arXiv:2602.05841v2

Generalized Perfect Matrices

Alexander Oertel, Achill Schürmann

math.MGmath.NT11H5552B7011D0911J0490C20

Paper metadata

Original submission: February 5, 2026
Latest revision: June 4, 2026 · version 2
Claimed MSC: 11H55, 52B70, 11D09, 11J04, 90C20
arXiv categories: math.MG, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 19 pages, 3 figures; revised version according to referee comments

Abstract

We generalize Voronoi's theory of perfect quadratic forms to generalized copositive matrices over a full-dimensional closed convex cone K, by introducing a K-copositive minimum and perfect K-copositive matrices. We consider a key feature of a given cone, which we call Interior Ryshkov (IR) property. Under this property the classical theory and its applications generalize nicely and we prove that rationally generated cones possess this IR property. For contrast, we give a detailed example of a simple cone without the IR property, showing various differences to the classical case. Moreover, this example yields connections to questions of number theory, in particular to Diophantine approximation and the Pell Equation. Finally, as an application, we give inner and outer polyhedral approximations for the generalized completely positive cone and a method to find rational certificates for (non-)membership in this cone.

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Audit summary

Audited against arXiv v2

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 18, 2026

01Statements3 reported findingsCorrect

The central existence and characterization results for generalized perfect matrices are correct. The proof of the cone-placement theorem uses the inverse congruence from the one printed, but that repair verifies the theorem without changing its statement. A noncentral introductory aside about completely positive cones needs the local correction recorded below and has no role in the results.

Theorem 2.11Correct

Existence of a perfect form adapted to a prescribed closed cone

Pages 5–7 · Theorem 2.11 · arXiv:2602.05841v2

Choose $U\in\mathrm{GL}_n(\mathbb Z)$ whose positive coordinate cone lies inside the given full-dimensional closed convex cone $K$ , and start with a perfect form $Q$ whose minimal vectors have the nonnegative representatives supplied by the cited copositive result. The arithmetically equivalent form $Q_K=U^{-T}QU^{-1}$ has minimal vectors $U\,\operatorname{Min}(Q)$ and hence places them in $K$ . Integral congruence preserves perfection. This independently verifies the statement; the manuscript prints the opposite congruence, which is recorded under proofs.

Theorems 4.1 and 6.1Correct

Rational cones and the outer characterization

Pages 9–16 · Sections 4 and 6 · arXiv:2602.05841v2

For rationally generated cones, simultaneous rational approximation can be performed inside the cone while retaining linear independence, which supplies the required integral transformations. The duality argument in the outer characterization correctly converts inequalities on cone-restricted minimal vectors into supporting hyperplanes of the generalized Ryshkov polyhedron.

Completely positive cone asideMinor formal correction · no status impact

The claimed equality criterion for $\mathrm{CP}^K$ is false

Page 3 · introductory discussion of $\mathrm{CP}^K$ · arXiv:2602.05841v2

The text says $\mathrm{CP}^K=\mathrm{PSD}$ only when $K=\mathbb R^n$ . If $K$ is a closed halfspace through the origin, for example $K=\mathbb R^{n-1}\times\mathbb R_{\geq0}$ , then every vector $v$ has either $v\in K$ or $-v\in K$ , and $vv^T=(-v)(-v)^T$ . Every positive-semidefinite matrix therefore has a rank-one decomposition using vectors in $K$ , so $\mathrm{CP}^K=\mathrm{PSD}$ although $K\neq\mathbb R^n$ . Deleting or weakening this unused introductory sentence is the complete local repair; no theorem or proof depends on it.

02Proofs2 reported findingsContains incorrect or incomplete proofs

The proof of Theorem 2.11 contains a reversed integral congruence. The theorem is repairable by replacing that congruence with its inverse transpose, but the proof is not correct as printed. The remaining central proof chains are complete.

Proof of Theorem 2.11Incorrect as written

The printed congruence moves minimal vectors in the wrong direction

Pages 6–7 · proof of Theorem 2.11 · arXiv:2602.05841v2

The chosen matrix $U$ satisfies $U\mathbb R_+^n\subset K$ . For the printed form $U^TQU$ , however, the minimal vectors are $U^{-1}\operatorname{Min}(Q)$ , and the construction gives no reason for those vectors to lie in $K$ . Replacing the displayed form by $U^{-T}QU^{-1}$ makes its minimal vectors $U\operatorname{Min}(Q)$ and proves the desired inclusion. This is a verified repair of the proof, so the theorem itself remains classified as correct.

Sections 4–6Correct and complete

Cone approximation and polyhedral duality

Pages 9–16 · arXiv:2602.05841v2

The approximation stays in the relative interiors needed for the cone constraints, the resulting integer matrices remain nonsingular, and the finiteness argument is applied only after normalizing the relevant minima. The supporting-functional argument then establishes both inclusions of the outer description.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2602.05841v2
Authors listed: Alexander Oertel, Achill Schürmann
Audit date: August 18, 2026

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Source: arXiv v2
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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

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...

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Generalized Perfect Matrices

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Existence of a perfect form adapted to a prescribed closed cone

Rational cones and the outer characterization

The claimed equality criterion for CPK\mathrm{CP}^KCPK is false

The printed congruence moves minimal vectors in the wrong direction

Cone approximation and polyhedral duality

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Authenticate with ORCID, then require an exact arXiv match

The claimed equality criterion for $\mathrm{CP}^K$ is false