Ordered Activation, Finite-Resolution Depth and Zero-Set Rigidity for Loxodromic Lorentz–Möbius Transitions

Earlier work established sharp depth bounds and asymptotic transition lattices for collinearity transitions of Möbius vector fields along Lorentz orbits. This paper answers three questions left open there. First, for the ordered Lorentz map Φ(s, τ) = eτYesXq0 from a null seed, the screen Gram matrix is independent of τ. Every isolated transition of the base orbit is therefore a complete transition line. Second, for two smooth flows we define an activation depth q by the first iterated bracket that leaves the surviving direction, and show that the Gram determinant vanishes to order 2q. In the Lorentz–Möbius setting the transverse parts of the brackets adXjY are exactly the derivatives of the transition scalar, so q equals the transition depth. The known depth bounds 2 and 4 then arise from universal real annihilating polynomials of degrees 3 and 5 for the adjoint action of X. Under a uniform factorization hypothesis the local area scales as ρq+1. Under an additional uniform covering law, q is recovered from a two-scale renormalized covering entropy. Third, for the loxodromic transition function F(s) = Im(Ae−(α+iβ)s + B + Ce(α+iβ)s) we derive second-order tail asymptotics and a canonical signature built from half-angle phases. We prove that, off the known blind locus, the ordered real zero set determines F up to real scale, translation, rescaling and orientation. Only the two asymptotic tails are needed.