A Curvature-Dependent Meta-Field Mechanism for Galactic Rotation and Regular Central Cores without Particle Dark Matter

Galactic rotation curves require an attractive contribution beyond that produced by baryonic matter. We develop a covariant alternative in which local fluctuations of a scalar meta-field about the stationary Tier-30 background modify the physical metric through a universal conformal coupling. The environmental restoring coefficient is sourced by the baryonic stress-energy trace, avoiding self-counting through the curvature of the already modified metric. In the static weak-field limit, high central density pins the field; declining density across the disk increases its correlation length, and the resulting radial relaxation adds positively to the circular velocity. Every smooth axisymmetric center remains regular, with vanishing central radial acceleration and finite Poisson-equivalent density. We solve the two-dimensional boundary-value problem for a fixed Milky Way stellar model composed of a Plummer bulge and Miyamoto-Nagai thin and thick disks, excluding the parent particle-dark-matter halo, and compare it with 30 circular-velocity measurements over 5.27-19.71 kpc. The best fit gives \( \gamma =1.77781\times {10}^{6}, V_{\ast }^{2}=1.35669\times {10}^{5}\; {\text{km}}^{2}\,{\text{s}}^{-2} \), and \( {\mu }_{0}=0 \). Using the unweighted RMS residual as the primary measure of point-by-point agreement, the best-fit response gives \( 2.760\ \mathrm{km}\ {\mathrm{s}}^{-1} \), compared with \( 58.254\ \mathrm{km}\ {\mathrm{s}}^{-1} \) for the same fixed baryonic components alone. The fitted parameters and RMS residual remain stable when the adopted systematic scale is varied over the published range. Because the full inter-bin covariance is unavailable, chi-squared values obtained by treating the two-percent systematic allowance as independent in every bin are reported only as conditional diagnostics rather than as a calibrated likelihood. Over the measured Milky Way interval, the axisymmetric calculation reproduces the rotation curve without including a particle-dark-matter halo. It also predicts a regular central dynamical core and gravitational slip with leading cancellation of the conformal contribution from the Weyl lensing potential, enabling independent tests with lensing, local-gravity measurements, and common-parameter analyses across galaxies.