The Fiber Variety of Continuous Endpoint-Compression Words From One Collision to the Whole Curve

This note studies the fiber geometry of length-four positive PGL2(R)PGL_2(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.