Structural reductions toward the quartic Hessian conjecture in dimension four

Summary

Post-publication scope clarification — 1 September 2026. The first version of this page used “the quartic Hessian conjecture in dimension four” ambiguously and omitted Zixiang Ni's directly relevant preprint. The immutable v0.1.0 research package has not been changed. This page now distinguishes the degree-at-most-four theorem candidate from full $HC_4$, and identifies the release's actual ternary-quintic scope.

The Hessian Conjecture asks when a polynomial potential with constant non-zero Hessian determinant has a polynomial inverse gradient map. In four variables, this page uses full $HC_4$ for the statement with no degree bound:

“Full” therefore means all finite polynomial degrees in dimension four. It does not mean only quartic polynomials.

This release begins instead with a ternary quintic $h_5$. Its binary-decimic and repeated-conic calculations therefore belong to a degree-five branch of full $HC_4$: the first degree beyond Ni's stated theorem, not a competing treatment of the degree-at-most-four case.

This release does not solve full $HC_4$ or the two-dimensional Jacobian Conjecture. It makes one difficult degree-five normal-layer route substantially more explicit.

The work associates a binary polynomial of degree ten—a binary decimic—to the repeated-conic boundary. Exact certificates rule out the case where that decimic has exactly five distinct roots. On one of the two remaining residual orbits, the tangent orbit, all 31 equations defining the required multiplicity-six locus are proved to vanish. On the harder secant orbit, three exact rational colon identities expose structure without proving the full saturation.

The closing test then selects sixteen particularly sparse source directions by a rule fixed before their transfer was computed. All sixteen survive exact lifting and rational reconstruction. They form sixteen explicit rational relations in the fourth Macaulay block. That is a real positive signal—but it is a statement about the source kernel, not yet a solution of the fourth target equation.

Candidate status: Anonymous · AI-assisted · unrefereed · exact computer-assisted structural results and bounded source-kernel theorem · internal editorial review and clean replay passed · no fourth-target membership, secant closure, full $HC_4$ or JC2 claim · Ni's degree-at-most-four result separately remains an unrefereed theorem candidate · no unaffiliated reconstruction, formal verification, external specialist review or peer review.

Summary for specialists

Restrict the ternary quintic normal layer to the Veronese conic $q=xz-y^2=0$. The resulting binary decimic $f_{10}$ controls the repeated-factor boundary of

For exactly five projective support points, the seven multiplicity partitions of ten are exhausted by exact characteristic-zero radical and unit-ideal certificates, excluding every non-zero residual line. On the tangent residual-quadratic orbit, an exact $SL_2$ profile identifies the 31 minimal generators of the coincident-root locus $X_{(6,1,1,1,1)}$. Seven highest-weight radical containments plus stabiliser-span calculations yield

The secant orbit leaves one chart. With $I=(F_1,\ldots,F_{17})$ and

the package gives exact rational quartics $h,h_2,h_3$ satisfying successive memberships

These are colon elements, not generators of the full colon or a proof that $I:M^\infty=(1)$.

For the degree-eight, character-two fourth source block, write the frozen matrix as $[B\mid C]$, with 85,688 rows and 36,587 selected pivot columns. The first sixteen free columns under the preregistered ordering by source support and global coordinate give $C_{16}$. Four shared LinBox solves over $\mathbb F_{173}$ produce a lift through $173^4$. Every one of the 585,392 coordinates reconstructs uniquely within the rational-reconstruction bound, and exact replay gives

over $\mathbb Q$. A separate producer-coordinated implementation bypasses the CSR rows, rebuilds the generator–multiplier products, and expands all sixteen residual polynomials to zero.

What changed

The technical mechanism: why the sixteen-column test matters

The fourth target had already lifted to very high $173$-adic precision, but rational reconstruction failed in the fixed gauge: 8,579 coordinates remained unresolved, and several preregistered recovery architectures failed their exact checks. Simply adding more digits had become a poor experiment.

The new test changes the question. It asks whether the source matrix contains rational directions along which the gauge may be moved. The sixteen columns were chosen only because they had minimum source support under a frozen ordering. Their successful exact reconstruction proves that a small rational kernel slice exists and can now be used in a targeted gauge-action calculation.

It does not prove that any linear combination of those directions makes the fourth target rationally reconstructible. That is the next experiment, not a corollary of this one.

Package map: where to inspect and replay the evidence

The shortest route through the package is:

1. read paper.pdf for the mathematical statements and dependencies;
2. inspect CLAIMS.json for the seven claim units and their scope limits;
3. open the terminal 16-column receipt for the frozen selection, dimensions, timings, hashes and zero-mismatch counts;
4. run python3 verify_package.py for the fast manifest and status checks; and
5. with SageMath available, run python3 verify_package.py --semantic-c16 to rebuild all sixteen rational polynomial identities.

The complete ZIP includes the positive certificates, failed and inconclusive routes, review reports, response matrix, environment record, source bridge, and research metrics. The direct-polynomial audit is implementation-diverse but producer-coordinated; it is not unaffiliated reproduction.

Who should care, and why

The most valuable next projects

1. The rational gauge-action test

Use the exact matrix $T$ with $BT+C_{16}=0$ to write the full sixteen-parameter action on the inhomogeneous fourth target. Derive a rational simplicity or height objective before optimization, then test whether some rational parameter choice yields an exactly reconstructible target representative. A negative result must be tied to a bounded objective class; a positive result must replay over $\mathbb Q$.

2. Close or falsify the secant saturation

The three colon elements show repeated structure but do not determine $I:M^\infty$. A direct saturation certificate, a finite module argument, or a structural countercomponent would be decisive. This remains the geometric bottleneck of the normal layer.

3. Reconstruct independently

The fastest assurance gain is an unaffiliated reconstruction of one load-bearing unit: the seven five-support branches, the seven tangent highest-weight families with stabiliser spans, or the sixteen direct polynomial syzygies. A separately authored implementation is more informative than another run of the supplied scripts.

Limitations and claim boundary

The release proves bounded structural statements inside one pinned degree-five normal-layer programme. It does not prove that the secant residual orbit is contained in the multiplicity-six nullcone, that the fourth target belongs to the relevant ideal over $\mathbb Q$, that the displayed colon elements generate a colon or saturation, that the full family lifts from the normal layer, or that full $HC_4$ or JC2 is true. It also does not supply an independent review of Ni's degree-at-most-four theorem candidate.

Those missing implications are the difference between a promising structural release and a solution.

@misc{hc4fivesupportstructuralreductions2026,
  title        = {Structural reductions toward the quartic Hessian conjecture in dimension four},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.22216400},
  url          = {https://doi.org/10.5281/zenodo.22216400},
  version      = {0.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/hc4-five-support-structural-reductions/}
}