arXiv:2606.03472v2

Planes in quadratic 4-space and associated shapes of lattices

Menny Aka, Konstantin Andritsch, Andreas Wieser

math.NT37A1711E1611G1511H55

Paper metadata

Original submission: June 2, 2026
Latest revision: August 17, 2026 · version 2
Claimed MSC: 37A17, 11E16, 11G15, 11H55
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 38 pages

Abstract

Let $Q=-x_1^2-x_2^2-x_3^2+x_4^2$ be the standard signature $(1,3)$ quadratic form. To each non-degenerate rational plane $L$ in the four-dimensional quadratic space $(\mathbb{Q}^4,Q)$ we can naturally attach a periodic geodesic on the Bianchi orbifold $\mathrm{SL}_2(\mathbb{Z}[i])\backslash \mathbb{H}^3$ which records the position of $L$ in the Grassmannian up to integer rotations. Moreover, each such plane $L$ defines a CM point and a periodic geodesic on the modular curve through restriction of $Q$ to $L$ and its orthogonal complement. Lastly, the local isomorphism between $\mathrm{SO}_{1,3}(\mathbb{R})$ and $\mathrm{SL}_2(\mathbb{C})$ gives rise to a further periodic geodesic on the Bianchi orbifold. In this article, we exhibit a natural coupling of all the above objects and prove simultaneous equidistribution under a Linnik-type splitting condition. The main ingredient is the classification of joinings of higher-rank diagonalizable actions on homogeneous spaces due to Einsiedler and Lindenstrauss.

AI-generated audit

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

01Statements2 reported findingsContains unsupported statements

The simultaneous equidistribution theorem under the square-free, odd, and auxiliary splitting hypotheses is verified. The separately stated Duke-type theorem for all nonsquare discriminants is not proved in the manuscript and is not obtained by projecting the simultaneous theorem, whose range of discriminants is strictly narrower. No counterexample to that Duke-type statement was found.

Theorem 1.2Correct

Simultaneous equidistribution of the four coupled objects

Pages 5 and 25–36 · Theorems 1.2 and 4.3 · arXiv:2606.03472v2

The adelic torus packet projects to the claimed coupled measure through Propositions 4.4 and 4.7. The individual factors equidistribute by the class-group estimates and the toric equidistribution input. At the auxiliary prime, the splitting condition supplies a second class- $\mathcal A'$ direction; the Einsiedler--Lindenstrauss joining classification then leaves only the product Haar measure, since the possible graph joinings are ruled out by the nonisomorphic factors or by the incompatible weights $a_s$ and $a_{2s}$ . The sequential form of Theorem 4.3 is uniform over the finitely many collections at each discriminant, so the final convex-combination argument is valid.

Theorem 1.1Not able to verify

Duke-type equidistribution for all nonsquare discriminants

Page 3 · Theorem 1.1 · arXiv:2606.03472v2

The theorem claims equidistribution of the full periodic-geodesic packet as $D\to\infty$ through every nonsquare $D$ . After its statement, the manuscript does not return to Theorem 1.1, give a proof, or cite an external theorem with this exact scope. Projection of Theorem 1.2 proves only the subsequences for which $D$ is odd and square-free and either $D$ or $-D$ is a nonzero square modulo a fixed odd prime. The toric-period method sketched later suggests a possible independent proof, but the paper does not verify the required order, packet, discriminant, and volume comparison for arbitrary nonsquare $D$ . Therefore the printed text does not support the full statement. This is an omitted proof obligation, not evidence that the theorem is false.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The Clifford-algebra construction and the main simultaneous-equidistribution chain are correct and complete at the stated level of reliance on standard toric-period and joining theorems. Theorem 1.1 has no proof or exact cited substitute, leaving one central statement unsupported. Several cross-references say “Theorem” where the referenced item is a lemma; these are harmless typos.

Theorem 1.1Incomplete as written

No proof is supplied for the all-nonsquare Duke-type theorem

Page 3, followed by Sections 2–5 · arXiv:2606.03472v2

A search through the complete manuscript finds no proof of Theorem 1.1 after its introductory statement. The proof of Theorem 1.2 explicitly assumes odd square-free discriminants and a local splitting condition, so it cannot serve as the missing argument. A repair requires either a precise reference proving equidistribution for the particular Bianchi geodesic packet attached to every nonsquare $D$ , or an added argument identifying that packet with the relevant adelic torus orbits and checking that their discriminants and volumes satisfy the toric equidistribution hypotheses as $D\to\infty$ .

Theorem 4.3 and deduction of Theorem 1.2Correct and complete

Adelic packet, marginal equidistribution, and joining classification

Pages 25–36 · Sections 4–5 · arXiv:2606.03472v2

The integral-plane reconstruction preserves discriminant and identifies the three associated binary forms. The class-group maps have only $D^{o(1)}$ index, which is within the volume threshold of Theorem 5.5. The limiting measure has Haar marginals, the real and $p$ -adic actions meet the higher-rank hypotheses, and the algebraic alternatives in every pair of factors are correctly eliminated. Finally, Proposition 4.7 identifies the projected packet with the average of the geometric measures, giving Theorem 1.2.

Theorem 5.5Correct and complete

Toric subcollection equidistribution input

Pages 32–34 · Theorem 5.5 and proof · arXiv:2606.03472v2

The lower volume bound limits the index of the subcollection to a small power of the torus discriminant. Finite Fourier expansion reduces its characteristic function, and the square-map restriction, to finitely many toric characters. Waldspurger's formula, uniform $\mathrm{GL}_2$ subconvexity, and local toric-integral bounds give a fixed power saving once the theorem's $\delta$ is chosen below that saving. The cited Clozel--Ullmo analysis also treats Eisenstein contributions, so the noncompact quotient does not leave an unaddressed continuous-spectrum case.

Clozel–Ullmo, Equidistribution de mesures algébriques ↗

Cross-references in Sections 4–5Typo

Several lemmas are called theorems

Pages 29–34 · references to 3.3, 4.5, 4.7, 5.2, 5.3, and 5.4 · arXiv:2606.03472v2

The cited numbered results are printed as Lemmas 3.3, 4.5, 5.2, 5.3, and 5.4 and Proposition 4.7, although the prose repeatedly calls them theorems. The intended references are uniquely determined by their numbers and content, so these naming mismatches do not affect any deduction.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2606.03472v2
Authors listed: Menny Aka, Konstantin Andritsch, Andreas Wieser
Audit date: August 18, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v2
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

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. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. 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

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

Planes in quadratic 4-space and associated shapes of lattices

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Simultaneous equidistribution of the four coupled objects

Duke-type equidistribution for all nonsquare discriminants

No proof is supplied for the all-nonsquare Duke-type theorem

Adelic packet, marginal equidistribution, and joining classification

Toric subcollection equidistribution input

Several lemmas are called theorems

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match