arXiv:2609.33788v1

Moments of the Cross-Sectional Siegel-Veech Transform

math.DS11H6037A1711B5737A44

Paper metadata

Original submission: September 27, 2026
Claimed MSC: 11H60, 37A17, 11B57, 37A44
arXiv categories: math.DS
Paper license: CC BY-NC-ND 4.0 ↗
Comments: 28 pages

Abstract

We study the Siegel--Veech transform on the Poincaré section for the horocycle flow consisting of lattice surfaces with a visible horizontal holonomy vector of length at most one. We compute its first moment and derive formulas for higher and factorial moments with respect to the section's induced ergodic invariant probability measure, which is not supported on a periodic orbit. We emphasize these formulas in the case of the square torus of unit area and, as an application, use its first-moment formula to recover the Boca--Zaharescu pair-correlation density for Farey fractions.

AI-generated audit

Audit summary

Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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 October 24, 2026

01Statements4 reported findingsNo statement issues found

The first, higher, and factorial moment formulas follow from the parabolic-orbit decomposition of visible holonomy vectors. Their modular specializations are correct, and the first-moment formula yields the stated Farey pair-correlation density. One auxiliary finite- $Q$ identity needs the harmless qualifier “for all sufficiently large $Q$ .”

Theorems 1.1–1.3No issue found

The cross-sectional first and second moments have the stated densities

PDF pp. 3–5 and 9–18 · arXiv:2609.33788v1

At each height $\zeta$ , visible vectors split into $\phi(\zeta)$ residue classes under the cusp translation. Summing the common parabolic shift tiles the horizontal line, and the change $y=\zeta/s$ supplies the factor $y^{-2}$ . For two vectors, fixing their heights, residue classes, and relative shift fixes their determinant and leaves the same one-dimensional unfolding. In the modular case, regrouping primitive residue classes by $n=Ak-Bj$ gives exactly the coefficient $\Phi_{j,k}(n)$ .

Theorems 5.1, 5.3, and 5.4No issue found

The higher and factorial moment formulas follow from the same determinant parametrization

PDF pp. 18–22 · arXiv:2609.33788v1

Using the first vector as the integration coordinate leaves one common parabolic shift and records every other vector by its height and determinant with the first. Lemma 5.2 correctly expresses every pairwise determinant as $(\zeta_\ell D_m-\zeta_mD_\ell)/\zeta_1$ , so the displayed nonvanishing conditions are exactly pairwise distinctness for visible positive-height vectors. The modular congruences count precisely the admissible primitive residue tuples.

Theorem 6.3No issue found

The first moment recovers the Boca–Zaharescu pair-correlation density

PDF pp. 22–27 · arXiv:2609.33788v1

Future section returns at the periodic BCZ point encode the normalized positive Farey differences. The test transforms are uniformly bounded, Riemann integrable, and converge uniformly as $N_Q/Q^2\to3/\pi^2$ , so periodic-orbit equidistribution applies. Inserting their defining function into the modular first-moment identity and changing $x$ to $\lambda=3y/(\pi^2x)$ produces the claimed totient sum and logarithm.

Lemma 6.2, equation (80)Suspected minor formal issue

The orbit-average identity is eventual, not valid for every $Q$

PDF p. 25 · arXiv:2609.33788v1

The periodic BCZ orbit average also contains wrap-around pairs with $i+r\geq N_Q$ , whereas $R_Q^{(2)}(H)$ sums only pairs inside one copy of $[0,1)$ . The proof itself shows these extra terms vanish once $N_Q/Q$ exceeds the upper endpoint of $\operatorname{supp}(H)$ . Thus equation (80) should read “for all sufficiently large $Q$ .” For example, at $Q=1$ a function with $H(1)\neq0$ makes the orbit average nonzero while the within-period pair sum is empty. The corrected eventual identity is all that Theorem 6.3 uses.

02Proofs3 reported findingsNo proof issues found

The cusp unfolding, determinant bookkeeping, integrability arguments, factorial restrictions, and Farey/BCZ limiting calculation were checked through their proof-critical steps. Apart from the local quantifier correction in Lemma 6.2, the proofs are complete.

Theorem 3.1 and Corollary 3.4No issue found in the reviewed proof

The first-moment unfolding is exact

PDF pp. 9–12 · arXiv:2609.33788v1

For fixed $s$ , consecutive cusp translates produce adjacent intervals of length $\alpha\zeta s$ , so their $t$ -integrals cover $\mathbb R$ exactly once. Then $ds=-\zeta y^{-2}dy$ cancels the prefactor $1/\zeta$ . Compact support bounds the contributing heights and proves absolute integrability for signed test functions. For the square torus, $\alpha=1$ , $J=\mathbb N$ , and $\phi=\varphi$ , yielding the density $2\Phi(y)/y^2$ .

Theorems 4.1 and 4.4No issue found in the reviewed proof

Relative shifts and determinant data are counted without multiplicity

PDF pp. 12–18 · arXiv:2609.33788v1

The change $(q_1,q_2)\leftrightarrow(q_1,q_1-q_2)$ is bijective, and determinant invariance determines the second transformed vector from the first. In the modular regrouping, a residue pair contributes to determinant $n$ exactly when $ak-bj\equiv n\pmod{jk}$ ; after that congruence, one free integer shift remains. Compact support bounds both heights and the determinant, so the signed formulas are absolutely integrable.

Lemmas 6.1–6.2 and Theorem 6.3Correct and complete after a minor formal correction

The Farey limit proof is complete after the stated eventual-identity correction

PDF pp. 24–27 · arXiv:2609.33788v1

A vector with transformed coordinates $(x,y)$ and $0<x\leq1$ returns after time $y/x$ , and the sum of BCZ return times is $Q^2(\gamma_{i+r}-\gamma_i)$ . Only finitely many primitive vectors can contribute uniformly for scaling constants near $3/\pi^2$ , which proves boundedness, Riemann integrability, and uniform convergence of the transforms. Wrap-around terms disappear for all sufficiently large $Q$ , exactly the regime needed before taking the weak limit.

03Novelty0 reported findingsNo novelty concerns found

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:2609.33788v1
Authors listed: Albert Artiles
Audit date: October 24, 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.

Moments of the Cross-Sectional Siegel-Veech Transform

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The cross-sectional first and second moments have the stated densities

The higher and factorial moment formulas follow from the same determinant parametrization

The first moment recovers the Boca–Zaharescu pair-correlation density

The orbit-average identity is eventual, not valid for every QQQ

The first-moment unfolding is exact

Relative shifts and determinant data are counted without multiplicity

The Farey limit proof is complete after the stated eventual-identity correction

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

The orbit-average identity is eventual, not valid for every $Q$