{"id":"7a97e619-284e-49f8-84df-55047245fa73","arxiv_id":"1908.10103","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Grassmannian cluster algebras, the non-degenerate quiver with potential is rigid and unique up to right-equivalence, and the auto-equivalence group of its category matches the cluster automorphism group.","lead":"This mathematics paper proves that each Grassmannian cluster algebra has a unique 'quiver with potential' object that behaves well under all mutations. It then shows that two symmetry groups, one from the algebra and one from its category, are the same.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.14's verification of Prop 2.13 condition (2) computes short(eta_alpha)+short(partial_alpha W)=4m, not the required strict >4m, leaving the uniqueness proof unsupported unless the quoted theorem actually uses a non-strict inequality.","rationale":"The Reader's weakest assumption concerns the connectedness of the geometric exchange graph and the omitted cases in Theorem 2.4. Those are plausible concerns, but the connectedness statement is very likely true in the literature and the omitted cases are diagrammatic repetitions. A more decisive, purely internal issue is the strictness mismatch in the proof of Theorem 2.14. The paper's own computation yields equality where the stated criterion demands a strict inequality. This directly affects the main uniqueness theorem and the subsequent application to cluster automorphism groups. The concern is checkable in one step by reading [17, Theorem 8.20] and, if necessary, re-deriving the length count. Because the issue is explicitly a proof gap that could be repaired by a correction of the inequality or by citing the correct version of the source theorem, the conditional verdict from the Reader remains appropriate; I do not see a reason to move to accept or reject on this basis alone. My answer to the agreement question is 'disagree' because the Reader identified different weak points, whereas the load-bearing gap here is an internal arithmetic inconsistency in the verification of Proposition 2.13 condition (2).","tokens_in":987,"tokens_out":1545,"duration_ms":186081,"concrete_test":"Consult [17, Theorem 8.20] and determine the exact inequality in condition (2). If it is '>=', then replace the two '>' conditions in Proposition 2.13 and the proof by '>=' and the argument goes through. If it is '>', recompute the m=1 case in Step I of Theorem 2.14: omega = alpha * partial_alpha(W_ini) gives short(eta_alpha) + short(partial_alpha(W_ini)) = 1+3 = 4 = length(omega), so condition (2) fails for the very first power, and the uniqueness claim is unproven as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central uniqueness theorem, Theorem 2.14, rests on verifying condition (2) of Proposition 2.13. In Step I of its proof, for a fundamental cycle omega of length 4, the authors express omega^m as eta_alpha * partial_alpha(W_ini) with eta_alpha = (alpha * partial_alpha(W_ini))^(m-1) * alpha. They then state that paths in eta_alpha have length 4m-3 and paths in partial_alpha(W_ini) have length 3. Hence short(eta_alpha) + short(partial_alpha(W_ini)) = 4m, which equals length(omega^m). However, Proposition 2.13, as quoted in the paper, requires short(eta_alpha) + short(partial_alpha(W)) > length(l). The proof does not establish the strict inequality; it establishes equality. The same issue recurs in Step II, where the added term p q * partial_alpha(W_ini) again has degree exactly length(l). Thus condition (2) is not verified as written. Even the simplest m=1 case gives omega = alpha * partial_alpha(W_ini) with 1+3 = 4 = length(omega). If [17, Theorem 8.20] actually requires '>' for every term, then Theorem 2.14, and consequently Corollary 0.3 and Theorem 3.5, are not supported by the argument given. If the original theorem uses '>=' instead, then the paper contains a substantive misquotation that should be corrected in a revision. This is a load-bearing internal gap, independent of the well-definedness and connectedness issues noted by the Reader.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper associates to each Postnikov diagram D a quiver with potential (Q(D),W(D)) and an iced version (Q(D),F(D),W(D)), and claims that mutations of these QPs are compatible with geometric exchanges of Postnikov diagrams (Theorem 0.1/2.4). This compatibility is used to define 'a QP of C[Gr(k,n)]' up to mutation-equivalence (Definition 2.5). The authors then prove rigidity and Jacobi-finiteness of these QPs (Theorems 2.7, 2.10) and, via a criterion of Geiß-Labardini-Schröer, uniqueness of the non-degenerate (hence rigid) QP on a Grassmannian cluster algebra quiver up to right-equivalence (Theorem 2.14). From uniqueness they derive Corollary 0.3 and, in Section 3, an isomorphism between the auto-equivalence group of the generalized cluster category and the cluster automorphism group of the associated cluster algebra (Theorem 3.5).","tokens_in":20158,"tokens_out":4158,"duration_ms":43051,"significance":"If the main results hold, this is a substantial contribution: it gives an explicit quiver-with-potential model for Grassmannian cluster algebras, proves rigidity and uniqueness of the potential, and extends the known isomorphism between cluster automorphisms and categorical auto-equivalences to the Grassmannian case. The paper also demonstrates a useful method: reducing a two-dimensional combinatorial problem to checking a list of local configurations and then invoking a general criterion. The authors are careful to distinguish the new results from existing rigidity proofs by Buan-Iyama-Reiten-Smith and Kulkarni. However, the central uniqueness theorem rests on a verification of a stated inequality that is not actually established, and other load-bearing steps are either only sketched or depend on unstated assumptions. The manuscript is therefore not yet ready for publication in its current form.","major_comments":[{"comment":"The proof of condition (2) of Proposition 2.13 is not correct as written. The paper states that paths in η_α have length 4m−3 and paths in ∂_α W_ini have length 3, so short(η_α)+short(∂_α W_ini)=4m. But condition (2) of Proposition 2.13 (quoted from [17, Theorem 8.20]) requires this sum to be strictly greater than length(l)=length(ω^m)=4m. The displayed computation gives equality, not strict inequality. The same issue occurs in Step II, where the additional term p q ∂_α W_ini has degree exactly length(l). Thus the verification does not satisfy the stated hypothesis of Proposition 2.13. This is load-bearing because Theorem 2.14, and consequently Corollary 0.3 and Theorem 3.5, depend on it. The authors must either prove the strict inequality, replace the quoted criterion by the correct version from [17] (if the correct version uses a non-strict inequality), or supply an alternative argument.","section":"§2.3, proof of Theorem 2.14, Step I"},{"comment":"The verification of condition (3) of Proposition 2.13 is too compressed to be valid as written. The proof says condition (3) 'follows immediately' from Proposition 2.12 and from the fact that every non-fundamental cycle has length greater than 4. These observations alone do not show that an arbitrary non-degenerate potential on Q_ini is right-equivalent to W_ini plus terms of degree greater than long(W_ini). One still needs to show that the degree-4 part of any non-degenerate potential is cyclically equivalent to the signed sum of fundamental cycles that defines W_ini, including control of coefficients and signs. This is a nontrivial step and must be spelled out.","section":"§2.3, proof of Theorem 2.14, condition (3)"},{"comment":"The definition of a QP of C[Gr(k,n)] as a QP mutation-equivalent to (Q(D),W(D)) depends on the class being independent of the choice of Postnikov diagram D. The paper does not state or prove that the graph of reduced Postnikov diagrams connected by geometric exchanges is connected for each (k,n). Since the compatibility result Theorem 2.4 is local, it cannot by itself establish well-definedness without this global connectivity assertion. The authors should either prove the connectivity or give a precise reference and explain how it follows.","section":"Definition 2.5"},{"comment":"The proof of Theorem 2.4 explicitly treats only one of the three local configurations shown in Figure 8 and says the others 'can be proved similarly.' Because Theorem 2.4 is the basis for the definition of the QP and IQP for Grassmannian cluster algebras, the omitted cases are load-bearing. The authors should at least state the structure of the remaining computations or give a clear symmetry/rotation argument that reduces all cases to the displayed one.","section":"§2.1, proof of Theorem 2.4"},{"comment":"Step 2 of the proof of Theorem 2.7 asserts that any fundamental cycle ω_2 can be connected to the bottom-left fundamental cycle ω_1 by repeatedly applying Lemma 2.8, but no proof is given that the dual graph of fundamental cycles sharing an arrow is connected. Since Lemma 2.8 only propagates rigidity from one fundamental cycle to another sharing an arrow, the argument requires this connectivity as an explicit premise. The authors should either prove the connectivity or cite a result that implies it.","section":"§2.2, proof of Theorem 2.7, Step 2"}],"minor_comments":[{"comment":"There is a typo in the abstract: 'Postnilov Diagram' should be 'Postnikov Diagram'.","section":"Abstract"},{"comment":"Several cross-references appear as 'Remarks ??, 2.9 and 2.11' or 'Remarks ??' with unresolved placeholders; these should be fixed.","section":"Remark 1.1 and text after Definition 2.1"},{"comment":"The name 'Buar-King-Marsh' is a typo for 'Baur-King-Marsh'.","section":"Introduction, paragraph after Theorem 0.1"},{"comment":"In the displayed potential after the pre-mutation, the term '−ξt' is used; it would help to define the path t explicitly at that point, since t appears both as a path and as a target map in the quiver conventions.","section":"§2.1, proof of Theorem 2.4"},{"comment":"The construction in Step 1 says 'we repeat above construction until a is never a vertex on a cycle l′', but the termination of this process is asserted rather than proved. A short termination argument using the finiteness of the quiver and monotonicity of the relevant coordinate would improve clarity.","section":"§2.2, Lemma 2.6"},{"comment":"The phrase 'unique rigid QP' in Theorem 2.14 is stronger than what is proved unless rigidity coincides with non-degeneracy for the QPs in question; the paper notes rigid implies non-degenerate, but the converse is not established. The statement should be phrased carefully as 'unique rigid among QPs on Q up to right-equivalence' only if the proof covers it, or a reference should be given.","section":"§2.3, Theorem 2.14"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the verification of the Geiß-Labardini-Schröer criterion in Theorem 2.14: the paper proves equality where the quoted theorem demands a strict inequality. This is a concrete, fixable gap, but it currently invalidates the uniqueness result and its application. The authors should also address the connectedness assumption in Definition 2.5 and the residual handwaving in Theorems 2.4 and 2.7. If the inequality issue can be resolved (e.g., by correcting the quoted criterion or strengthening the proof), the paper would be a solid contribution; as it stands, the central claim is not supported by the argument given."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThe paper does two things. First, it gives an explicit QP associated to a Postnikov diagram and proves mutation of the QP matches geometric exchange. That part is mostly a detailed write-up of a known picture, with the initial diagram spelled out. Second, it proves uniqueness of the non-degenerate QP on the quiver of a Grassmannian cluster algebra up to right-equivalence, and uses that to get AutT(C) ≅ Aut(Aφ(T)). The uniqueness result is genuinely new, as the authors acknowledge rigidity was already known from Buan–Iyama–Reiten–Smith and Kulkarni. The explicit initial QP description is useful and is what carries the new argument.\n\nThe main soft spot is real. In the proof of Theorem 2.14, Step I, the authors claim short(ηα) + short(∂αW_ini) > 4m, but their own calculation gives 4m−3 + 3 = 4m, equality. They are quoting Proposition 2.13 with a strict inequality. So condition (2) is not verified as written. This is a load-bearing gap: Theorem 2.14 is the foundation for Corollary 0.3 and Theorem 3.5. It may be fixable—if the original Geiß–Labardini–Schröer theorem actually uses ≥, then it is a misquotation that can be corrected; if the strict inequality is genuinely required, the argument needs a different idea—but as written it does not go through.\n\nThere are smaller issues. Definition 2.5 silently assumes the geometric exchange graph of reduced Postnikov diagrams is connected, so the QP of C[Gr(k,n)] is well-defined; that fact is standard but should be stated or cited. Theorem 2.4 proves one local configuration and says the rest \"can be proved similarly\"; for a claim that is this central, that is not shameful, but it is an omission a referee will want filled. The paper also has unresolved cross-references and a typo or two, indicating it was not fully polished for submission.\n\nThe citation pattern is fine: prior work gets credit, the new claims are flagged as new. The proof is a derivation from definitions, no fitted parameters.\n\nWho is this for? People working on cluster categories, Jacobian algebras, and Grassmannian cluster algebras. The application to automorphism groups is a nice bonus. The paper deserves a serious referee—the uniqueness result is important if the gap is repairable—but the referee should focus on the strict inequality issue first.\n\nI would not cite the central theorem until the gap is addressed, but I would send it out.","headline":"Useful and partly new uniqueness result for Grassmannian cluster algebra QPs, but the proof of the key theorem has a strict-vs-equality inequality gap that must be fixed.","tokens_in":20640,"tokens_out":3449,"would_cite":false,"duration_ms":33370,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F60","16G20","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a Grassmannian cluster algebra, the quiver alone determines the potential, and this uniqueness identifies the symmetry groups of its category and cluster algebra.","keywords":["quivers with potentials","Grassmannian cluster algebras","Postnikov diagrams","rigidity","Jacobian algebras","generalized cluster categories","cluster automorphism groups","mutation"],"falsifier":"Exhibit two reduced $(k,n)$-Postnikov diagrams for the same pair $(k,n)$ that are not connected by geometric exchanges, which would make the QP in Definition 2.5 ill-defined. Alternatively, find a quiver $Q$ in the Grassmannian mutation class and a non-degenerate potential $W'$ on $Q$ that is not right-equivalent to $W$ or $W^{\\mathrm{op}}$; even a single 2-cycle appearing after an iterated mutation of $(Q,W)$ would disprove rigidity.","tokens_in":19616,"feed_emoji":"📐","tokens_out":9617,"duration_ms":85386,"temperature":0.7,"pith_summary":"Associated to any Grassmannian $\\mathbb{C}[\\mathrm{Gr}(k,n)]$ there is a natural quiver with potential, built from a Postnikov diagram $D$: the quiver is dual to the diagram, and the potential is the alternating sum of the fundamental cycles corresponding to its oriented regions. The paper proves that mutating this quiver with potential matches, up to right-equivalence, the geometric exchange operation on $D$, so the quiver with potential is independent of the diagram chosen and deserves to be called the quiver with potential of the Grassmannian cluster algebra. It is shown to be rigid (no mutation ever creates a 2-cycle) and Jacobian-finite, and to be the unique non-degenerate, hence unique rigid, quiver with potential on its underlying quiver. From this uniqueness, any quiver in the mutation class determines its potential up to right-equivalence, which in turn implies that the auto-equivalence group of the generalized cluster category $\\mathcal{C}_{(Q,W)}$ is isomorphic to the cluster automorphism group of the associated Grassmannian cluster algebra with trivial coefficients.","feed_headline":"Quiver determines potential for Grassmannian cluster algebras","feed_subtitle":"Planar Postnikov diagrams fix a unique rigid potential, aligning category symmetries with cluster automorphisms.","key_machinery":"The load-bearing object is the Postnikov diagram $D$ together with its dual quiver $Q(D)$: vertices are the alternating oriented regions into which the strands cut the disk, and arrows record crossings of strands. The potential $W(D)$ is the signed sum of the fundamental cycles of $Q(D)$, one for each oriented region, taken clockwise with positive sign and anticlockwise with negative sign; the iced version adds frozen boundary arrows. The decisive mechanism is the compatibility theorem: applying the quiver-with-potential mutation at the vertex corresponding to an alternating quadrilateral cell $R$ gives, after reduction and up to right-equivalence, exactly the quiver with potential of the geometrically exchanged diagram $\\mu_R(D)$. Uniqueness is then produced by the general uniqueness criterion from [17], which says a QP satisfying a finiteness condition, a Jacobian-ideal condition on long cycles, and a perturbation condition is the unique non-degenerate QP on its quiver; this criterion is checked directly on the initial diagram $D_{\\mathrm{ini}}$.","core_discovery":"The central claim is that the Grassmannian cluster algebra $\\mathbb{C}[\\mathrm{Gr}(k,n)]$ has a canonical quiver with potential $(Q,W)$, well-defined up to mutation and right-equivalence, and that $(Q,W)$ is rigid, Jacobi-finite, and unique: every non-degenerate potential on the same quiver $Q$ is right-equivalent to $W$, and every non-degenerate potential on the opposite quiver $Q^{\\mathrm{op}}$ is right-equivalent to $W^{\\mathrm{op}}$. The proof constructs $W(D)$ from a Postnikov diagram as the signed sum of fundamental cycles, proves mutation compatibility with geometric exchange, and then applies a uniqueness criterion provided in [17] to the initial diagram. The consequence the authors draw is that, within the mutation class of a Grassmannian cluster algebra, the quiver determines the potential, so the cluster automorphism group of the algebra coincides with the auto-equivalence group of its generalized cluster category.","pith_inferences":["The same uniqueness route should work for any cluster algebra whose quivers arise from dimer models with a signed fundamental-cycle potential, including the dimer-model, unipotent-group, and double-Bruhat-cell settings the authors list, so quiver-determines-potential may hold for a broad class of Jacobian-finite QPs beyond Grassmannians.","Because the iced QP is neither rigid nor Jacobi-finite, the frozen-vertex version cannot carry the same uniqueness theorem; a separate theory would be needed to decide whether iced quivers also determine their potentials.","A practical test of the uniqueness theorem: take the initial quiver $Q_{\\mathrm{ini}}$ for a small pair like $(k,n)=(3,7)$, enumerate non-degenerate potentials on $Q_{\\mathrm{ini}}$ by computer, and check right-equivalence to $W_{\\mathrm{ini}}$; the criterion predicts that every sufficiently long-cycle perturbation is absorbed by a right-equivalence.","The category-algebra isomorphism suggests that cluster automorphism groups of Grassmannian cluster algebras can be computed by classifying auto-equivalences of the generalized cluster category, a problem where tilting theory supplies tools that are not available on the algebra side."],"forward_implications":["For any two reduced Postnikov diagrams representing the same $(k,n)$, the associated quivers with potentials are mutation-equivalent, so the QP of a Grassmannian cluster algebra is a well-defined invariant rather than an artifact of a diagram choice.","The QP $(Q,W)$ is rigid and non-degenerate: no iterated mutation produces a 2-cycle, and the Jacobian algebra $P(Q,W)$ is finite-dimensional.","If $Q'$ is a quiver in the mutation class and $Q'\\cong Q$, then any QP $(Q',W')$ over $Q'$ is right-equivalent to $(Q,W)$; if $Q'\\cong Q^{\\mathrm{op}}$, it is right-equivalent to $(Q^{\\mathrm{op}},W^{\\mathrm{op}})$.","The generalized cluster category $\\mathcal{C}_{(Q,W)}$ has the same symmetry group as the cluster algebra: $\\mathrm{Aut}_T(\\mathcal{C}_{(Q,W)}) \\cong \\mathrm{Aut}(\\mathcal{A}_{(Q,W)})$ for trivial coefficients.","The authors conjecture the same quiver-determines-potential principle for all mutation classes of non-degenerate QPs, which would give the category-algebra isomorphism for every generalized cluster category; this paper establishes that case for Grassmannians."],"supporting_citations":[{"why":"Supplies the Postnikov-diagram quiver $Q(D)$, the cluster structure on Grassmannians, and the quiver-mutation/geometric-exchange compatibility that the paper upgrades to potentials.","marker":"[28]"},{"why":"Introduces quivers with potentials, their mutation, reduction, and right-equivalence; the paper performs potential mutation in this formalism.","marker":"[12]"},{"why":"Provides the uniqueness criterion that Theorem 2.14 verifies to prove $(Q,W)$ is the unique non-degenerate QP on its quiver.","marker":"[17]"},{"why":"Defines mutation and reduction for iced quivers with potentials, giving the IQP machinery used for the frozen version $(\\bar{Q}(D),F(D),\\bar{W}(D))$.","marker":"[27]"},{"why":"Establishes mutation of cluster-tilting objects and potentials and gives prior rigidity results; the paper builds on this framework for mutation equivalence and the generalized cluster category.","marker":"[8]"},{"why":"Supplies the surface QP techniques, including the sign-change argument used in Lemma 2.3 and the rigidity strategy adapted to the grid-like initial quiver.","marker":"[25]"},{"why":"Gives derived equivalences from mutations of quivers with potentials, used in Proposition 3.4 to lift cluster automorphisms to auto-equivalences.","marker":"[23]"},{"why":"Constructs the generalized cluster category $\\mathcal{C}_{(Q,W)}$ whose auto-equivalence group is compared with the cluster automorphism group.","marker":"[1]"},{"why":"Realizes the iced Jacobian algebra as the endomorphism algebra of a cluster-tilting object in a Frobenius category and shows invariance under geometric exchange, grounding the IQP definition.","marker":"[6]"}],"fun_headline_variants":["Quiver alone fixes unique rigid potential for Grassmannian cluster algebras","Grassmannian cluster algebras: quiver determines unique potential","Unique rigid potential from Postnikov diagrams for Grassmannian cluster algebras","Quiver forces unique potential, symmetries match in Grassmannian cluster algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the assumption that any two reduced $(k,n)$-Postnikov diagrams are connected by geometric exchanges, so the mutation class of the quiver with potential does not depend on which diagram one starts from; the proof of the local compatibility theorem also presents one configuration in detail and asserts the remaining configurations are similar.","fun_headline_variants_meta":{"raw":{"variants":["Quiver alone fixes unique rigid potential for Grassmannian cluster algebras","Grassmannian cluster algebras: quiver determines unique potential","Unique rigid potential from Postnikov diagrams for Grassmannian cluster algebras","Quiver forces unique potential, symmetries match in Grassmannian cluster algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000498,"raw_usage":{"total_tokens":2436,"prompt_tokens":940,"completion_tokens":1496,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":556,"completion_tokens_details":{"reasoning_tokens":1420}},"tokens_in":556,"tokens_out":1496,"duration_ms":9923,"temperature":1.0,"reasoning_tokens":1420,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:52:23.844377+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit two reduced $(k,n)$-Postnikov diagrams for the same pair $(k,n)$ that are not connected by geometric exchanges, which would make the QP in Definition 2.5 ill-defined. Alternatively, find a quiver $Q$ in the Grassmannian mutation class and a non-degenerate potential $W'$ on $Q$ that is not right-equivalent to $W$ or $W^{\\mathrm{op}}$; even a single 2-cycle appearing after an iterated mutation of $(Q,W)$ would disprove rigidity.","supporting_citations":[{"cited_title":"Scott, Grassmannians and cluster algebras, Proc","cited_arxiv_id":null,"evidence_quote":"Supplies the Postnikov-diagram quiver $Q(D)$, the cluster structure on Grassmannians, and the quiver-mutation/geometric-exchange compatibility that the paper upgrades to potentials."},{"cited_title":"Derksen, J","cited_arxiv_id":null,"evidence_quote":"Introduces quivers with potentials, their mutation, reduction, and right-equivalence; the paper performs potential mutation in this formalism."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the uniqueness criterion that Theorem 2.14 verifies to prove $(Q,W)$ is the unique non-degenerate QP on its quiver."},{"cited_title":"Pressland, Mutation of frozen Jacobian algebras, J","cited_arxiv_id":null,"evidence_quote":"Defines mutation and reduction for iced quivers with potentials, giving the IQP machinery used for the frozen version $(\\bar{Q}(D),F(D),\\bar{W}(D))$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes mutation of cluster-tilting objects and potentials and gives prior rigidity results; the paper builds on this framework for mutation equivalence and the generalized cluster category."},{"cited_title":"Labardini-Fragoso Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions, Selecta Math","cited_arxiv_id":null,"evidence_quote":"Supplies the surface QP techniques, including the sign-change argument used in Lemma 2.3 and the rigidity strategy adapted to the grid-like initial quiver."},{"cited_title":"Keller, D","cited_arxiv_id":null,"evidence_quote":"Gives derived equivalences from mutations of quivers with potentials, used in Proposition 3.4 to lift cluster automorphisms to auto-equivalences."},{"cited_title":"Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential","cited_arxiv_id":null,"evidence_quote":"Constructs the generalized cluster category $\\mathcal{C}_{(Q,W)}$ whose auto-equivalence group is compared with the cluster automorphism group."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Realizes the iced Jacobian algebra as the endomorphism algebra of a cluster-tilting object in a Frobenius category and shows invariance under geometric exchange, grounding the IQP definition."}],"review_version":1}