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)\mathrm{PGL}_2(\mathbb{R})PGL2(R) word products from a single observed collision matrix. In the $A$-chart, the full fiber is shown to be a rational curve, with positive coefficient locus exactly characterized by $u>1$, $v>0$, and $w>uv$. 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.