math-ph — Mathematical Physics
Era statistics for the vacuum Bianchi IX flow
We prove the Gauss statistical law for eras of the exact vacuum Bianchi IX flow with zero cosmological constant. Eras are defined by transverse events along the solution, and their lengths count curvature transitions. Along almost every trajectory, the era phases have the Gauss distribution and every finite block of era lengths has its Gauss frequency. The tail decays inversely with era length; the limiting truncated mean grows logarithmically with the cutoff, while the full mean diverges. One invariant Borel exceptional set is null on every smooth transverse section and in the four-dimensional constraint manifold. The proof combines estimates for exact return branches with absolute continuity of their coding measure. Transverse contraction controls the accumulated phase error within each era, uniformly over unbounded era lengths and arbitrarily unequal small curvature variables. A density with strictly expanding section flux then forces almost every recurrent orbit to enter a statistical set on a small section.