This note studies the fiber geometry of length-four positive PGL2 (R) PGL₂ (R) PGL2 (R) word products from a single observed collision matrix. In the AAA-chart, the full fiber is shown to be a rational curve, with positive coefficient locus exactly characterized by u>1u>1u>1, v>0v>0v>0, and w>uvw>uvw>uv. The work also clarifies why the square-root parametrization appearing in an earlier GMA4 formulation is a coordinate artifact rather than an intrinsic feature of the fiber. Symbolic checks and figure-generation scripts are included for reproducibility.
Hiroyuki Shioiri (Wed,) studied this question.