Galois Covers of M23 over Q: A Positive-Dimensional Approach

The article develops a positive-dimensional Hurwitz-space approach to the Mathieu group M23, combining exact braid-orbit reconstruction, admissible-cover geometry, explicit collision tails, arithmetic boundary analysis, and independently executable computational certificates.

Starting from the ordered Nielsen class (2A,2A,3A,5A), the work reconstructs a single 980-state pure-braid orbit and the associated genus-119 Hurwitz curve, then passes through the full 11,760-state ordering action to a degree-2940, genus-43 reduced j-line cover with 406 cusps. The geometry is extended through the connected 21,456-state five-branch component (2A,2A,2A,2A,3A), all six equal-2A width-five collision directions, explicit rational admissible tails, six-branch comparison data, and ten-involution refinements connecting the positive-dimensional construction with the explicit three-point M23 realization of Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang (arXiv:2608.08538).

A central theme is the explicit compatibility between finite combinatorics, boundary geometry, and arithmetic structure. The release includes exact braid words, clutching certificates, factorization-orbit decompositions, reduced monodromy data, real-boundary computations, finite étale target algebras, dual-five and width-11 structures, comparison-morphism data, recorded outputs, and SHA-256 integrity manifests. The finite computational claims may be independently reconstructed from the released source by the accompanying verification suite, while the manuscript proves the geometric deductions from those certified inputs.

The four-, five-, six-, and ten-branch descriptions are developed as parts of one connected geometric picture. Width-five collision strata recover the original four-branch component through admissible-cover clutching. Explicit six-branch refinements connect the two five-branch constructions. At the ten-involution level, exact braid and factorization data connect the positive-dimensional geometry to refinements contracting to the published three-point tuple. The arithmetic analysis further records the real boundary, rational boundary points, reduced j-line behavior, finite étale projector structure, and the descent data supporting the comparison morphism.

The resulting package is intended both as a reproducible computational record and as a geometric companion to the recent resolution of the inverse Galois problem for M23. Rather than presenting only isolated numerical outputs, the release records the surrounding Hurwitz geometry and the exact transitions through which the different branch descriptions, collision strata, arithmetic structures, and comparison constructions fit together.

Verification: from the archive root, bash RUN_ALL_CERTIFICATES.sh replays the public certificate suite from source and compares fresh outputs with the recorded results. sha256sum -c SHA256SUMS.txt verifies file integrity. An optional independent character-theoretic cross-check using GAP/CTblLib is supplied as computations/INDEPENDENT_GAP_CROSSCHECK.g. See ENVIRONMENT.md for the tested environment, STRUCTURE.md for the package map, and README.md for usage.

Licenses: article and documentation — CC BY-NC-ND 4.0; source code — GPL-3.0-only; computational data and recorded outputs — CC BY-NC 4.0. See LICENSES.md for exact scope.