Testing Modified Inertia in Galaxy Clusters: An Anisotropy-Dependent Mass Normalization

A scoped sharpening of Milgrom's modified-inertia-vs-modified-gravity discriminator, applied to galaxy clusters. The MOND cluster residual (~factor 2 after the boost) is shared by all realizations and is the paradigm's central open problem. Modified-inertia (MI) and modified-gravity (MG) agree on circular orbits but diverge on non-circular ones -- this divergence is Milgrom's, established for discs via the global deep-MOND parameter Q (Milgrom 2012, PRL 109 251103) and the general MG virial theorem (Milgrom 2014, PRD 89 024016). This note applies it to PRESSURE-SUPPORTED clusters and extracts one definite, previously-unstated consequence: in modified inertia the dynamical-mass normalization eta in G M a0 = eta sigma^4 is SYSTEM-DEPENDENT -- it slides with the cluster's orbital anisotropy -- and with a definite SIGN: radial anisotropy RAISES eta and WORSENS the residual, because Milgrom's non-local inertial response is set by the orbit's Fourier acceleration amplitude (pericentre-dominated), not by residence time. Every physical acceleration average entering the amplitude functional rises with eccentricity; the pure apocentre value (which would help) enters no term. On the framework footing a0 = c^2 sqrt(Lambda/32pi) = 9.36e-11 m/s^2, an eRASS1-calibrated deep-MOND ensemble gives eta ~ 2.15 (isotropic) rising to ~2.8-3.0 (radial anisotropy beta ~ 0.3-0.5); the mean-field (circular) value is a FLOOR, so pure modified inertia CANNOT reach closure (eta=1) by orbital structure -- a genuine sharpening that removes anisotropy as a candidate cure and shows the effect runs the wrong way. In modified gravity (AQUAL/QUMOND/AeST) eta is anisotropy-INDEPENDENT by Milgrom's virial theorem; in LambdaCDM the dynamical mass is the true mass, so no anisotropy dependence. Hence a measured correlation 'eta rises with radial anisotropy beta' is a signature no MG theory and no dark-matter model can produce -- an MG-impossible cluster test. It is galaxy-safe by construction: disc orbits are single-frequency, so the anisotropy-keyed effect is identically zero in the rotationally-supported RAR (Milgrom 2022, Eq 29). A second, related MI cluster signature (an infall-phase velocity-dispersion excess, exactly zero in the memoryless MG/AeST field theories) shares the same trajectory-dependent origin. Neither is decisive with present data -- these are near-term (eROSITA+spectroscopy / ELT-era) tests limited by resolved-anisotropy precision and cluster-kinematic systematics. SCOPE/CREDIT: the parent divergence and machinery are entirely Milgrom's (2012, 2014, 2022); the contribution is the cluster-anisotropy realization and its sign, with explicit galaxy-safety and a falsifiable eta(beta) statement. It does NOT solve the cluster problem -- it shows pure MI cannot close the residual through orbital structure, which continues to require a mass component (in AeST, the ghost-condensate dust, whose amount is a free initial condition, 'not (classically) predicted', Skordis-Zlosnik 2021). Companion to 'Scale Without Law' (DOI 10.5281/zenodo.21016309). The author retracted all earlier 'theory of everything' claims (2026-06-23); this note makes none.