µ(x) = x/(1 + x), derived from resource allocation on a causal graph (Hess 2026), is tested against 175 galaxies from the SPARC database and 5 galaxy clusters. At galaxy scales, pure MOND (no dark matter halo) wins 86/175 galaxies (49%), outperforming both CDM (22%) and an earlier structural-entropy halo model that assumed a concentrated Hernquist profile (29%). The earlier halo model is superseded by a three-scale picture in which the structural entropy is diffuse at kiloparsec scales and only accumulates to the full cosmological dark-matter ratio 7−ϕ at the Hubble scale. At cluster scales, the derived µ-function reduces the standard MOND mass deficit from a factor of ∼2 to 10–36%, largely resolving the long-standing cluster problem. The specific functional form x/(1+x) provides more gravitational boost in the deep-MOND regime than the traditional x/√1 + x2, explaining the improvement. These results, combined with the cosmological prediction Ωcdm/Ωb = 7−ϕ= 5.382 (Planck-consistent to 0.3σ), establish a complete three-scale picture: pure MOND at galaxy scales, MOND with a residual diffuse entropy contribution at cluster scales, and the full structural entropy ratio at cosmological scales. DISCLAIMER Generative AI was used to assist with literature screening / coding support / draft language revision. All AI-assisted outputs were independently checked by the author, and the author takes full responsibility for the final analysis and text. This is encompassing all the work that has been done and will be done. All code is under MIT licensing. All research papers are under Creative Commons License. All code, outputs and notes are included in the reproducibility bundle zip file.