Dark Matter as a Reversible Phase Transition: The \(c\sqrt{\alpha}\) Threshold and Torsional Restructuring in Particle Collisions

Abstract In Generalized Mass as Twisted Time (G-MaTT), dark matter is not an exotic substance — it is ordinary matter in a desynchronized phase state, analogous to ice versus water. The transition is governed by a single, geometrically fixed threshold: \[v_{\text{crit}} = c \sqrt{\alpha / w}, \quad \alpha^{-1} = 137.035999206.\] Above this velocity, phase slip exceeds the coherence margin \(\sqrt{\alpha}\), suppressing the Twist-Untwist Threshold (TUT) weight \(\mathcal{W}_{\text{TUT}} = \exp[-(\Delta\phi)^2/\alpha]\) and decoupling electromagnetic/weak interactions — while gravity, sourced by fractional entanglement, persists. Crucially, this transition is fully reversible: when \(v < c\sqrt{\alpha/w}\), braids re-lock, restoring full coherence. We extend this to particle accelerators: high-energy collisions do not “create particles from energy”, but induce torsional shock and topological fission of \(\hat{M}_\mu\)-knots, followed by re-coherence below threshold. Higher-order scales (\(c\alpha\), \(c\alpha^{3/2}\)) define internal stability regimes — not new thresholds. This unified picture resolves galactic rotation curves, predicts dark→visible conversions in cluster cores, and offers falsifiable signatures at the LHC, DUNE, and CMB-S4.