The Fiber Variety at All Lengths Closed-Form Fibers and a Length-Parity Law for Endpoint-Compression Words

This preprint studies the fiber variety of continuous endpoint-compression words in PGL2(R)PGL_2( R)PGL2(R). For the generator family represented by matrices Ma=(0a11)M_a={pmatrix}0&a\\1&1{pmatrix}Ma=(01a1), it gives a closed-form rational chart for the fiber over a generic AAA-chart image at every flexible length L≥4L≥4L≥4, with generic fiber dimension max⁡(0,L−3)max(0,L-3)max(0,L−3). The length-five case is worked out explicitly as a rational surface, including the positive-coefficient locus and the nonemptiness criterion v<w<uvv<w<uvv<w<uv. A parity law shows that sign⁡(w−uv)=sign⁡((−1)L∏iai)sign(w-uv)=sign((-1)^L∏_i a_i)sign(w−uv)=sign((−1)L∏iai), so positive words alternate determinant sign by length. Symbolic and numerical scripts verify the closed-form identities, positive-locus criterion, mixed-sign parity law, fiber-dimension law, and figures.