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June 4, 20260 citationsOpen Access

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

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HSHiroyuki Shioiri

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.

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Cite This Study

Hiroyuki Shioiri (2026) studied this question.

synapsesocial.com/papers/6a2116fad499ed480b16fe8ahttps://doi.org/10.5281/zenodo.20518224
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