Press release · 1 September 2026 · version 0.1.0-candidate
Structural reductions toward the quartic Hessian conjecture in dimension four
Exact five-support and tangent-orbit reductions, a three-step secant colon chain, and sixteen rational fourth-block source syzygies sharply narrow one HC4 normal-layer programme without proving HC4 or JC2.
Summary
The Jacobian Conjecture asks when a polynomial map with constant non-zero Jacobian has a polynomial inverse. One influential reformulation studies special homogeneous polynomials whose Hessian matrices are nilpotent. The quartic case in dimension four—HC4—is an early unresolved boundary of that programme.
This release does not solve HC4 or the two-dimensional Jacobian Conjecture. It makes one difficult 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, HC4 or JC2 claim · 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
| part of the programme | exact contribution | still missing |
|---|---|---|
| five-support boundary | all seven exactly-five-root partitions excluded | six-or-more support and full double-conic packet |
| tangent residual orbit | all 31 multiplicity-six target generators vanish | transfer or separate proof on the secant orbit |
| secant chart | three successive rational localized-colon elements | full colon equality or saturation |
| fixed fourth target | congruence lift through $173^{96}$ | rational target identity |
| sixteen-column test | sixteen exact rational source syzygies | a gauge action that simplifies the inhomogeneous target |
| global HC4 programme | sharply smaller normal-layer frontier | polynomial-level and full-family closure |
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:
- read
paper.pdffor the mathematical statements and dependencies; - inspect
CLAIMS.jsonfor the seven claim units and their scope limits; - open the terminal 16-column receipt for the frozen selection, dimensions, timings, hashes and zero-mismatch counts;
- run
python3 verify_package.pyfor the fast manifest and status checks; and - with SageMath available, run
python3 verify_package.py --semantic-c16to 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
| likely audience | what should interest them | useful next action |
|---|---|---|
| researchers on Hessian-nilpotent and Jacobian problems | a repeated-factor boundary split into exact support and residual-orbit statements | check the normal-layer reductions and attack the secant orbit |
| classical invariant theorists | explicit use of the binary-decimic multiplicity-six locus and its seven highest-weight families | reconstruct the radical containments or find a conceptual proof |
| computational algebra researchers | a large sparse exact system turned into a small rational kernel slice with fail-closed p-adic recovery | independently reimplement the C16 audit or improve the gauge objective |
| formalisation researchers | sharply indexed statements, exact certificates and explicit missing implications | formalise one support partition, the tangent stabiliser step, or the source-kernel identity |
| AI-assisted mathematics researchers | prospective selection, measured compute, preserved failures and explicit assurance boundaries | study or challenge the claim–evidence architecture |
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 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 HC4 or JC2 is true.
Those missing implications are the difference between a promising structural release and a solution.
Searchable mathematical objects
Exact text is provided alongside typeset mathematics so people and automated research tools can find and compare the objects without inferring a stronger assurance state.
Five-support exclusion
- Exact searchable text
No clean nonzero binary-decimic restriction with exactly five projective roots satisfies the frozen double-conic normal-layer equation in characteristic zero.- Object type
- statement
- Relation to release
- claimed-result
- Scope
- Exactly five support points on the clean double-conic normal layer; computer-assisted exact certificates.
Tangent nullcone containment
- Exact searchable text
All 31 generators of the multiplicity-six binary-decimic locus lie in the radical of the tangent normal-layer ideal.- Object type
- statement
- Relation to release
- claimed-result
- Scope
- Tangent residual orbit only; no secant transfer.
Sixteen rational source syzygies
- Exact searchable text
For the frozen fourth block there is a rational matrix T satisfying B T plus C16 equals zero.- LaTeX
BT+C_{16}=0- Object type
- identity
- Relation to release
- claimed-result
- Scope
- Sixteen selected source columns only; not the inhomogeneous fourth target.
Secant fourth-target obstruction
- Exact searchable text
Prove rational fourth-target membership or a full secant colon or saturation statement.- Object type
- obstruction
- Relation to release
- open-problem
- Scope
- Remaining secant chart; required before normal-layer closure, HC4, or JC2.
Media
The audio briefing is provided in the header above. Download the MP3 briefing · read the transcript.
Open directions for follow-up research
Also available in machine-readable form for research agents and follow-up projects.
- Use the exact 16-dimensional rational source-kernel slice to derive and test a rational gauge action on the inhomogeneous fourth target.
- Prove or falsify full colon equality or saturation on the remaining secant chart.
- Transfer multiplicity-six nullcone containment from the tangent residual orbit to the secant residual orbit.
- Close the six-or-more-support and polynomial-level lifting gates required for the global quartic Hessian programme.
- Obtain unaffiliated specialist reconstruction and a separately authored implementation while preserving corrections as versioned successors.
Verification status
Anonymous, AI-assisted, unrefereed computer-assisted structural-reductions candidate. The exact five-support, tangent-orbit, colon-element and 16-column source-kernel claims have producer-side rational or characteristic-zero replay. The direct-polynomial audit bypasses the sparse row encoding but remains producer-coordinated. The 16-column selection rule was frozen before transfer computation. A fixed-gauge fourth target lifts through 173^96, but rational recovery failed under the preregistered hierarchy; this is not evidence of non-membership. No unaffiliated rerun, independent reimplementation, proof-assistant formalization, external specialist review, journal peer review, absolute priority determination or impact evaluation has occurred. The release does not prove HC4 or JC2.
Cite
BibTeX
@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/}
}Also: cite.bib · paper.json · this page as Markdown