MathAudit

Paper metadata

Original submission
July 5, 2026
Claimed MSC
11H55
arXiv categories
math.NT
Comments
8 pages

Abstract

We prove a new asymptotic lower bound for the number of perfect positive definite quadratic forms in n variables which is close to the optimal one.

AI-generated audit

Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 20, 2026
01Statements3 reported findingsCorrect

The lower bound En≥exp⁡(12n2log⁡n−O(n2))E_n\geq\exp(\tfrac12 n^2\log n-O(n^2)) for inequivalent extremal, hence perfect, positive definite forms is correct. The repeated lattice-subscript mismatch in Lemma 4 is typographical.

Theorem 1Correct

The asymptotic count of extremal forms has the stated exponent

Pages 1–8 · Theorem 1 and its proof · arXiv:2607.04239v1

The construction from the root lattice produces perfect eutactic forms, and quotienting the admissible sign data by the relevant lattice automorphisms costs only an exponential exp⁡(O(n2))\exp(O(n^2)) factor. Stirling's formula then gives the leading term 12n2log⁡n\tfrac12 n^2\log n.

Full paper, version 1 ↗
Lemmas 1–3Correct

The admissible overlattices give perfect extremal forms

Pages 4–6 · Lemmas 1–3 and their corollaries · arXiv:2607.04239v1

The group-theoretic uniqueness in Lemma 1 forces an intermediate lattice of index pp to be AnA_n. Lemma 2 then bounds how many admissible vectors can describe an equivalent form, while Lemma 3 transfers the minimal vectors of AnA_n to the new lattice and proves extremality. The corollary's perfection conclusion follows from the same spanning system of minimal-vector equations.

Lemma 4Typos · no status impact

Three occurrences of one lattice subscript are typos

Page 7 · proof of Lemma 4 · arXiv:2607.04239v1

The proof counts points of p−1Anp^{-1}A_n in the unit ball, but three displayed occurrences write ApA_p. Replacing those three subscripts by nn makes the translated fundamental-parallelepiped argument agree with the statement of Lemma 4. The floor ⌊pn−1/2⌋\lfloor p^{n-1}/2\rfloor in Lemma 5 is already the correct integer count and needs no repair.

02Proofs1 reported findingCorrect

The lattice construction, extremality check, equivalence count, and asymptotic evaluation are correct and complete after the local lattice-subscript repair.

Sections 2–4Correct and complete after the stated repairs

The construction supplies enough inequivalent extremal forms

Pages 2–8 · lattice construction and counting argument · arXiv:2607.04239v1

The minimal-vector equations determine the form, the eutaxy coefficients have the required positivity, and the automorphism bound is small enough on the logarithmic scale. Lemma 5 already uses the required floor for its odd cardinality, and the corrected AnA_n subscript in Lemma 4 makes the volume estimate apply exactly as used.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2607.04239v1
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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