A Newton–Mach Effective Cosmology on an FRW Background: Derivation, Effective-Fluid Representation, and H0-Motivated Structure

This version updates and extends NMuniverse_ver3. The revised manuscript improves the internal consistency of the Newton–Mach (NM) cosmological framework, clarifies the logical status of the FRW embedding, and sharpens the homogeneous-level effective-fluid reconstruction on a spatially flat FRW background. A central new point of this version is the added phenomenological motivation for the rational value alpha = 50/3. In particular, the manuscript now includes the effective counting argument summarized by Eqs. (41a)–(41d): an effective primordial degree count N_total = 3^6 = 729, a topological correction on the spherical holographic screen S^2 giving N = 727, an axisymmetric spherical-harmonic projection leading to N_ax = 25, and the corresponding effective electromagnetic count N_EM = 50. This yields the effective identification α= l_pu / l_P = N_EM / 3 = 50/3. Within the Planck-time constraint of the model, this version therefore highlights more explicitly that the working hypothesis α= 50/3 leads to the representative Newton–Mach value H0^(NM) ≃ 73.4951 km s^-1 Mpc^-1, which is numerically very close to the recent H0DN local-distance-network determination H0 = 73.50 ± 0.81 km s^-1 Mpc^-1. In addition, the revised text makes the theoretical status of the construction more explicit. The manuscript does not claim a new covariant field equation replacing Einstein gravity. Instead, it formulates a specific NM-type homogeneous background evolution on an FRW spacetime and reconstructs the corresponding effective perfect-fluid description. The paper also shows that, at the homogeneous FRW level, the reconstructed effective density and pressure satisfy the continuity equation identically. More broadly, this work studies an effective cosmological model in which the Friedmann–Robertson–Walker (FRW) metric of general relativity is retained as background geometry, while the time evolution of the scale factor a(t) is governed by a Newton–Mach type law derived from (i) Newtonian radial dynamics, (ii) the Mach-type relation c^2 = G M(r) / r, (iii) the local equivalence principle in an FRW background, and (iv) the Planck force F_P = c^4 / G. For the physical radius r(t) = a(t) chi_0 of a representative comoving sphere, the model gives dr/dt = c √(1 - 2 ln(r/r0)), or equivalently da/dt = H0 √(1 - 2 ln a). This version should be understood as a refined phenomenological update of NMuniverse_ver3. It strengthens the presentation of the FRW-compatible NM background ansatz, clarifies the role of the effective-fluid reconstruction, and adds a compact holographic-topological motivation for α= 50/3 and N = 727. At the same time, it does not claim that the framework is already a complete alternative to standard cosmology. Detailed confrontation with CMB, BAO, supernova, large-scale structure, perturbations, and local-gravity constraints remains future work.