Positive Image Geometry of Endpoint-Compression Words A Parity-Reachability Theorem for Matrix Word Observables

This preprint studies the positive image geometry of endpoint-compression words in PGL2(R)PGL_2( R)PGL2(R). For the generator family Ma=(0a11)M_a={pmatrix}0&a\\1&1{pmatrix}Ma=(01a1) with positive coefficients, it gives an exact classification of the AAA-chart image regions. After the length-two boundary seed, the image stabilizes by parity: odd lengths L≥3L≥3L≥3 fill u>1, v>0, v<w<uvu>1,\ v>0,\ v<w<uvu>1, v>0, v<w<uv, while even lengths L≥4L≥4L≥4 fill u>1, v>0, w>uvu>1,\ v>0,\ w>uvu>1, v>0, w>uv. The proof uses a one-step recurrence, a determinant parity law, and a constructive backward parametrization. Symbolic, numerical, high-precision boundary, and robustness checks are included.