Reducible incidence divisors and the isolation of affine slices in binary-form factorisation spaces

Summary

When does a naturally defined geometric object break into pieces? This paper answers that question exactly, for a family of hypersurfaces that arise when you track where two polynomials share a root.

The answer has a classical elegance. The hypersurfaces in the family are indexed by linear functionals, and the paper shows that a divisor in the family is reducible — breaks into components — precisely when its functional lies on a specific ruled surface: the tangent developable of the rational normal curve, the sweep of all tangent lines to the most fundamental curve in projective geometry. On that surface, the divisor splits into two or three identifiable pieces; off it, the divisor stays whole.

The second half of the manuscript is a process of elimination aimed at one of algebra's oldest open questions, the Jacobian conjecture. It shows that a large family of candidate geometric stages are never ordinary affine space — each is blocked by a precisely computed defect — and, assuming a result from a companion release, isolates exactly one slice that could in principle host the kind of polynomial map the conjecture is about. Everything else is screened out.

Summary for specialists

For consecutive degrees $(m, m+1)$ with $m \geq 2$, the manuscript classifies reducibility in the marked-common-root incidence system: $D_\ell = \{\ell(P^2 A'B') = 0\}$ is reducible precisely when $[\ell]$ lies on the tangent developable of the rational normal curve of evaluation functionals. Rank-one functionals give three reduced components, first-jet functionals two; genuine two-point secants and higher catalecticant ranks give irreducible divisors.

For the associated slices, the results are: $X_\ell^{m,m+1} \not\cong \mathbb{A}^{2m+1}$ for all $m \geq 2$ and nonzero $\ell$, with precise defects — Grothendieck class $\mathbb{L}^{2m+1} - \mathbb{L}^{2m}$ in the rank-one case, diagnostic Hodge coefficients $-2$ and $-1$ for rank-two secants and first-jets. Non-adjacent degrees ($|r-s| \geq 2$) yield non-contractible slices via finite cyclic actions. Conditionally on the upstream cubic classification from the companion exotic-spheres release, the tangent nonosculating linear–quadratic slice is the unique positive-bidegree affine source permitting a nonzero constant Jacobian determinant.

The proofs are conventional algebraic geometry — catalecticant stratification, rational normal curves, Hodge–Deligne polynomials, cyclic group actions — with deterministic SymPy audits of selected consequences rather than a certificate architecture. The conditional isolation theorem should be read with its hypothesis in full view: the upstream classification it leans on is itself an unrefereed candidate.

Technical summary

The proofs are conventional algebraic geometry throughout — this is the trilogy's least computational instalment. The classification theorem works in the linear system $\{\ell(P^2 A'B') = 0\}$ on the space of marked factorisations of binary forms of consecutive degrees: the functionals $\ell$ are stratified by catalecticant rank, the rational normal curve of evaluation functionals and its tangent developable are identified inside the dual space, and reducibility of $D_\ell$ is shown to occur exactly on that developable, with component counts (three for rank-one, two for first-jet, one otherwise) read off from the stratification. The non-isomorphism results compute exact classes in the Grothendieck ring — the rank-one defect is $\mathbb{L}^{2m+1} - \mathbb{L}^{2m}$ — and expand Hodge–Deligne polynomials whose coefficients at diagnostic positions ($-2$ for rank-two secants, $-1$ for first-jets) are incompatible with affine space. Non-adjacent degree pairs ($|r-s| \geq 2$) are handled separately: finite cyclic group actions on the slices obstruct contractibility, which is weaker than the motivic defect but suffices for non-isomorphism.

The isolation theorem stacks these exclusions: given the upstream cubic classification (a result of the companion exotic-spheres release, itself unverified), every positive-bidegree slice except the tangent nonosculating linear–quadratic one is eliminated as a source for maps with nonzero constant Jacobian determinant. Deterministic SymPy audits (verify_paper.py, verify_rank_two.py) check selected numerical consequences — component counts and defect coefficients in low degree — but the theorems themselves are prose proofs, and the release's internal review disposition ("minor revision for candidate publication") is recorded in the repository.

Who should care, and why

The most valuable next projects

1. Test the classification against classical knowledge

The central theorem lives in well-mapped territory: rational normal curves, their tangent developables, catalecticant stratifications. A specialist can likely confirm, refute, or antedate the classification with classical methods in days. Any of the three outcomes is decisive for the release.

2. Discharge the conditional hypothesis

The isolation theorem — the screening programme's payoff — is conditional on the cubic classification from the companion exotic-spheres release, which is itself unrefereed. Verifying that upstream classification (or reproving the isolation theorem without it) would convert a conditional statement into the programme's first unconditional structural result.

3. Resolve the surviving slice

The programme has narrowed its search to a single named slice. The sharpest possible follow-up is binary: construct a nonzero-constant-Jacobian map on the tangent nonosculating linear–quadratic slice, or prove none exists. Either outcome closes a branch — and an exclusion would retire this family as a source of Jacobian-conjecture candidates altogether.

Specialist audience candidates

The natural specialist readers are projective and affine algebraic geometers working on secant varieties, catalecticants, and the geometry of rational normal curves; motivic-obstruction specialists; and Jacobian-conjecture researchers following structured screening programmes. This identifies intellectual proximity, not a prediction of endorsement.

The strongest pitch to them is:

A natural incidence family turns reducible exactly on the tangent developable of the rational normal curve — and the same analysis eliminates every slice but one as a stage for constant-Jacobian behaviour.

The screening programme

Across this trilogy of releases, the strategy is consistent: rather than attacking the Jacobian conjecture head-on, map the geography of factorisation-space slices and eliminate, with exact obstructions, every stage where a counterexample or interesting Keller map cannot live. This release performs the elimination step. What remains — one specific slice — is now a sharply posed target: construct the map there, or exclude it and close the branch.

What is in the evidence package

The deposit contains the manuscript (PDF and TeX), deterministic audit scripts (verify_paper.py, verify_rank_two.py), a claim-to-evidence map (AI_INDEX.md/.json), assurance-boundary documents (STATUS.md, ASSURANCE.md), SHA-256 manifests, and immutable source snapshots, archived on Zenodo as v1.0-candidate with a v1.0.1 revision (same-day metadata and attribution fixes).

@misc{reducibleincidencedivisors2026,
  title        = {Reducible incidence divisors and the isolation of affine slices in binary-form factorisation spaces},
  author       = {OpenAI Codex 5.6 Sol and Anthropic Fable 5},
  year         = {2026},
  doi          = {10.5281/zenodo.21647616},
  url          = {https://doi.org/10.5281/zenodo.21647616},
  version      = {1.0-candidate (v1.0.1 archived as 10.5281/zenodo.21653119)},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidence-press.pages.dev/releases/reducible-incidence-divisors/}
}