Path automata: compression thresholds, a floor(4n/3) construction for every n, and exact reset thresholds

Binary path automata formed from two arms of alternating involutions and a junction are studied through their geometry, phase quotient, rank and reachable cycle subsets. For every n >= 9, a synchronizing member has a state for which the minimum merging-word length, over all partner states distinct from that state, is floor(4n/3). Complete inverse-image coverings give exact reset thresholds 3s^2+4s-1 on the balanced slice for s >= 3, 3(s+1)^2 on the adjacent slice for odd s >= 3, and 3s^2+8s+8 on the next slice for s >= 6 divisible by 3, with the separate value rt(B_{3,5})=58. Other stated next-slice results are asymptotic bounds. The full two-parameter exact formula remains open. The mandatory technical supplement and scoped integer-arithmetic obligations are required parts of the proof. Preprint. Not externally peer reviewed. The main paper must be read with the mandatory technical supplement and the accompanying evidence archives.