Baryonic Virial Equilibrium in the Coma Cluster (Abell 1656): Resolving the Velocity Dispersion Paradox Without Non-Baryonic Dark Matter via a 3D Spatial Integral Kernel Operator

Abstract Since foundational dynamical analyses of the Coma Cluster (Abell 1656), the severediscrepancy between the classical Newtonian Keplerian velocity falloff predicted from visiblebaryons (∼ 594.01 km/s at R = 3.0 Mpc) and the observed outer-halo velocity dispersionplateau (∼ 910.00 km/s) [1, 2] has served as a primary pillar for non-baryonic Cold DarkMatter (CDM). Extending the 1D radial spectral integral framework established for galacticrotation curves in Paper I [3], we formulate a three-dimensional Spatial Integral KernelOperator (3D-SIKO) over a 1, 000-node native discrete manifold (R ∈ [0.1, 3.0] Mpc, radialinteraction cutoff rcut = 0.48 Mpc, mean coordination degree 34.4) governed by a 5Dtrans-brane restoring stress-energy tensor TEPR (R, w) [4]. We demonstrate that standardunconditioned variational PDE solvers suffer catastrophic gradient blowup on multi-scalecluster potentials (RMSE = 14, 851, 929.00 km/s). By enforcing sequential native-footprintf = Mbar × 10−11 /1000.0), an equilibrium residual conditioning divisormass conditioning (Mof 1000.0, and a hard-wired Heaviside-linear positivity projector (P+ : TEPR ≥ 0), thestabilized 3D-SIKO solver converges in 13.19 s. Using strictly baryonic inputs (central cDgalaxies NGC 4874/4889 modeled via Hernquist potentials [5] plus β-model X-ray intraclustergas [1, 6]), the operator resolves the full radial dispersion profile with RMSE = 17.08 km/s(R2 = 0.9040) and matches the R = 3.0 Mpc outer halo boundary at 909.86 km/s versus the910.00 km/s observational benchmark (99.98% accuracy) while maintaining exact adherence(0.0000% error) to the universal bulk attenuation constant λ = 8.2390. Intellectual Property, Copyright & Commercial Licensing Notice: © 2026 Sudhir Raj, Researcher (Equilibrium-Research.org), Equilibrium-Research.org Group, Headington, England. Manuscript text and figures may be distributed for non-commercial academic review and scientific citation with full attribution to the author and https://equilibrium-research.org. The underlying thermodynamically-constrained variational operator architecture, sequential bespoke-footprint conditioning pipeline, 1000.0 equilibrium residual conditioning divisor, hard-wired Heaviside-linear positivity projector, and pre-calibrated spectral operator state tensors described herein are the subject of United Kingdom Patent Application No. GB2625201.5 (Patent Pending). Nothing in this academic publication grants any implied or express license under any patent rights. All commercial implementation and Field-of-Use licensing rights are strictly reserved. Commercial licensing inquiries: [email protected]