# Verification report This report is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Problem: AIM-ARITHMETIC_GEOMETRY-0067. Date: 5 September 2026. Title: A Seventeen-Dimensional Component of the Hilbert Scheme of Eighteen Points on an Integral Curve. ## Result and exact scope Theorem 1.1 gives an explicit integral projective curve C and a length-18 subscheme Z with full Hilbert tangent dimension 17. A corrected truncated unit family defines an open A^17 in Hilb^18(C), dense in a rational irreducible component of exact dimension 17. It works over every algebraically closed field. Corollary 1.2 separately applies Kass's theorem in characteristic zero to give a component of dimension d-1 for every d>=18. Characteristic independence of the first theorem is not transferred without proof to that cited corollary. The source question is the existential Problem 20 in the official 2010 AIM Components of Hilbert Schemes booklet, page 3. Integrality and projectivity make the example valid even under the narrower curve conventions left unstated in that question. The known eight-generator numerical semigroup is explicitly credited to Herzog--Kumashiro--Stamate; the canonical data and Kass's general theorem are prior work. A bounded search did not locate this exact Hilbert application. This is not certification of absolute priority. ## Completed verification gates | Gate | Outcome and evidence | | --- | --- | | Original problem and context | PASS: official four-page source, Problem 20 and opening irreducible-component conventions, plus workshop report. | | Source integrity and rights | PASS: 52 retained source/provenance objects, exact coverage and hashes, source-specific rights labels. | | Integral projective curve | PASS: two endpoint charts cover the monomial image; normalization P1; smooth infinity and no extra singularities or embedded points. | | Colength and inverse canonical ideal | PASS: explicit conductor block, complete quotient list of 18 values, and nine coefficientwise colon witnesses. | | Full tangent identification | PASS: localization/completion with finite target; local canonical self-Hom/self-Ext; no fixed-support tangent restriction or false global Ext vanishing. | | Independent assembled core proof | PASS: full source-aligned existence proof read adversarially in `full_assembled_proof_audit.md`. | | Independent exact finite checks | PASS: 683-equation and independently generated 361-equation systems, both with 90 unknowns, rank 73, and identical gap-shift basis. Both copied submission programs rerun byte-identically. | | Complete finite relation bounds | PASS: reduction by t^13 and independent annihilator cutoff at degree 84. No sampled cutoff is treated as a proof. | | All-field rank certificate | PASS: equality graph and 73-edge spanning forest, not extrapolation from modular ranks. | | Flat A17 family | PASS: full-tail truncated preimage, free rank-18 quotient, stabilizer checked over arbitrary base algebras, unique triangular normal form. | | Exact component and rationality | PASS: dimension lower bound 17 plus full tangent upper bound; monomorphism between smooth loci is etale and therefore an open immersion. | | Kass consequence | PASS: g=17, deg omega=32, deg omega dual=-33, d0=34, threshold 18; correct integral/projective and characteristic-zero hypotheses. | | Separate literature review | COMPLETED WITH LIMITS: positive prior inputs credited; no direct prior application found in bounded primary-source and antecedent-text searches. | | Final LaTeX transfer and scope audit | PASS: entire final `main.tex` compared independently with the audited core and unit-family proofs; no substantive gap or scope inflation. | | Compilation and cross references | PASS: Tectonic 0.17.0 and BibTeX 0.99d, eight letter pages, no TeX errors, undefined references/citations, missing glyphs, or overfull/underfull boxes. | | Final visual review | PASS: all eight final pages rendered at 125 dpi and directly inspected by the primary reviewer. No clipped mathematics, broken tables, overlap or unreadable references. | | Standalone upload archive | PASS: source and verification ZIPs extracted into a fresh temporary directory; TeX compiled without repository dependencies; both programs reran byte-identically. Rebuilt PDF has identical text and page content streams on all eight pages. | | Formal proof-assistant verification | NOT PERFORMED: no Lean, Coq, Isabelle, or comparable formal certificate. | | External human peer review | NOT PERFORMED: independent AI-assisted internal lanes are not independent human referees. | | External publication | NOT PERFORMED: no DOI, arXiv identifier or public acceptance is asserted. | The finite certificate proves a finite exact Hom calculation whose completeness is established analytically. It does not replace the written geometric, source or novelty arguments. Conversely, a clean PDF is a typography/build gate, not an additional mathematical correctness theorem. ## Adversarial checks and preserved corrections 1. An embedded-point example was not used to answer the integral-curve version. The final curve is genuinely integral and projective. 2. A low-dimensional punctual stratum alone does not prove existence of a global low-dimensional component. The complete Hilbert tangent bound controls every component through the explicit point. 3. Local canonical self-Ext is not global projective self-Ext. The proof reduces to the origin because the quotient has finite support. 4. Canonical trace is not automatically the colon ideal. The exact inverse ideal is checked coefficient by coefficient against all low values. 5. The algebraic unit family is not the global polynomial product uI: extra zeros of u could create other support. It is the preimage modulo t^52 k[t], with the full tail, and is flat over arbitrary base rings. 6. The stabilizer and normalization are functorial over nonreduced rings. Pointwise injectivity and a characteristic-zero logarithm were not used to infer rationality or eliminate infinitesimal stabilizers. 7. The exact dimension statement uses the flat family's matching lower bound. Tangent dimension alone gives only an upper bound. 8. Degrees of arbitrary torsion-free tensor products were not assumed additive. The global calculation uses only line-bundle twisting. 9. Kass's Hilbert formula is n_d, not d, and his finite representation type assumption belongs to a different theorem. The applied threshold and characteristic restriction are explicitly verified. 10. Known fold/plane special cases and the exact known semigroup were not advertised as discoveries. Failure to find a prior application is distinguished from an exhaustive priority proof. ## Build and visual record The first successful build had one underfull paragraph following a long proof heading. Shortening that heading removed the diagnostic. The final build has zero listed warning/error categories; all pages were rerendered and inspected after the final source change. PDF metadata identifies Alper Ferudun and the complete title, and the affiliation is Mercury Software GmbH. The paper is English throughout. Page review covered: 1 title/author/abstract and construction; 2 theorem, corollary and conductor data; 3 curve and canonical-colon lists; 4 full Hom sequence and global/local warning; 5 flat family and stabilizer; 6 open immersion and Kass degrees; 7 complete syzygy bounds and claim limits; 8 AI disclosure and all five bibliography entries. The separate build record and visual manifest bind these checks to the exact PDF and images. The two added Stacks source downloads initially encountered sandbox DNS restrictions. An authorized network-enabled retry succeeded, and the source manifest was regenerated to cover 52 objects. This environment event was not a LaTeX or mathematical failure. ## Audit trail The permanent dossier `problems/AIM-ARITHMETIC_GEOMETRY-0067` retains `proof.md`, `proof_audit.md`, `full_assembled_proof_audit.md`, `canonical_degree_audit.md`, `unit_orbit_audit.md`, `kass_bridge_audit.md`, `literature_kass_hk_bridge.md`, `manuscript_transfer_audit.md`, `literature.json`, `novelty_search.md`, both exact certificates and all earlier failed/superseded routes. The manuscript-transfer audit records the hash of the actual TeX reviewed. No disagreements about validity or scope remain between these internal reviews. New evidence of prior art or a concrete mathematical gap must still lead to an appropriate revision rather than reliance on this label.