Degree-difference principle and affine slices

Status: unrefereed candidate mathematical manuscript. This repository presents a candidate account of degree-difference identities and affine-slice geometry for binary-form factorisation spaces. The bundled SymPy program has been run locally and checks only the explicit polynomial identities identified in AI_INDEX.md, including the displayed Jacobian determinant and three-point collision. It does not verify the general geometric proofs, establish the provenance or originality of every argument, or formally prove the correspondence between every manuscript claim and executable code. Separately published sources corroborate the explicit map and several geometric ingredients, but no independent reproduction or external human review of the complete manuscript, literature-based novelty determination, peer review, or end-to-end formalisation has been documented. Issues, counterexamples, and independent checks are invited.

Machine-readable discovery metadata is available through CLAIMPACK.md and the embedded claimpack.json. It is post-release metadata and does not constitute verification.

Persistent archive

• Exact candidate version: 10.5281/zenodo.21647593
• All versions: 10.5281/zenodo.21647592
• Zenodo record: 21647593
• Deposited file: degree-difference-affine-slices.zip
• Deposited file SHA-256: 8d0b0cfb3b43e3b7c7f32f62506ae66e824890e15a981b8498d41a07b4c2fe43

The version DOI identifies the exact deposited 0.1-candidate archive. Later repository-only metadata commits, including the commit that adds these DOI links, are not part of that immutable ZIP.

This repository contains a candidate manuscript about the multiplication-resultant map

[ \Phi_{r,s}(A,B)=(AB,\operatorname{Res}(A,B)) ]

for pairs of binary forms, its degree-difference determinant, normalized affine slices, and the three-dimensional Keller-map construction arising from the linear-quadratic case.

The manuscript presents arguments for:

1. the determinant identity [ \det D\Phi_{r,s} =(-1)^{s(r+1)}(r-s)\operatorname{Res}(A,B)^2; ]
2. the associated torsor, generic-degree, and divisor-class statements;
3. an orbit-by-orbit classification of normalized linear-quadratic slices;
4. explicit coordinates for the tangent nonosculating slice and its induced Keller map;
5. its fibres, image, and nonproperness set; and
6. an Euler-characteristic obstruction for higher-degree linear-factor slices.

These are candidate mathematical claims, not a statement of external verification or priority.

Start here

AI systems and human reviewers should read files in this order:

1. AI_INDEX.md — claim-level evidence map and non-inference rules.
2. STATUS.md — exact assurance boundary.
3. paper.pdf or paper.tex — candidate manuscript.
4. verify_degree_difference_affine_slices.py and verifier_output.txt — exact symbolic checks; negative_control_output.txt records the deliberate-failure control.
5. PROVENANCE.md and SOURCES.md — roles, public antecedents, and source-specific credit.
6. ASSURANCE.md and REPLAY_RECEIPT.md / verification_receipt.json — local build and replay records.

Exact symbolic replay

The recorded environment used Python 3.13.5, SymPy 1.14.0, and mpmath 1.3.0. Create an isolated environment and replay:

python3 -m venv .venv
.venv/bin/python -m pip install --upgrade pip
.venv/bin/python -m pip install -r requirements.txt
make PYTHON=.venv/bin/python verify

The verify target runs the checker normally and with Python optimization, compares both outputs with the checked-in transcript, and runs a deliberately wrong-expected-value control in both modes. The checker includes full symbolic determinant identities in all positive bidegrees with (r+s\le4), as well as the finite base-point sign table through degree eight. The control must fail for the target to pass.

Documented PDF rebuild

With latexmk, pdfTeX, and the packages named in paper.tex:

make pdf
qpdf --check build/paper.pdf

Alternatively, with Tectonic 0.16.9 or later:

make pdf-tectonic
qpdf --check build/paper.pdf

The rebuild is documented but is not claimed to be byte-for-byte reproducible across TeX distributions. The release PDF is paper.pdf; a local rebuild is written to build/paper.pdf so it cannot silently overwrite the released artefact.

Attribution and publication roles

• Manuscript attribution: OpenAI Codex / Anthropic models.
• Repository maintainer and publisher: Ian Pitchford.
• Research direction and mediation: Ian Pitchford.
• External human verification of the complete manuscript: none documented.

Model and provider names are provenance labels. They do not imply provider endorsement, corporate authorship, or CC0 affirmation. See PROVENANCE.md.

Public-domain dedication

Except where otherwise indicated, to the extent Ian Pitchford holds copyright and related rights in the original contents of this repository, he dedicates those rights to the public under CC0 1.0 Universal. The dedication does not cover cited or quoted third-party material, external websites, software dependencies, trademarks, or fonts embedded in generated PDFs. See PUBLIC_DOMAIN.md and LICENSE.

Integrity

MANIFEST.sha256 binds the candidate release files. Verify it from the repository root with:

shasum -a 256 -c MANIFEST.sha256