Arithmetic Reconstruction from Doubled Mixmaster Winding

This paper analyzes what can be rigorously extracted from doubled Mixmaster Weyl‑channel winding. It shows that, although reversed-ray generators satisfy a pointwise parity relation on diagonal class‑A frames, noncommuting time‑ordered evolution prevents identifying opposite rays with inverse propagators through Bianchi‑IX walls. Under a fixed‑Cartan Kasner‑epoch closure and a stated power‑law specialization, the homogeneity‑anchored pair on each real Kasner plateau becomes a fixed‑phase multiple of a square, giving explicit local scaling laws under golden‑orbit assumptions. Independently of any transport closure, squaring the exact one‑wall axis‑slice curves doubles their winding increments to 2π and 4π, and this doubled word uniquely reconstructs era lengths, continued‑fraction digits, and full bounce‑axis dynamics up to a single Z₂ ambiguity. No global pair memory or noncommuting transport through Bianchi‑IX walls is claimed.