--- title: "Reducedness of the Hilbert scheme of eight points in affine four-space" date: 2026-09-07 version: "0.1.0-candidate" doi: 10.5281/zenodo.22643443 pdf: https://github.com/ipitchford/hilbert-eight-points-reducedness/releases/download/v0.1.0-candidate/paper.pdf repository: https://github.com/ipitchford/hilbert-eight-points-reducedness archive: https://zenodo.org/records/22643443 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Reducedness of the Hilbert scheme of eight points in affine four-space ## Summary Eight points can collide in ways that ordinary pictures cannot distinguish. The Hilbert scheme records not only their positions but also algebraic structure at collisions. This unrefereed proof candidate claims that, for eight points in four-dimensional affine space over a characteristic-zero algebraically closed field, the Hilbert scheme has no hidden nilpotent structure. It remains singular and reducible: reduced does not mean smooth. ## Summary for specialists The claimed theorem is reducedness of $\operatorname{Hilb}^{8}(\mathbb A^4_k)$ for algebraically closed $k$ of characteristic zero. The candidate reduces the question to two monomial local rings, separates the translated homogeneous component by exact annihilator identities, and identifies the full associated graded algebras of their Pfaffian quotients. No positive-characteristic or general-length assertion is included. ## Technical account The closed support of the nilradical, a Borel fixed-point argument and sixteen explicit triangular coordinate frames reduce the global question to B15 and B16. Original commutator identities and an ambient equivariant Grassmannian projection establish local component separation. For B16, an integral open chart and a rank-1907 finite-module comparison produce an integral core. B15 requires two open charts and the corrected parameter $\ell_1=c_7+c_{79}$. A zero fiber modulo 32003 gives an upper bound; a Cohen–Macaulay, flat comparison fiber modulo 65521 and its completion give the matching lower bound 9923. Regularity of that parameter closes the remaining tangent-cone kernel in every degree. This is not an inference from the last computed degree. The antecedent component and Pfaffian descriptions are due to Cartwright, Erman, Velasco and Viray. Hu supplies important local-structure and reduction methods. Their results are inputs; the candidate's claimed contribution is the source-specific reducedness argument, not rediscovery of those inputs. ## Evidence, assurance and limitations The complete package includes exact ring presentations, parameter forms, coordinate maps, written proof modules, all four finite bases, native software-diverse counterparts and direct coefficient-comparison records. Those records cover 55,557,191 terms. The public aggregate replay checks saved receipts and their exact dependencies; it does not silently recompute all large bases. Original code, deeper replay instructions and complete outputs allow inspection beyond the aggregate. The supplied review saw the overview and elementary checks, not the full archive, owing to an upload limit. Internal model editorial review is a different, producer-coordinated process. Neither establishes unaffiliated whole-proof validation. Formal verification, exhaustive novelty clearance and independent external reproduction remain unestablished. ## Who should care, and why | Audience | Potential use | Required caution | |---|---|---| | Hilbert-scheme specialists | Inspect a proposed answer to a scheme-structure question beyond component classification. | Scrutinize the source-specific identifications and all-degree argument. | | Computational algebraists | Reproduce large finite fibers and inspect a mixed-prime finite-module bridge. | Software agreement and hashes do not replace the source-to-claim argument. | | Research-tool builders | Reuse explicit replay levels and semantic corruption controls. | Workflow assurance is not mathematical truth or measured productivity. | ## Why the problem matters Knowing the irreducible components of a space does not determine its scheme structure. Nilpotents can record infinitesimal information invisible in its underlying set. This candidate targets that distinction in a small but already reducible Hilbert scheme. Its proposed finite-to-global method may also be useful elsewhere, provided every hypothesis is proved anew. ## How to inspect or reproduce the recorded checks Download and extract the full versioned archive. Read the manuscript and README, then run `python3 verify_release.py`. Use `--manifest-only` for a byte-integrity check. The wrapper keeps assertions enabled in its legacy children even if invoked with `python -O`. Follow the detailed instructions in a disposable copy for new direct comparisons or native Gröbner runs; those runs regenerate timed receipts and must not overwrite the frozen record. ## The most valuable next projects - Obtain unaffiliated scrutiny of the ambient Pfaffian transport and the B15 purity/open-coverage argument. - Independently rebuild the source-to-fiber encoding and recompute the full bases in a documented environment. - Explore other applications of the mixed-prime lemma without transferring its flatness or dimension hypotheses by analogy. ## What is in the evidence package The manuscript, original proof modules, exact source code and full finite outputs are accompanied by structured claims, an assurance map, a review response, internal editorial reports, environment instructions, negative controls, licences and a complete hash manifest. Immutable GitHub and Zenodo assets are the reference versions; the page and audio are communication aids. ## Open directions for follow-up research - Unaffiliated scrutiny of ambient Pfaffian transport and B15 purity/open coverage. - Independent reconstruction of source encodings and full native computations. - Investigate positive characteristic or other lengths only as new problems with their hypotheses checked anew. ## 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:hilbert-eight-points-reducedness - Attempt and metric receipts: ep-attempt:hilbert-eight-points-reducedness-assurance-publication: published / positive; scope assurance-through-publication; target Review-repaired candidate, exact archive replay, internal editorial gate, immutable public assets and canonical Evidence Press readback, or a preserved integrity blocker.; active forecast 150 minutes (120-210); Fermi components Manuscript and source revision: 1 x 35/45/65 minutes low/central/high (Existing full evidence; substantive exposition repairs.); Package replay and editorial gate: 1 x 35/45/65 minutes low/central/high (Large frozen bases, one five-role internal gate.); Public identities and media: 1 x 25/30/40 minutes low/central/high (Established publication tools.); Seals and canonical readback: 1 x 25/30/40 minutes low/central/high (Two new-slug deployment cycles.); positive-signal/closure probabilities 0.9/0.8 within 240 active minutes; observed active-agent/human/compute/wait/blocked/rework minutes 19/unknown/unknown/3/0/0; cycles positive/negative/inconclusive 0/0/0; falsification gates 1; architectures tested/rejected 0/0; result target-closed; target reached true; forecast error -131 minutes; ratio 0.12666666666666668; inside interval false; positive-signal/target-closure Brier scores 0.01/0.04; missing telemetry activeHumanMinutes: No contemporaneous human active-time instrument.; computeMinutes: No complete scoped substantive-computation wall clock; ordinary CI/build time is not research compute.; appended measurement corrections measurement.agentRuns -> metrics.outcome.agentRuns: Terminal receipt records six agent runs; original intake snapshot retained. (reason: Prospective snapshot predates five completed internal reviewers.); measurement.reworkMinutes -> metrics.outcome.reworkMinutes: See outcome varianceReason: actual replay-instruction, audio and governance repairs are disclosed; zero is not an absence-of-rework claim. (reason: Intake snapshot was unknown; terminal timing records an explicitly incomplete zero instrumented lower bound.). 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, communication - Decision object: certificate — Source-specific local reducedness proof candidate and exact finite computational evidence. Scope: Hilb^8(A^4_k), algebraically closed characteristic-zero k; two local models B15 and B16. - 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: assurance, publication - Semantic bridge: explicit — Original border-basis equations, ambient equivariant projection, component separation and all-degree graded arguments connect finite computations to the claimed reducedness statement. Remaining risks: Written mathematical bridges require unaffiliated specialist scrutiny.; Aggregate replay checks saved records, not every large computation afresh.; Finite-field identities alone do not prove characteristic-zero reducedness.; Novelty and priority search is bounded.. - Human judgement gates: Check global-to-local reduction and ambient transport.; Check purity, open coverage and mixed-prime freeness hypotheses.; Preserve source attribution, component rights and assurance boundaries. - Next assurance action: Unaffiliated scrutiny and independent source-to-computation reconstruction. - Claim ceiling: Unrefereed, author-claimed complete computer-assisted proof in characteristic zero only. No external whole-proof validation, formal verification, exhaustive novelty clearance or priority determination. - Aim-scoped impact evidence: - science: NO_IMPACT_EVIDENCE — Inspectable reducedness proof candidate in Producer-coordinated mathematical publication; design none; comparator No matched workflow comparator.; estimand No discovery-speed, effort, reuse or impact effect estimated.; no real-world effect evidence asserted - Parent handoffs: depends-on-claim https://doi.org/10.2140/ant.2009.3.763; inherited claim: Component and Pfaffian descriptions for eight-point Hilbert schemes.; inherited ceiling: Established antecedent is not validation of this candidate reducedness argument. ## Verification status Unrefereed, author-claimed complete computer-assisted proof in characteristic zero only. No external whole-proof validation, formal verification, exhaustive novelty clearance or priority determination. ## References 1. Cartwright, Erman, Velasco and Viray (2009). Hilbert schemes of 8 points. Component and Pfaffian antecedents. 2. Hu (2025). On singular Hilbert schemes of points: Local structures and tautological sheaves. Corrected reduction and local-structure methods. 3. Kustin. Generic matrix-vector Northcott perfectness input, introduction (0.1)–(0.2); see the citation audit for the precise application boundary. 4. The Stacks Project, Tag 00R4. Miracle flatness.