Symmetry-Forbidden Corridors and the Origin of the Dark-to-Visible Matter Ratio: Geometric Selection with Empirical Validation on the Radial Acceleration Relation

This work explores a geometric hypothesis for the cosmological dark-to-visible matter ratio based on a symmetry-forbidden configuration in flat-space tiling. The crystallographic restriction theorem excludes five-fold rotational symmetry from periodic planar lattices, creating a structural gap between the last allowed (four-fold) and first forbidden (five-fold) aperture settings. Within a Thales semicircle partition framework, we construct a stall-to-transit ratio R_0 = \frac{1-\eta_\star}{\eta_\star} at a corridor entry threshold \eta_\star. Using an empirically identified threshold \eta_\star \approx 0.218 and a curvature-motivated dimensional correction factor 3/2, the construction yields R = 5.38, within 1% of the Planck 2018 value \Omega_{\mathrm{DM}}/\Omega_b = 5.33 \pm 0.13. The threshold and dimensional factor are not derived from the crystallographic theorem and constitute the two open inputs of the construction. The result is therefore a geometrically motivated hypothesis rather than a parameter-free derivation. The underlying Thales partition geometry is independently evaluated using the radial acceleration relation (RAR) across 153 SPARC galaxies with zero fitted parameters in the geometric mapping. Five quantitative features emerge: (i) the MOND scale lies near the apex of the coupling manifold; (ii) the induced arc-length metric provides empirical geometric regularization in forward simulations (0/600 boundary violations versus 391/600 in flat dynamics); (iii) connection terms account for 5–42% of effective acceleration gradients; (iv) the inferred arc-length potential is substantially smoother than its flat counterpart; and (v) a π/8 flat-measure baseline defines an empirically populated coupling corridor spanning 0.04a_0–3a_0. The framework does not identify a dark matter particle, derive the MOND scale, or claim a closed first-principles derivation of the cosmic ratio. Rather, it establishes the Thales partition as a consistent geometric description of baryonic–dark coupling and demonstrates that a symmetry-forbidden corridor, supplemented by two identified open inputs, can reproduce the observed cosmic mean ratio to percent-level accuracy.