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 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 as the only stage in this family that can host the kind of polynomial map the conjecture is about. Everything else is screened out.

In July 2026 a counterexample to the Jacobian conjecture was announced externally, and the surviving slice is understood to be exactly where it lives. That turns the isolation theorem into an explanation rather than a search: among all the obvious ways to build such a map from binary-form factorisations, the known counterexample sits on the one slice that could work, and every higher-degree variation of the same construction is provably obstructed. The release neither establishes the counterexample nor depends on it — it accounts for why the counterexample exists where it does, and why its natural continuations do not.

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.

That surviving slice is affine three-space, and it is the one now understood to underlie the externally announced July 2026 counterexample to the Jacobian conjecture — the resultant–factor route of NASQRET, discussed by Speyer et al. and digested by Tao. The release does not establish the counterexample and the counterexample does not depend on it; what the isolation theorem adds is the statement that, within this factorisation architecture, the counterexample occupies the unique affine-space case and its higher-degree analogues fail for identifiable geometric reasons. A rigorous proof that the counterexample equals this slice under the stated coordinate transformations is not part of the release and stands as the sharpest open task.

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. That surviving slice is the affine three-space the July 2026 resultant–factor counterexample is understood to inhabit, so the theorem reads as an isolation of that counterexample within the family rather than a search for a hypothetical one. 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

The surviving slice is no longer an open target to be filled or emptied: a counterexample to the Jacobian conjecture was announced externally in July 2026, and it is understood to live exactly there. The sharpest work is now to pin down and independently check the structure around it.

1. Prove the equivalence to the known counterexample

The isolation theorem now rests on identifying the tangent nonosculating linear–quadratic slice with the 2026 resultant–factor counterexample. That identification is an interpretation, not yet a proof. Establishing rigorously that the known counterexample equals this slice under the stated coordinate transformations would move the release's headline reading from plausible to certified — and is the single most valuable thing anyone could do with it.

2. Independently verify the classification

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 tangent-developable classification with classical methods in days. Any of the three outcomes is decisive for the release.

3. Discharge the conditional hypothesis

The isolation theorem 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 an unconditional structural result.

4. Establish novelty against the classical literature

Fragments of the reducibility classification may already exist in the nineteenth- and twentieth-century work on binary forms, apolarity, catalecticants, secant varieties, and tangent developables. A literature-wide priority determination — absent from the release by its own account — is needed before any part of the classification can be called new.

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 isolates, as the unique affine-space stage for constant-Jacobian behaviour, the single slice on which the known Jacobian counterexample lives.

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 Keller map cannot live. This release performs the elimination step. What remains — one specific slice — is now understood to be the home of the externally announced 2026 counterexample.

So the programme's contribution has shifted from searching for a possible counterexample to classifying and explaining the architecture that produced the actual one. Its strongest defensible claim is not that it helped refute the Jacobian conjecture. It is narrower and more structural: the known counterexample may be the unique surviving member of a natural factorisation-space mechanism, while every apparent higher-degree continuation of that mechanism is obstructed. That reading is conditional on the companion classification and unverified against the literature — but it is a sharper and more useful statement than the release's original one, and a legitimate target for a substantive companion paper if the proofs and the novelty survive review.

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://evidencepress.org/releases/reducible-incidence-divisors/}
}