Pure fibres of the four-site matching map
Year: 2026
Type: preprint
Abstract: The four-site matching map sends six pair tensors on four three-dimensional complex vector spaces to the sum of their products over the three perfect matchings. Over a general output its fibre is known to be a single orbit of a three-dimensional torus. We study the fibre over a pure tensor, that is, a nonzero product of four vectors. We give complete parametrizations of this fibre, organized by nine support and rank patterns after quotienting each local space by its target line. The fibre has dimension 19 and exactly twelve irreducible components of that dimension, each isomorphic to A^18 x G_m. On the open subset where all six pair tensors are nonzero, the dimension is 15, again with twelve components of maximal dimension. The corresponding space of three ordered nonzero summands, one decomposable across each of the three pair partitions, has dimension 12. The proof combines a projected-support obstruction, elementary tensor syzygies, and an incidence calculation for isotropic planes. The parametrizations allow zero blocks and drops of whole rank; lower-dimensional components are not classified. Files. paper.pdf is the manuscript (20 pages). paper.tex is its LaTeX source, and atlas.tex is the source of Section 3, input by paper.tex. check_examples.py is the checking program: in exact rational arithmetic it checks 20 examples with rational entries, covering all nine projected support and rank types, each under all 24 permutations of the four sites. These are finite checks of examples; they do not prove that the nine types are exhaustive or that the parametrizations are complete, and they do not check the dimension or component counts. results.json is the program's saved output. It was written on Windows and has CRLF line endings, and the checksum in manifest.json is of that file; a run on Linux writes the same lines with LF endings, hence a file with a different checksum. manifest.json gives the sizes and SHA-256 checksums of the other three program files and the Python version the program was tested with. README.md says how to run the program and what it does and does not establish. FILES.sha256 gives the SHA-256 checksums of the other seven files, with the program's files under the folder reproducibility/. Prior work. The closest earlier results are credited in Section 1 and in the references of the paper. Disclosure. The author directed the project, takes responsibility for the paper, and used OpenAI Codex to produce the results, proofs, checking program and text, and Anthropic Claude to check and revise the paper; the paper's closing section "Proof artifacts and AI use" gives the details. The attribution to OpenAI Codex rests on the project's records and the author's account; which model versions were used at each step has not been verified. The author has not verified the mathematics; no human has checked the proofs, and no human peer review or formal proof-assistant verification is claimed. No AI system is an author. The author publishes under a pseudonym. All checks described in that section were made by AI agents; none of them is journal peer review, a formal verification or a certificate of originality or priority. Corrections are welcome at [email protected].
Author: Ignoramus
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225780005
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