Resolving the Coma Cluster Mass Discrepancy Without Dark Matter: Multi-Layer Field Overlap and Apparent Ruler Scaling in the Planck Gravity Theoretical System

This preprint presents a quantitative, exploratory application of the Planck Gravity (PG) theoretical system to the Coma cluster (Abell 1656), addressing the classic galaxy-cluster missing-mass discrepancy first identified by Fritz Zwicky (1933) without invoking non-baryonic dark matter. Developing directly from the foundational discrete-spacetime framework ("The Planck Gravity Theoretical System: A Discrete Mechanical Model of Gravity, Motion, Light, Event Horizons, and Cosmological Effects", DOI: 10.5281/zenodo.22995823) and its galactic-scale evaluations (NGC 6822, DOI: 10.5281/zenodo.23042315; DDO 126, DOI: 10.5281/zenodo.23345031), this study models the gravitational vacuum as a discrete network of elementary Planck cells where the microscopic dwell time (t_dwell) remains invariant while the apparent spatial ruler scale varies locally as l_app(r) = l_p * σ(r). Key Quantitative Results & Methodology:- Baryonic Mass Inventory: Integrates a spherical β-model intracluster gas profile (r_c = 0.29 Mpc, β_g = 0.75, M_gas(< 2.5 Mpc) = 1.2 × 10^14 M_sun) and a Hernquist stellar profile (a = 0.3 Mpc, M_stars(< 2.5 Mpc) = 0.2 × 10^14 M_sun), yielding a total baryonic mass of 1.4 × 10^14 M_sun within 2.5 Mpc.- X-Ray Hydrostatic Equilibrium (HSE) Extraction: Using a polytropic intracluster medium (k_B T_0 = 8.2 keV, γ_p = 1.15, μ = 0.60), the HSE estimator yields enclosed gravitating masses of 4.75–93.69 × 10^13 M_sun across six sampled radii (0.2 to 2.5 Mpc), corresponding to HSE-to-baryon mass ratios of 6.69–9.52 and apparent ruler-scaling factors σ_PG(r) = [M_HSE(r)/M_baryon(r)]^(1/2) of 2.59–3.08.- Radial Scaling Law & Universal Geometry: Fitting the phenomenological scaling function σ_local(r) = 1 + α * (r/R_σ)^p / [1 + β(r/R_σ)] yields α = 2.56132, R_σ = 0.14896 Mpc, p = 0.61812, and β = 0.50125 (in-sample RMSE = 0.03777). Notably, the exponent p ≈ 0.618 closely parallels the p = 2/3 scaling found in dwarf galaxies, motivating a cross-scale geometric scaling hypothesis.- Multi-Layer Field Overlap: Compares local cluster gas density against the cosmic mean baryon density (ρ_b,cosmic = 6.223 × 10^9 M_sun Mpc^-3), deriving a local field weight of 98.99% at 2.5 Mpc and a formal cluster-to-cosmic transition radius of 19.31 Mpc. This work establishes a transparent, falsifiable research baseline and locks in the four Coma cluster scaling parameters for independent weak-lensing and outer-gas observational tests. Part of the Planck Gravity Theoretical System Series:- Foundational System: https://doi.org/10.5281/zenodo.22995823 - Kinematic Framework (V15): https://doi.org/10.5281/zenodo.23013678- Cell Density & NGC 6822: https://doi.org/10.5281/zenodo.23042315- DDO 126 Evaluation: https://doi.org/10.5281/zenodo.23940519