A Parameter-Free Disk Formation Timescale from $S^1_{\mathrm{Sol}}$ Temporal Geometry, ApJL Letter, Preprint

The James Webb Space Telescope has revealed ordered, rotationally supported disk galaxies at redshifts $z \sim 6\text{--}8$, well before the $\gtrsim 1 \text{ Gyr}$ of dynamical evolution typically required for disk settling in $\Lambda\text{CDM}$. We derive a parameter-free disk formation timescale $\tau_{\text{align}} = 2 t_{\text{dyn}}$ from the temporal geometry of the twelve-dimensional Riemannian framework $\mathcal{M}_{12} = (S^1_{\text{Sol}} \times S^3) \times K_8$, in which the temporal sector $S^1_{\text{Sol}}$ is a Sol 3-manifold with hyperbolic $\text{SL}(2, \mathbb{Z})$ monodromy. The monodromy eigenvalues $\phi = (1 + \sqrt{5})/2$ and $\phi^{-1}$ generate an Anosov flow on the internal $K_8$ shape modulus; integration over the internal space yields a rank-2 anisotropic correction to the four-dimensional shear tensor that couples to protogalactic angular momentum through standard tidal torque theory. The $(0,0)$ component of the twelve-dimensional Einstein equation, evaluated in the FLRW background, then fixes the proportionality constant: $T_{\text{Sol}} = 2 \ln \phi \times t_{\text{dyn}}$, whence $\tau_{\text{align}} = T_{\text{Sol}} / \ln \phi = 2 t_{\text{dyn}}$. The factor of 2 is not chosen; it is exact, a consequence of the golden ratio identity $\phi^2 = 1 + \phi$. At $z = 8$ this enables ordered disk settling in protocluster environments ($\delta \sim 10\text{--}50$, $\tau_{\text{align}} \approx 66\text{--}150 \text{ Myr}$) and forbids it in voids ($\tau_{\text{align}} \gtrsim 340 \text{ Myr}$). Three falsifiable predictions are stated. The cosmological corollaries ($S^3$ halo suppression, BAO sound horizon, CMB low-multipole suppression) are addressed in the companion long-form Paper X.