What question did this study set out to answer?

This note aims to investigate the fiber geometry associated with length-four positive PGL2 word products derived from collision matrices.

June 4, 2026Open Access

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

Key Points

This note aims to investigate the fiber geometry associated with length-four positive PGL2 word products derived from collision matrices.
Analyzed the fiber structure using length-four word products in the context of collision matrices.
Characterized the positive coefficient locus in the AAA-chart.
Included symbolic checks and scripts for figure generation to ensure reproducibility.
Demonstrated that the full fiber is a rational curve.
Identified the positive coefficient locus characterized by specific conditions (u>1, v>0, w>uvw).
Clarified that the square-root parametrization is a coordinate artifact, not an intrinsic feature.

Abstract

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

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

Key Points

Abstract

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