--- title: "Structural reductions toward the quartic Hessian conjecture in dimension four" date: 2026-09-01 version: "0.1.0-candidate" doi: 10.5281/zenodo.22216400 pdf: https://github.com/ipitchford/hc4-five-support-structural-reductions/releases/download/v0.1.0-candidate/paper.pdf repository: https://github.com/ipitchford/hc4-five-support-structural-reductions archive: https://zenodo.org/records/22216400 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Structural reductions toward the quartic Hessian conjecture in dimension four ## Summary The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 $$ \det\operatorname{Hess}(h_5)=q^4\ell. $$ 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 $$ N_6\subseteq\sqrt{I_{\mathrm{tan}}}. $$ The secant orbit leaves one chart. With $I=(F_1,\ldots,F_{17})$ and $$ M=f_9f_{10}(2f_9^2+5f_{10}g_0), $$ the package gives exact rational quartics $h,h_2,h_3$ satisfying successive memberships $$ h\in I:M,\qquad h_2\in(I,h):M,\qquad h_3\in(I,h,h_2):M. $$ 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 $$ BT+C_{16}=0 $$ 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: 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 | 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 ### 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. ## Open directions for follow-up research - 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. ## Research process, metrics and reusable methods This is prospective process metadata under the Evidence Press operating model and research-metrics policy. It records the intended handoff, measured scope and claim boundary; it is not evidence that the method accelerated this work. - Work ID: ep-work:hc4-five-support-structural-reductions - Attempt and metric receipts: ep-attempt:hc4-five-support-structural-reductions: published / partial; scope research-through-publication; target Complete and replay the preregistered 16-column test, then publish an exact claim-bounded HC4 structural-reductions candidate with deterministic package, five-role internal review, immutable archive identity, reader-first media, composite sealing, hosted CI, guarded zero-cost deployment and exact canonical readback.; active forecast 390 minutes (240-600); Fermi components 16-column system export, modular solve, replay and first p-adic lift gate: 1 x 25/50/90 minutes low/central/high (The nearest 114-column canonical-section solve took about 189 seconds, but the row-normalization export and independent replay are new engineering surfaces.); Evidence recovery, theorem-boundary synthesis and manuscript/package construction: 1 x 70/120/190 minutes low/central/high (The frozen campaign is extensive and exact but distributed across receipts; one concise manuscript must reconstruct dependencies without upgrading bounded evidence into HC4.); Five-role internal editorial review, one repair batch and confirmation: 1 x 45/80/130 minutes low/central/high (The candidate has several adjacent claim boundaries and one new experiment, making scientific scope review more demanding than a short finite theorem.); Public GitHub repository, deterministic archive, Zenodo identity and cross-service byte readback: 1 x 35/55/85 minutes low/central/high (Established automation is available, but a new repository, DOI-bearing rebuild and immutable asset comparison are required.); Evidence Press authoring, provenance-bound media, operating records, composite seals, hosted CI, deployment and readback: 1 x 65/85/105 minutes low/central/high (Recent releases use stable generators and deployment tooling, while this new slug still requires method assignment, audio, art, Open Graph image, thumbnail and two append-only ledger closures.); positive-signal/closure probabilities 0.72/0.6 within 600 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 75/unknown/8/16/0/10; cycles positive/negative/inconclusive 1/0/0; falsification gates 4; architectures tested/rejected 1/0; result target-closed; target reached true; forecast error -315 minutes; ratio 0.19; inside interval false; positive-signal/target-closure Brier scores 0.0784/0.16; missing telemetry activeHumanMinutes: No instrument captured human direction, mediation or review time at the prospective research-through-publication boundary.; deduplicatedModelTokens: No active fork-aware goal counter exposed exact task-local model-token usage for the complete prospectively registered interval, so tokens are not reconstructed from conversation context.; uncachedInputTokens: The runtime does not expose an uncached-input token counter.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: Retain measurement.agentRuns as the observed zero in the first public snapshot and record four audited root-and-helper agent runs, with maximum parallelism four, in metrics.outcome.agentRuns and metrics.outcome.maxParallelAgents. (reason: The first live active-attempt snapshot recorded agentRuns as zero before the terminal task-tree audit. The immutable snapshot cannot be silently replaced, while a terminal research-metrics outcome requires a positive audited count.). Work ledger: https://evidencepress.org/api/work-ledger.json. Metrics policy: https://evidencepress.org/api/research-metrics-policy.json - Intended aims: science - Artifact roles: research-output, evidence-assessment, method-demonstration, communication - Decision object: executable-open-problem — A sharply bounded secant fourth-target and rational gauge-action research handoff following exact five-support, tangent and source-kernel reductions. Scope: The clean normal-layer five-support theorem, tangent nullcone containment, three secant colon elements, fixed-gauge recovery failure and sixteen rational source syzygies; not fourth-target membership, secant closure, HC4 or JC2. - Reusable methods: Certificate-first, proof-carrying research (certificate-first); Structural compression (structural-compression); Adversarial scientific controls (adversarial-controls); Productive failure and stop receipts (productive-failure); Assurance as a vector (assurance-vector); Agent-readable research objects (agent-readable-research-object). Registry: https://evidencepress.org/api/method-registry.json - Targeted clocks: discovery, assurance, publication - Semantic bridge: explicit — The package maps the quartic Hessian normal layer through binary-decimic support and residual-orbit geometry to an unresolved secant saturation, then isolates an exact rational 16-direction source-kernel slice for the next gauge-action test. Remaining risks: The normal-layer reductions have not been reconstructed by an unaffiliated specialist.; Source-kernel freedom may not move the inhomogeneous fourth target into a rationally simple gauge.; Tangent containment need not transfer to the secant residual orbit.; Normal-layer closure would still require polynomial-level and full-family lifting.. - Human judgement gates: Check the invariant-theoretic and normal-layer reductions as mathematics rather than inferring them from successful replay.; Keep source syzygies separate from inhomogeneous target membership and colon elements separate from colon equality.; Keep tangent-orbit containment separate from the unresolved secant orbit.; Require external specialist assessment before any HC4, JC2, first, priority or field-acceptance language. - Next assurance action: Obtain an unaffiliated reconstruction of the tangent and secant reductions, then use the exact 16-direction kernel slice to prove or decisively falsify rational fourth-target membership. - Claim ceiling: An anonymous unrefereed computer-assisted structural-reductions candidate with exact characteristic-zero and rational certificates, a prospectively frozen 16-column source-kernel theorem, fail-closed negative routes and internal editorial closure; not fourth-target membership, secant closure, HC4, JC2, independent validation, formal verification, external specialist or editorial peer review, absolute priority or demonstrated workflow impact. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Faster or more reliable resolution and assurance of difficult algebraic conjectures in AI-assisted invariant theory, exact algebra and p-adic reconstruction around a quartic Hessian programme; design none; comparator No matched conventional research, theorem-review or publication workflow was registered.; estimand No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake or citation was estimated.; no real-world effect evidence asserted ## 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. ## References 1. Zhao, W. (2004). Hessian Nilpotent Polynomials and the Jacobian Conjecture. 2. van den Essen, A., & Zhao, W. (2007). Two Results on Homogeneous Hessian Nilpotent Polynomials. 3. Brouwer, A. E., & Popoviciu, M. (2010). The invariants of the binary decimic. 4. Chipalkatti, J. V. (2003). On equations defining coincident root loci. 5. Lee, H., & Sturmfels, B. (2016). Duality of Multiple Root Loci.