Generic Two-Chart Cover and the Coefficient-Collapse Boundary Nondegenerate Tail Fibers for Endpoint-Compression Words

This preprint proves a generic two-chart cover theorem for endpoint-compression word fibers in PGL2(R)PGL_2( R)PGL2(R). Building on GMA6–GMA8, it shows that, on the nondegenerate tail locus and away from the visible image-side walls u=1u=1u=1 and uv=wuv=wuv=w, the first fiber chart boundary Δ=0Δ=0Δ=0 and the second-chart pole locus do not meet on the generic AAA-chart fiber. The coefficient-side collapse det⁡N=0 N=0detN=0 is identified as a rank-one boundary rather than a new generic interior component. The result is set-theoretic and generic; special image-wall fibers and embedded components are left open. Symbolic verification scripts reproduce the Groebner/saturation checks.