MathAudit

Paper metadata

Original submission
April 27, 2025
Latest revision
February 27, 2026 · version 4
Claimed MSC
arXiv categories
math.AG, math.QA, math.RT
Comments
17 pages, latex; added sections 2.6 and 3.8 on applications to symplectic singularities and quantum cluster algebras

Abstract

M. Kontsevich conjectured and T. Bitoun proved that if M is a nonzero holonomic D-module then the p-support of a generic reduction of M to characteristic p>0 is Lagrangian. We provide a new elementary proof of this theorem and also generalize it to q-D-modules. The proofs are based on Bernstein's theorem that any holonomic D-module can be transformed by an element of the symplectic group into a vector bundle with a flat connection, and a q-analog of this theorem. We also discuss potential applications to quantizations of symplectic singularities and to quantum cluster algebras.

Dependence graphs

Proof lineage

On Lagrangianity of pp-supports of holonomic DD-modules and qq-DD-modules

A statement-restricted graph for the proved $D$-module, $q$-difference, and quantum-cluster assertions in arXiv:2504.19329v4. It follows every proof-critical non-book source recursively and stops at books or monographs. Historical antecedents that are reproved in the focal paper, definitional pointers, alternative citations, and the explicitly conjectural symplectic-singularity discussion are excluded.

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AI-generated audit

Audit summary

Audited against arXiv v4

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 August 27, 2026
01Statements4 reported findingsSuspected unsupported statements

The characteristic-zero Bitoun theorem, its smooth-variety corollary, and the q-analog of Bernstein's theorem are correct after the verified local proof repair and literal notation corrections below. The q-support theorems are also established when the residue characteristic is zero or exceeds the rank of the transformed bundle. Their stated fixed-positive-characteristic scope is not verified when that rank is at least the characteristic. Independently, the cluster theorem omits the full-order condition needed for quantum mutations to identify the centers of adjacent charts at roots of unity: for N=ord⁡(q)N=\operatorname{ord}(q) one needs gcd⁡(N,dk)=1\gcd(N,d_k)=1. Both limitations propagate to Theorem 3.14.

Theorem 2.9 and Corollary 2.11No issue found

The characteristic-zero p-support conclusions are correct

PDF pages 5-6 · Theorem 2.9 and Corollary 2.11 · arXiv:2504.19329v4

After spreading the symplectic transformation of Theorem 2.8 over a localization of the finitely generated coefficient ring, reduction gives a symplectic automorphism of the characteristic-pp Weyl algebra and hence of its center; it preserves Lagrangianity. The transformed module is a vector bundle with a flat connection of a fixed rank nn. Outside the finitely many residue characteristics p≤np\leq n, each reduction is liftable and Corollary 2.6 applies. For the smooth-variety statement, the transfer-module calculation for a closed embedding gives exactly the displayed cotangent correspondence for pp-supports; its local verification and source locator are recorded below. The false sentence in the proof of Lemma 2.3 has the verified replacement recorded in the Proofs dimension, so it does not change these theorem outcomes.

Theorem 3.3No issue found

The q-analog of Bernstein's theorem is established

PDF pages 8-10 · Lemma 3.1 through Theorem 3.3 · arXiv:2504.19329v4

The orbit lemma patches local freeness over the Laurent polynomial algebra, while Lemmas 3.4-3.6 reduce one noncommuting variable at a time after an integral symplectic change of coordinates. The Zariski density of Sp⁡(2s,Z)\operatorname{Sp}(2s,\mathbb Z) in Sp⁡(2s,C)\operatorname{Sp}(2s,\mathbb C) supplies the integral point in the nonempty open set used in Lemma 3.4. Bernstein's inequality excludes the free-submodule alternative for a nonzero holonomic module, completing the induction. The index and product-order errors listed in the Proofs dimension are uniquely repairable and do not affect the result.

Section 3.7Correct in the supported rank range

The dimension-rr definition is consistent after forcing finite Casimir support

PDF pages 14-15 · paragraph preceding Theorem 3.13 · arXiv:2504.19329v4

When the skew form has rank 2r2r on an ss-torus, a Lagrangian inside the whole Poisson variety would ordinarily have dimension s−rs-r, not rr. Here the stronger holonomicity bound forces the missing s−2rs-2r Casimir directions to be zero-dimensional. After the finite isogeny used in the manuscript, the algebra is finite over the tensor product of a rank-2r2r quantum torus and a central Laurent algebra CC in s−2rs-2r variables. If the CC-support has dimension dd, Bernstein's inequality on the nondegenerate factor gives GKdim⁡M≥r+d\operatorname{GKdim}M\geq r+d. Thus GKdim⁡M≤r\operatorname{GKdim}M\leq r forces d=0d=0: a finite-length module is supported on finitely many Casimir fibers. On each 2r2r-dimensional symplectic leaf, a coisotropic support of total dimension at most rr must have dimension exactly rr and is Lagrangian. The wording is compressed, but the apparent dimension mismatch is therefore resolved. This does not repair the separate fixed-positive-characteristic rank gap inherited from Theorem 3.11.

Theorems 3.11, 3.13, and 3.14Not independently verified in the two stated regimes

The fixed-positive-characteristic rank range and non-admissible cluster orders are unsupported

PDF pages 13-16 · Theorems 3.11, 3.13, 3.14 and their reductions to Proposition 3.9 · arXiv:2504.19329v4

Theorem 3.11 is stated over an algebraically closed field of any characteristic. Its proof transforms MM into a free Laurent-polynomial module of some fixed rank nn and then invokes Proposition 3.9. In residue characteristic p>0p>0, that proposition explicitly assumes n<pn<p: its trace argument divides by joint-eigenspace multiplicities mk≤nm_k\leq n. Over a characteristic-zero base, one may discard the finitely many residue characteristics p≤np\leq n, so the argument works. Over a base field of fixed characteristic pp, every specialization has the same characteristic, and when n≥pn\geq p no generic-open restriction makes the hypothesis true. Example 3.10 shows that the rank restriction cannot simply be erased from Proposition 3.9, although it does not provide a fixed generic-qq counterexample to Theorem 3.11. Thus this regime is not established, and Theorems 3.13-3.14 inherit the limitation. Theorem 3.14 has a second, independent scope failure. If B~TΛ=(D∣0)\widetilde B^T\Lambda=(D\mid0), the two quantum exchange summands in direction kk commute with factor qdkq^{d_k}. At a torsion value with N=ord⁡(q)N=\operatorname{ord}(q), the classical mutation formula for their NNth powers requires ord⁡(qdk)=N\operatorname{ord}(q^{d_k})=N, equivalently gcd⁡(N,dk)=1\gcd(N,d_k)=1. Otherwise quantum-binomial cross terms can survive, so the centers of adjacent quantum charts do not glue by the displayed classical mutation. Parity is not an intrinsic extra condition for Etingof's parameter qq; MNTY's half-parameter Frobenius convention imposes odd half-parameter order as a sufficient additional hypothesis. Whenever some relevant dk>1d_k>1, which the theorem allows, the characteristic-zero torsion orders divisible by a prime factor of that dkd_k form an infinite, hence Zariski-dense, subset of Gm\mathbb G_m and cannot be removed by the asserted generic Zariski-open restriction. If every relevant dk=1d_k=1, this particular obstruction disappears. A repair in the general stated scope is verified only after restricting to an explicitly full-order torsion family.

02Proofs8 reported findingsSuspected incomplete or incorrect proofs

The q-support proof omits the entire fixed-characteristic range in which the transformed rank is at least the characteristic, and the cluster proof additionally omits the root-of-unity admissibility needed for adjacent chart centers to mutate classically. Separately, Lemma 2.3 uses a false statement about the ideal pSpS, but a one-line local-ring argument gives a verified repair. The spread-out symplectic reduction in Theorem 2.9 is recoverable, and chartwise supports glue after explicitly restricting to admissible torsion orders. The unnormalized quantum-torus monomial transformations compose only projectively, but every use has a verified normalization-free repair. Nine literal rank, exponent, locator, and algebra-symbol errors have unique harmless corrections.

Proof of Lemma 2.3Suspected incorrect as written · an apparently sound repair

The claim that pSpS surjects onto the residue field is false

PDF page 3 · final sentence of the proof of Lemma 2.3 · arXiv:2504.19329v4

The proof obtains p(ja~pi−1,pj−ib~pi,pj−1)=0p\bigl(j\widetilde a_{pi-1,pj}-i\widetilde b_{pi,pj-1}\bigr)=0 in the local Artinian ring SS and then says that pSpS surjects onto the residue field kk. In fact pp lies in the maximal ideal because kk has characteristic pp, so the natural image of pSpS in kk is zero. The needed coefficient identity nevertheless follows: if pc=0pc=0 and cc were a unit, then p=0p=0 in SS, contrary to liftability. Hence cc is a nonunit, therefore belongs to the maximal ideal, and its reduction in kk is zero. Applying this to the parenthesized coefficient proves the displayed equality and the closedness of the curvature form. Repair classification: Verified repair.

Proofs of Theorems 3.11, 3.13, and 3.14Suspected incomplete as written · no repair proposed in fixed characteristic $p$ for rank $n\geq p$

Proposition 3.9 is applied outside its rank hypothesis

PDF pages 13-16 · proofs of Theorems 3.11 and 3.14 and the paragraph preceding Theorem 3.13 · arXiv:2504.19329v4

After Theorem 3.3, the transformed module is free of rank nn over the Laurent polynomial algebra. Its reductions are liftable q-connections, but Proposition 3.9 proves Lagrangianity in positive characteristic only when n<pn<p. The sentence 'the result follows from Proposition 3.9' does not address n≥pn\geq p when the base field already has characteristic pp. The proof of Proposition 3.9 uses mk≠0m_k\neq0 in the residue field for every joint-eigenspace multiplicity; when a multiplicity is divisible by pp, equation (3.2) yields no logarithmic-derivative identity. The quantum-torus reduction and the cluster theorem call the same result, so they do not restore the missing case. Repair classification: No repair supplied for this regime.

Compressed reduction in Theorem 2.9Correct and complete after immediate-consequence closure

The omitted spread-out symplectic twist is formally recoverable

PDF page 5 · proof of Theorem 2.9 · arXiv:2504.19329v4

Adjoining the finitely many coefficients of the symplectic matrix and localizing the model spreads the twist and the finite locally free connection simultaneously. Reduction of the twist is an automorphism of the Weyl algebra whose induced map on the center is symplectic, so it transports the pp-support and preserves Lagrangianity. This compression creates no additional proof obligation beyond the stated source inputs.

Proof of Theorem 3.14Suspected incomplete as written · verified only on an admissible torsion subfamily

Chartwise support gluing fails without root-order admissibility

PDF pages 15-16 · definition of cluster pp-support and proof of Theorem 3.14 · arXiv:2504.19329v4

The sentence that the central supports 'glue together' assumes that specialization of a quantum mutation induces the classical mutation between the spectra of the chart centers. For a compatible pair B~TΛ=(D∣0)\widetilde B^T\Lambda=(D\mid0), its two exchange summands A,BA,B satisfy AB=qdkBAAB=q^{d_k}BA. With N=ord⁡(q)N=\operatorname{ord}(q), the identity (A+B)N=AN+BN(A+B)^N=A^N+B^N holds in the required form only when qdkq^{d_k} still has order NN, equivalently gcd⁡(N,dk)=1\gcd(N,d_k)=1. If qdkq^{d_k} has smaller order, quantum-binomial cross terms survive; for example when its order is N/2N/2, the middle term in (AN/2+BN/2)2(A^{N/2}+B^{N/2})^2 remains. Even NN is not by itself an obstruction for this parameter when the full-order condition holds. MNTY's oddness condition belongs to its half-parameter Frobenius convention and is a sufficient additional hypothesis there. If some relevant dk>1d_k>1, orders divisible by a prime factor of dkd_k give a Zariski-dense characteristic-zero torsion family, so no Zariski-open 'generic torsion' set removes them; when all relevant dk=1d_k=1, this obstruction vanishes. Once one explicitly restricts to full-order specializations in the general case, localization of the same Serre-quotient object identifies the central supports on overlaps, a finite chart cover gives a common good-specialization condition, and Lagrangianity glues. That verifies a restricted theorem but not the stated one. Repair classification: Verified only after a substantive full-order restriction.

Proof of Corollary 2.11Correct after a verified local calculation

The closed-embedding support identity follows from the transfer module

PDF page 6 · proof of Corollary 2.11 · arXiv:2504.19329v4

The equality for the pp-support of i∗Mi_*M is asserted rather than derived in the focal manuscript, so it is proof-critical. It is exactly Bitoun's Proposition 2.4.1. It can also be checked independently in local coordinates in which the smooth closed subvariety is cut out by normal coordinates y1,…,ycy_1,\ldots,y_c. The transfer module identifies i∗Mi_*M locally with M⊗k[∂y1,…,∂yc]M\otimes k[\partial_{y_1},\ldots,\partial_{y_c}]. In characteristic pp, each central function yjpy_j^p acts by zero, the tangential central operators have the original support, and the normal operators ∂yjp\partial_{y_j}^p are unrestricted. Thus the central support is precisely the inverse image of the original support under cotangent restriction, followed by the inclusion over X(1)X^{(1)}. Base change of the transfer module is compatible with reduction, and the cotangent correspondence preserves vanishing of the symplectic form and the dimension shift. This discharges the omitted calculation.

Symplectic monomial changes in Section 3.3Suspected incorrect as written · an apparently sound repair

The cited transformations are projective rather than a literal unnormalized group action

PDF pages 8-10 · paragraph before Theorem 3.3 and its proof · arXiv:2504.19329v4

The manuscript says that Sp⁡(2r,Z)\operatorname{Sp}(2r,\mathbb Z) acts on the quantum torus by automorphisms. Sabbah constructs an automorphism for each integral symplectic matrix, but with the displayed unnormalized monomial lifts, compositions agree only up to powers of qq; this is a projective action, or an action after choosing a compatible normalization or central extension. The proof needs only the existence of an automorphism realizing each exponent-lattice transformation. The scalar qq-factors do not change finiteness, Gelfand-Kirillov dimension, or monomial exponent separation, and consecutive twists may be replaced by the single automorphism attached to the product matrix. Thus the literal group-action claim is overstated but all uses have a verified repair. Repair classification: Verified repair.

Section 2 notationSuspected typo

Four rank, tensor, and cross-reference symbols need unique corrections

PDF pages 2, 4-5 · Section 2.1, Lemma 2.5, and proof of Theorem 2.8 · arXiv:2504.19329v4

For a rank-nn connection, M∇M_\nabla is generated by f1,…,fnf_1,\ldots,f_n, not f1,…,frf_1,\ldots,f_r. In the displayed exterior-power formula in Lemma 2.5, the final factor is 1⊗(n−j)1^{\otimes(n-j)}, not 1⊗(n−j−1)1^{\otimes(n-j-1)}; the printed exponent is even −1-1 when j=nj=n. The proof of Theorem 2.8 must choose gg by Lemma 2.7(ii), not nonexistent part (ii) of Lemma 2.3. Finally, its concluding finite-generation sentence concerns gMgM, not the original symbol MM. Each correction is forced by the definitions and the surrounding proof.

Section 3 notationSuspected typo

Five q-difference and cluster symbols need unique corrections

PDF pages 9-11 and 14-15 · proofs of Lemma 3.4 and Theorem 3.3, Section 3.4, Theorem 3.13, and the quantum-cluster paragraph · arXiv:2504.19329v4

In Lemma 3.4, the recurrence coefficients belong to Ds−1,q(R[xs±1])\mathcal D_{s-1,q}(R[x_s^{\pm1}]), not Dr−1,q(R[xr±1])\mathcal D_{r-1,q}(R[x_r^{\pm1}]). At the end of the induction for Theorem 3.3, the product carrying MM to the already displayed module g2g1Mg_2g_1M is g2g1g_2g_1, not g1g2g_1g_2. In Section 3.4, a rank-nn connection acts on k(x1,…,xr)n\boldsymbol{k}(x_1,\ldots,x_r)^n, not the power indexed by the root order NN. Theorem 3.13 must say a holonomic DΛ,q\mathcal D_{\Lambda,q}-module, not Dr,q\mathcal D_{r,q}; otherwise it merely repeats Theorem 3.11 and cannot imply Theorem 3.14. In the quantum-cluster paragraph, Uq(Λ,B~)\boldsymbol U_q(\Lambda,\widetilde B) is a flat deformation of the classical algebra U(Λ,B~)\boldsymbol U(\Lambda,\widetilde B), not of itself. These are uniquely determined notation repairs.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

04Sources7 reported findingsSuspected incomplete or incorrect source use

The proof-critical Sabbah, Bitoun, Lam, Berenstein-Zelevinsky, Matherne-Muller, and Muller-Nguyen-Trampel-Yakimov inputs were traced to exact sources. Their portions actually needed by the focal argument are supported after explicit repairs: Sabbah's monomial transformations are projective before normalization; Bitoun supplies the closed-immersion formula; Berenstein-Zelevinsky Appendix Proposition A.2(2) is false but the separately used intersection statement A.2(3) has a direct proof; and the public Matherne-Muller fallback has a gcd/lcm and exponent error in its Hartogs lemma but a verified correction establishes the codimension-two implication used by MNTY Proposition 4.6. MNTY Theorem A and Proposition 4.6 therefore remain supported, and their later false Azumaya-locus result is unrelated to the focal use. However, the focal cluster theorem omits the condition ord⁡(qdk)=ord⁡(q)\operatorname{ord}(q^{d_k})=\operatorname{ord}(q) required when applying quantum mutation to root-of-unity chart centers. MNTY's half-parameter formalism additionally assumes odd half-parameter order. This is an incomplete use of the quantum-cluster sources and cannot be repaired by a generic-open torsion restriction. Losev's disclosed error remains unrelated to the exact material cited.

Sabbah, Sections 1-2Applicable and sufficient

The q-difference structural inputs are applicable

Target PDF pages 8-10 · Sections 3.1-3.3 · arXiv:2504.19329v4

Sabbah's Proposition 1.2.1 gives simplicity of the quantum torus when qq is not a root of unity, Section 1.3 gives an automorphism for each integral symplectic coordinate change, Corollary 2.1.2 gives Bernstein's inequality, Corollary 2.5.2 gives finite-dimensional generic localization, and Theorem 2.7.1 relates rational q-connections to holonomic modules. Two qualifications matter. First, the unnormalized monomial automorphisms in Section 1.3 compose only up to powers of qq, so the focal phrase 'the group acts' requires normalization or projective-cocycle bookkeeping; existence of each twist is all the proof uses. Second, Theorem 2.7.1(2) has a compressed statement, but its proof chooses a finite lattice M′M' whose localization is the prescribed rational connection, exactly the existence required by the focal converse. The cited source is written in characteristic zero, but the proof-critical algebraic steps used here depend on qq having infinite multiplicative order and on the quantum-torus division and homological arguments, not on characteristic-zero differentiation. They remain valid when qq is not a root of unity in positive characteristic. The later rank obstruction comes from the focal Proposition 3.9, not from an overextension of Sabbah's theorem.

C. Sabbah, Systèmes holonomes d'équations aux q-différences, author PDF ↗
AGM and prior main-theorem attributionsCorrect attribution; no unresolved source dependency

The cited claims are independently established in the focal paper

Target PDF pages 4-6 · Lemma 2.7 through Corollary 2.11 · arXiv:2504.19329v4

The focal Lemma 2.7 supplies the complete incidence-dimension proof of the Lagrangian-projection lemma attributed to Bernstein, and Theorem 2.8 then derives the finite projection and finite module generation directly. Thus AGM Lemma 3.15 and Corollary 3.16 are historical locators rather than omitted proof inputs. Bitoun and van den Bergh are also cited for the earlier main p-support theorem and its generalization, while the focal manuscript proves its affine theorem. This finding concerns only those prior main-theorem attributions; the separate closed-immersion support formula used in Corollary 2.11 is proof-critical and is audited in the next finding through Bitoun's Proposition 2.4.1 and a local calculation.

Aizenbud-Gourevitch-Minchenko, exact final arXiv version 2 ↗
Bitoun, Proposition 2.4.1Applicable and sufficient

The closed-embedding formula used in Corollary 2.11 is the cited result

Target PDF page 6 · proof of Corollary 2.11 · source Section 2.4, Proposition 2.4.1 · arXiv:2504.19329v4

Bitoun's exact Inventiones version, Proposition 2.4.1 on printed pages 787-788, states that direct image along a closed immersion commutes with base change and gives p-supp⁡(H0fs,+Ms)=(fs′)π∘(fs′)d−1(p-supp⁡(Ms)).p\text{-}\operatorname{supp}(H^0 f_{s,+}M_s)=(f_s')_\pi\circ(f_s')_d^{-1}(p\text{-}\operatorname{supp}(M_s)). This is the focal identity written with π\pi and ι\iota. Bitoun derives it from the classical transfer-bimodule description, ultimately citing Borel et al., Algebraic D-Modules, equation (4) on page 259; the local normal-coordinate computation is reproduced in the Proofs dimension, so no unexamined source step remains. Bitoun's Appendix Lemma A.0.6 on printed page 817 supplies the corresponding symplectic-form comparison. Van den Bergh is cited alongside Bitoun for the general reduction, but the exact formula and sufficient locator are in Bitoun.

T. Bitoun, On the p-supports of a holonomic D-module, Inventiones Mathematicae 215 (2019), exact VOR ↗
Berenstein-Zelevinsky, Appendix Proposition A.2Suspected issue in source statement · used consequence appears repaired

A false ideal statement has a local repair for the only used intersection consequence

Target PDF pages 14-16 · Section 3.8 and proof of Theorem 3.14 · arXiv:2504.19329v4

In the exact Advances in Mathematics version, Appendix Proposition A.2(2), printed pages 453-454, claims that every two-sided ideal II of an integral based quantum torus TT is generated by I∩Z(T)I\cap Z(T). This is false. With t=q1/2t=q^{1/2} and T=Z[t±1]⟨X±1,Y±1⟩/(XY−t2YX)T=\mathbb Z[t^{\pm1}]\langle X^{\pm1},Y^{\pm1}\rangle/(XY-t^2YX), the proper two-sided ideal I=(t−1,X−1)I=(t-1,X-1) satisfies I∩Z(T)=(t−1)I\cap Z(T)=(t-1) but is strictly larger than (t−1)T(t-1)T. The proof's assertion that every two-sided ideal is graded fails when eigencharacters coalesce after t=1t=1. The downstream Laurent argument uses only Proposition A.2(3): for relatively prime central elements z1,z2z_1,z_2, one has Tz1∩Tz2=Tz1z2Tz_1\cap Tz_2=Tz_1z_2. This follows directly by choosing coset-monomial representatives that make TT free over its center and comparing coefficients. Thus Theorem 5.1 and Corollary 5.2, together with the BFZ commutative input, remain supported despite the false broader appendix claim.

Berenstein-Zelevinsky, Quantum cluster algebras, exact Advances in Mathematics version ↗
Matherne-Muller and MNTY Proposition 4.6Applicable after an apparently sound repair of the public fallback

The codimension-two chart input survives a verified fallback repair

Target PDF pages 15-16 · cluster-locus paragraph · MNTY Proposition 4.6 · arXiv:2504.19329v4

MNTY Proposition 4.6 uses the algebraic Hartogs implication in Matherne-Muller Lemmas 4.3.1 and 4.4.2. The exact IMRN version was inaccessible; the only public arXiv version 1/author manuscript has two errors in the final converse of Lemma 4.3.1 on page 9. If vi=ν(ai)>0v_i=\nu(a_i)>0 and ν(a1)=1\nu(a_1)=1, its dd must be lcm⁡(vi)\operatorname{lcm}(v_i), not the gcd, and the displayed akda_k^d must be aka_k. With x=a1−d∏i>1gid/vi,x=a_1^{-d}\prod_{i>1}g_i^{d/v_i}, the identity ak=(fk/gk)a1vka_k=(f_k/g_k)a_1^{v_k} gives x=(fk/ak)d/vk∏i>1, i≠kgid/vi∈S[ak−1],x=(f_k/a_k)^{d/v_k}\prod_{i>1,\,i\ne k}g_i^{d/v_i}\in S[a_k^{-1}], while ν(x)=−d\nu(x)=-d, proving the required codimension-one converse. BFZ supplies the remaining upper-bound equality, so MNTY Proposition 4.6 and Theorem A remain valid for the focal use. The false upper Azumaya-locus inclusion later in MNTY Theorem 6.1 is independent and is not used here.

Matherne-Muller, Computing upper cluster algebras, public arXiv version 1 fallback ↗
Root-of-unity mutation scopeSuspected incorrect or insufficient source use

The cluster-support gluing omits the admissible-order hypothesis

Target PDF pages 15-16 · definition of cluster pp-support and Theorem 3.14 · arXiv:2504.19329v4

The generic-qq quantum Laurent theorem identifies the fraction-field charts, but it does not by itself identify their centers after every torsion specialization. For B~TΛ=(D∣0)\widetilde B^T\Lambda=(D\mid0) and N=ord⁡(q)N=\operatorname{ord}(q), the central classical mutation intrinsically requires ord⁡(qdk)=N\operatorname{ord}(q^{d_k})=N, equivalently gcd⁡(N,dk)=1\gcd(N,d_k)=1, for every relevant diagonal entry. MNTY's root-of-unity Theorem B uses the half-parameter ϵ1/2\epsilon^{1/2} and imposes both coprimality and odd half-parameter order, a sufficient convention-dependent Frobenius hypothesis rather than evidence that parity alone is necessary for Etingof's qq. The focal manuscript imposes no full-order condition and calls the chart-center gluing automatic. Whenever some relevant dk>1d_k>1, orders divisible by a prime factor of that entry form a Zariski-dense characteristic-zero torsion family, so the missing scope cannot be supplied by the phrase 'generic torsion point'; if all relevant dk=1d_k=1, this particular obstruction vanishes. An explicitly restricted full-order family is required in the general case allowed by the theorem.

Muller-Nguyen-Trampel-Yakimov, exact Advances in Mathematics version, Theorem B ↗
Losev-Etingof, arXiv:1501.01260v4Applicable for the exact cited material

The disclosed Theorem 1.1 error is not used by the focal manuscript

Target PDF pages 6-7 · Section 2.6 · arXiv:2504.19329v4

The current arXiv record for the cited paper explicitly says that the proof of its Theorem 1.1 has a crucial error. Section 2.6 of the focal manuscript does not invoke that theorem. It uses the definition of holonomic modules and Lemma 5.1 from the joint appendix concerning isotropic subvarieties under a symplectic resolution; the appendix proves that lemma directly from pullback of the symplectic form. The focal Section 2.6 then states Conjecture 2.14 and a proposed Hamiltonian-reduction route rather than claiming a theorem from the erroneous result. The source defect therefore has no downstream effect here.

Losev-Etingof, exact current arXiv version 4 ↗
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Paper
arXiv:2504.19329v4
Authors listed
Pavel Etingof
Audit date
August 27, 2026
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