MathAudit

Paper metadata

Original submission
September 22, 2026
Latest revision
October 19, 2026 · version 2
Claimed MSC
22E40, 37A25, 37D40, 20F67, 28D05
arXiv categories
math.DS, math.GT
Paper license
CC BY 4.0 ↗

Abstract

Let Γ<SL(d,R)Γ<\mathsf{SL}(d,\mathbb R) be a torsion-free, Zariski-dense Borel-Anosov subgroup, and let φ\varphi be a linear functional positive on the limit cone of ΓΓ. For every χ∈Hom⁡(Γ,R)χ\in \operatorname{Hom}(Γ, \mathbb R), we construct twisted (φ,χ)(\varphi,χ)-conformal densities and prove their existence, uniqueness and ergodicity. For every non-trivial normal subgroup Γ0⊲ΓΓ_0\lhdΓ with nilpotent quotient, we classify the ergodic φ\varphi-conformal densities of Γ0Γ_0 in terms of characters of Γ/Γ0Γ/Γ_0. Up to scaling, they are naturally parametrized by Hom⁡(Γ/Γ0,R)\operatorname{Hom}(Γ/Γ_0,\mathbb R), and their dimensions are precisely the critical exponents of the corresponding twisted Poincaré series. We also prove a shadow lemma for twisted densities of normal subgroups, with estimates uniform over densities with fixed parameters, and a twisted Hopf-Tsuji-Sullivan theorem.

AI-generated audit

Audit summary

Audited against arXiv v2

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 23, 2026
01Statements2 reported findingsNo statement issues found

The existence, dimension, shadow, comparison, uniqueness, and ergodicity assertions for twisted conformal densities are stated with the needed Anosov, positivity, normality, and nilpotence hypotheses. The classification in Theorem 1.1 and its growth-form consequences preserve the character and scaling identifications. No mathematical statement requiring correction was found.

Theorem 1.1No issue found

Ergodic densities on nilpotent covers are classified by quotient characters

PDF pages 3–4 and 33–37 · Theorem 1.1 and Propositions 6.1–6.5 · arXiv:2609.27125v2

The lower central series of the nilpotent quotient lifts to a finite normal tower. Ergodicity and quasi-invariance produce a character at each extension step; the invariant-mean argument forces the intermediate characters to vanish until the final ambient extension. Conversely, a twisted ambient density is shown ergodic for the normal subgroup by propagating invariant sets up the same tower and applying uniqueness. The mass-ratio formula identifies the resulting character and makes the correspondence injective modulo positive scaling.

Theorems 3.8, 3.16, 4.6, and 5.9No issue found

The supporting twisted Patterson–Sullivan theory has the claimed scope

PDF pages 16–33 · Sections 3–5 · arXiv:2609.27125v2

Coarse additivity of the Cartan projection and boundary Busemann estimates give uniform two-sided shadow bounds. Patterson's construction at the twisted critical exponent supplies a nonzero density, the shadow differentiation argument upgrades mass comparison to measure comparison, and the Hopf argument gives the conservative/ergodic alternative. The same parameters and character signs are used consistently across these steps.

02Proofs2 reported findingsNo proof issues found

The proof chain from the Anosov boundary geometry through twisted Patterson construction, the shadow lemma, differentiation, comparison, Hopf–Tsuji–Sullivan theory, and the nilpotent induction was checked. The use of right-invariant means, character cocycles, critical exponents, and measure-class invariance is internally consistent. No incorrect or materially incomplete proof was found.

Theorem 3.16 and Corollary 4.7No issue found in the reviewed proof

Existence and uniqueness use matching critical data

PDF pages 23–29 · Theorem 3.16, Theorem 4.6, and Corollary 4.7 · arXiv:2609.27125v2

The modified Poincaré series diverges at its critical exponent and normalized orbit measures have nonzero boundary limits. Their transformation rule yields the prescribed twist. Uniform shadow estimates and differentiation show that domination of one basepoint mass implies domination of the full density, so equal normalized masses force equality.

Lemmas 6.3 and Propositions 6.4–6.5No issue found in the reviewed proof

The nilpotent-tower induction closes in both directions

PDF pages 35–37 · Section 6.3 · arXiv:2609.27125v2

The logarithmic mass increment is bounded on the quotient and its invariant mean is an ambient character vanishing on the normal subgroup. On a lower-central-series term it agrees with the provisional twist, but every real character annihilates the commutator term, forcing that twist to vanish. For the converse, measure-class domination propagates invariance of a positive-measure set to the ambient group, after which uniqueness makes its restriction a scalar multiple of the original density.

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:2609.27125v2
Authors listed
Krishnendu Gongopadhyay, Neelanjan Mondal
Audit date
October 23, 2026
  1. 01Establish identityMatch an authenticated scholarly identity to this paper.
  2. 02Submit evidenceIdentify the finding and give a formal mathematical response.
  3. 03Re-evaluateA separate agent checks the response and records a disposition.
Recommended production method

Authenticate with ORCID, then require an exact arXiv match

MathAudit should accept the identity only when ORCID OAuth authenticates the claimant's iD and this exact arXiv paper appears in arXiv's public authority feed for that iD. A matching name alone is not sufficient.

ORCID OAuth and arXiv authority-record lookup are not connected in this local prototype.

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.

This demonstration does not send, store, or compare the address.

Structured response preview

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

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.