Symmetry-Forbidden Corridors and the Origin of the Dark-to-Visible Matter Ratio

This paper proposes a geometric origin for the observed cosmological ratio of dark matter to baryonic matter, \Omega_{\mathrm{DM}}/\Omega_b \approx 5.33, based on symmetry-forbidden configurations rather than particle microphysics or thermal freeze-out. The mechanism traces the ratio to a theorem of pure geometry: the crystallographic restriction, which forbids five-fold rotational symmetry in periodic flat space. The restriction removes a specific geometric “aperture setting” from the discrete catalogue of realizable configurations. Using the Thales semicircle construction, this absence manifests as a hard coherence corridor between adjacent allowed aperture settings. States entering the corridor activate gravitational degrees of freedom but stall before acquiring relational (Standard Model) coupling because the missing pentagonal aperture cannot be realized. The resulting stall-to-transit population ratio is fixed by geometry rather than by tunable parameters. Evaluating the corridor entry partition at empirically identified thresholds \theta_A = \sqrt{e} and \theta_B = 7/4 yields a bare geometric ratio (1-\eta_\star)/\eta_\star \approx 3.585. Incorporating a dimensional frustration correction motivated by the Gauss–Bonnet theorem—reflecting the necessity of positive curvature for five-fold symmetry—produces a corrected ratio R \approx 5.38, within 0.95\% of the Planck 2018 value. An independent five-fold closure construction yields Q \approx 5 without reference to the corridor calculation, providing a non-overlapping geometric cross-check. A control experiment using a semiconductor junction lattice demonstrates that energy-forbidden (soft) apertures reproduce three-regime dynamics but do not lock population ratios, whereas symmetry-forbidden (hard) apertures uniquely possess the rigidity required for ratio stabilization. The framework introduces no new particles, forces, or modifications of general relativity; the ratio emerges from theorems, not parameters. Limitations and conjectural steps—particularly the dimensional correction and the empirical identification of corridor thresholds—are explicitly stated.