{"id":"6efef962-09bb-4039-a30d-704d5d14a1be","arxiv_id":"2505.22770","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For algebras R⊗kQ with R local and Q acyclic, BHM mutation of complete τ-exceptional sequences coincides with classical mutation, and the braid group acts transitively.","lead":"This paper proves that for algebras built from a local commutative ring and an acyclic directed graph, two different mutation procedures for module sequences agree, and the braid group acts transitively on the resulting complete sequences. The result adds a new class of algebras to the short list where such symmetry is known to hold.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.16 is exactly as secure as the imported bijection [Non25, Thm. 5.10]: if induction is not surjective on τ-exceptional sequences, the comparison with classical mutation has no ground.","rationale":"The reader identified Theorem 2.5 and Theorem 1.8 as the weakest assumption, and my review agrees. I examined the internal proof for independent gaps: Proposition 3.17's left-regular case follows from Lemma 3.18 and the BHM results, and the left-irregular case is a line-by-line translation of BHM Prop. 6.7 using the technical lemmas from [Non25]. I did not find an internal mathematical contradiction. Lemma 4.4's proof is terse but defensible using wide subcategories and projective dimension at most one. Proposition 4.5 relies on standard tensor-hom isomorphisms for the induction functor. The only serious risk is external: the paper's main theorem is conditional on a network of results in [Non25], which is cited but not proved here and is not yet a published, independently verified source. Since the reader's CONDITIONAL verdict already reflects exactly this dependency, my stress-test does not change the verdict.","tokens_in":20199,"tokens_out":20768,"duration_ms":228504,"concrete_test":"Verify the two imported theorems in the smallest nontrivial case, R=k[x]/(x^2) and Q=A_2. Enumerate all complete τ-exceptional sequences over Λ=RQ and all complete exceptional sequences over kQ, and confirm that the induction functor gives a bijection as asserted by Theorem 2.5. Also compute every τ-perpendicular subcategory of mod Λ and confirm that each is equivalent to mod R⊗Q' for an acyclic quiver Q' as asserted by Theorem 1.8. A failure in this case would refute the structural ground of Theorem 3.16; success would provide direct evidence for the dependency.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of the main theorem is not self-contained at its two structural joints. Theorem 3.16 starts by writing a complete τ-exceptional sequence over Λ as Λ⊗kQ M for a unique exceptional sequence M over kQ; this is Theorem 2.5, imported from [Non25]. The termwise comparison in the proof then requires that the mutated first entry in J(M_{i+2},...,M_n) equals Λ⊗Y, where the ambient category is identified with mod R⊗Q' via Theorem 1.8, also imported. Corollary 3.13 (mutation completeness) likewise uses Theorem 1.8. The proof of Proposition 3.21 additionally invokes [Non25, Prop. 1.8(iii), Cor. 4.11, Lemma 6.22], so the dependency is broader than just two quoted theorems. If any of these structural results fail, the identification of BHM mutation with classical mutation collapses. This is not an internal inconsistency: the paper's arguments from those points onward are coherent. But it is a load-bearing external dependency, and [Non25] is an unreviewed preprint whose statements are asserted without proof here.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies τ-exceptional sequences over Λ = R ⊗_k kQ for a finite-dimensional commutative local k-algebra R and an acyclic quiver Q. It recalls from the author's preprint [Non25] that induction Λ⊗_{kQ}– gives a bijection between exceptional sequences over kQ and τ-exceptional sequences over Λ (Theorem 2.5), and that every τ-perpendicular subcategory of mod Λ is equivalent to mod(R ⊗_k Q') for an acyclic Q' (Theorem 1.8). Using Buan–Hanson–Marsh mutation, the author proves that every τ-exceptional pair over Λ is left and right mutable (Proposition 3.8, Corollary 3.13); the main theorem (Theorem 3.16) shows BHM mutation φ_i on complete τ-exceptional sequences over Λ coincides, via induction, with the classical Crawley-Boevey–Ringel mutation σ_i on complete exceptional sequences over kQ. This gives a transitive braid-group action on complete τ-exceptional sequences in mod Λ (Corollary 3.22). Section 4 identifies τ-exceptional sequences with R-exceptional RQ-lattices and compares the braid actions; Section 5 gives a worked example.","tokens_in":20367,"tokens_out":10497,"duration_ms":117518,"significance":"If the main theorem is correct, it is a meaningful extension of the hereditary case: it provides a new class of algebras—beyond hereditary, τ-tilting finite, and rank two—that are mutation complete and on which the braid group acts transitively. The comparison between BHM mutation and classical mutation is natural, and the paper's own arguments are detailed and coherent. The worked example and the connection to Crawley-Boevey's R-exceptional sequences add value. The main caveat is the heavy, load-bearing reliance on structural results imported from the same-author preprint [Non25], which are not re-verified here; the significance is therefore conditional on those results.","major_comments":[{"comment":"The central claim is conditional on unverified imported results. Theorem 2.5 ([Non25, Thm. 5.10]) supplies the bijection between (τ-)exceptional sequences over kQ and τ-exceptional sequences over Λ, and Theorem 1.8 ([Non25, Thm. 4.10]) identifies every τ-perpendicular subcategory of mod Λ as mod(R ⊗_k Q'). Both are used at load-bearing points: Theorem 3.16 uses Theorem 2.5 to write a complete τ-exceptional sequence as Λ⊗_{kQ}M and Theorem 1.8 for every ambient perpendicular category; Corollary 3.13 uses Theorem 1.8; Proposition 3.9 uses Theorem 2.5 and Proposition 1.7; Proposition 3.21 invokes [Non25, Prop. 1.8(iii), Cor. 4.11, Lemma 6.22]. The arguments after these imports are coherent, but if any of the imported statements failed, the identification φ_i(Λ⊗M) = Λ⊗σ_i(M) would lose its ground. The revision should either include proofs of the imported structural results, or state explicitly that the theorems are proved modulo [Non25] and give a precise account of which statements of [Non25] are assumed.","section":"§1–§2, Theorems 1.8 and 2.5; used throughout §3"},{"comment":"In the left irregular case, the proof asserts \"As shown in the Proof of [BHM24, Prop. 6.2], we have that Ext^1_{kQ}(B,C) ≠ 0\". This assertion is load-bearing because Proposition 3.21, on which the irregular case rests, is stated only under the hypotheses Ext^1_{kQ}(B,C) ≠ 0 and C ∉ proj kQ. The manuscript does not derive these hypotheses for the irregular pair in the relevant ambient category. Since this is a necessary condition to apply Proposition 3.21, the step should be proved explicitly, or Proposition 3.21 should be reformulated so that the zero-extension case is also covered.","section":"§3, proof of Proposition 3.17 (left irregular case)"}],"minor_comments":[{"comment":"In the statement of Proposition 3.21, the hypothesis is printed as \"Ext1_Λ(B,C), 0\" and \"C < proj kQ\"; it should read \"Ext^1_{kQ}(B,C) ≠ 0\" and \"C ∉ proj kQ\".","section":"§3, Proposition 3.21"},{"comment":"In the proof of Theorem 3.16, the notation \"J(M_{i+1},···, M_n)\" should be \"J(M_{i+2},···, M_n)\" (or the corresponding induced sequence in mod Λ), to match Definition 3.10.","section":"§3, proof of Theorem 3.16"},{"comment":"Definition 3.10(c) says that M is left i-irregular if (M_i, M_{i+1}) is left regular; the word \"regular\" should be \"irregular\".","section":"§3, Definition 3.10(c)"},{"comment":"In the proof of Proposition 3.21(iii), the expressions \"Hom_kQ(Λ⊗_{kQ} C, X)\" and \"Hom_kQ(Λ⊗_{kQ} E', X)\" should use Hom_Λ rather than Hom_kQ.","section":"§3, Proposition 3.21(iii), proof"},{"comment":"There are several typographical slips to correct: \"(Y1,,···, Yn)\" in Theorem 2.4, \"τ-exeptional\" in Corollary 2.6, and \"arizes\" in the introduction.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of a representation-theory journal and the main idea is appealing. The main risk is the unverified dependence on [Non25], a preprint by the same author; if the editor can obtain an independent check of Theorems 1.8 and 2.5, or the author includes their proofs, the paper would be much stronger. The typographical issues in Section 3 should also be corrected in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Key point: this paper proves a real theorem, Theorem 3.16, that BHM mutation of complete τ-exceptional sequences over Λ = R⊗kQ agrees with classical mutation of exceptional sequences, and the braid group acts transitively. That is a meaningful extension of the hereditary case to a family that includes non-hereditary, τ-tilting infinite algebras.\n\nWhat is genuinely new: the mutation coincidence itself, the mutation-completeness of Λ (Corollary 3.13), and the comparison with Crawley-Boevey's R-exceptional sequences in Section 4. The proof of Proposition 4.5 is a neat, short argument that R-exceptional sequences and τ-exceptional sequences coincide here.\n\nThe paper is well-written and the proof of Proposition 3.17, once the imports are granted, is coherent. The author is careful to cite [Non25] for the two load-bearing structural results: the bijection between (τ-)exceptional sequences over kQ and τ-exceptional sequences over Λ (Theorem 2.5) and the description of τ-perpendicular subcategories as module categories over R⊗Q′ (Theorem 1.8). These are not proved here. That is the soft spot. If either failed, the main comparison would lose its ground. The reader's conditional verdict is right: this is a standalone paper only in a weak sense. It is honest about the dependency, but a referee cannot fully verify the main theorem without the same-author preprint.\n\nMinor issues: there are typos in Proposition 3.21 (a stray comma in Ext^1_Λ(B,C), 0) and in the statement/terminology in Definition 3.10(c) where \"left regular\" appears in the irregular clause. These are cosmetic.\n\nMy bottom line: the math is very likely correct, and the paper deserves a serious referee. I would send it out, with a note to the referee that they should check the dependence on [Non25], and ideally ask the author to either wait for that preprint to be accepted or include the needed statements with proofs in an appendix. The result is useful for people working on τ-tilting theory and mutation.","headline":"Real new result, but the main theorem leans on an unreviewed same-author preprint; worth refereeing.","tokens_in":20961,"tokens_out":2668,"would_cite":true,"duration_ms":30441,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","16G70","16G10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that BHM mutation of $\\tau$-exceptional sequences over $\\Lambda=R\\otimes_k kQ$ agrees with the induced classical mutation, so the braid group acts transitively on complete $\\tau$-exceptional sequences.","keywords":["tau-exceptional sequences","exceptional sequences","mutation","braid group","quivers over local algebras","tau-perpendicular categories","RQ-lattices","hereditary algebras"],"falsifier":"Take $\\Lambda=R\\otimes_k kQ$ with $R=k[x]/(x^2)$ and $Q$ a three-vertex wild quiver, list all complete $\\tau$-exceptional sequences by induction from $kQ$, and compute the BHM mutation $\\varphi_1$ on any one of them directly from the pair formulas; if the first entry of $\\varphi_1(\\Lambda\\otimes_{kQ} M)$ is not isomorphic to $\\Lambda\\otimes_{kQ}$ applied to the first entry of $\\sigma_1(M)$, the theorem fails.","tokens_in":19945,"feed_emoji":"🔄","tokens_out":6665,"duration_ms":65272,"temperature":0.7,"pith_summary":"This paper establishes a precise compatibility between two notions of mutation for a family of algebras built from an acyclic quiver $Q$ and a local commutative finite-dimensional algebra $R$: the algebra $\\Lambda=R\\otimes_k kQ$. The main theorem says that the BHM mutation $\\varphi_i$ of a complete $\\tau$-exceptional sequence induced from $kQ$ equals the induced classical mutation $\\Lambda\\otimes_{kQ}\\sigma_i(M)$ at every position $i$. Since classical mutation over the acyclic quiver is generated by a transitive braid group action, the same braid group action is transitive on complete $\\tau$-exceptional sequences over $\\Lambda$. This matters because $\\Lambda$ is generally neither hereditary, nor $\\tau$-tilting finite, nor of rank two, the classes where mutability of $\\tau$-exceptional pairs was previously known; the paper also proves every $\\tau$-exceptional pair over $\\Lambda$ is mutable in both directions.","feed_headline":"Braid group acts transitively on tau-exceptional sequences","feed_subtitle":"For Λ=R⊗kQ, BHM mutation lifts classical mutation, so complete sequences are all related by braid moves.","key_machinery":"The load-bearing device is the induction functor $\\Lambda\\otimes_{kQ}-$ together with the bijection it induces between $(\\tau-)$exceptional sequences in $\\operatorname{mod} kQ$ and $\\tau$-exceptional sequences in $\\operatorname{mod}\\Lambda$. Around it sit two structural facts: every indecomposable $\\tau$-rigid $\\Lambda$-module is induced from a $\\tau$-rigid $kQ$-module, and every $\\tau$-perpendicular subcategory of $\\operatorname{mod}\\Lambda$ is equivalent to $\\operatorname{mod}(R\\otimes_k Q')$ for an acyclic quiver $Q'$. Mutation of $\\tau$-exceptional pairs is then compared by checking the BHM left and right regularity definitions against the classical exchange pair, using the explicit formulas for $\\varphi$ and the uniqueness property of complete $\\tau$-exceptional sequences to identify the mutated entry.","core_discovery":"On the paper's own terms: given a complete $\\tau$-exceptional sequence $\\Lambda\\otimes_{kQ} M$ in $\\operatorname{mod}\\Lambda$, its BHM mutation at position $i$ is exactly the induction of the classical mutation $\\sigma_i(M)$ of the underlying complete exceptional sequence in $\\operatorname{mod} kQ$. The proof transfers the pair-wise mutation comparison from the hereditary case: one rewrites each $\\tau$-exceptional pair as an induced pair, separates the left-regular and left-irregular cases, and uses the fact that every $\\tau$-perpendicular subcategory of $\\operatorname{mod}\\Lambda$ is again of the form $\\operatorname{mod}(R\\otimes_k Q')$ for an acyclic quiver $Q'$. The conclusion is that the BHM mutation and the classical mutation are the same operation after applying the induction functor, and consequently the braid group acts transitively on complete $\\tau$-exceptional sequences in $\\operatorname{mod}\\Lambda$.","pith_inferences":["The same mechanism may extend to any finite-dimensional algebra whose $\\tau$-perpendicular subcategories are all module categories of acyclic-quiver algebras with a compatible induction functor, giving braid transitivity for that broader class; this generalization is not claimed in the paper.","The compatibility suggests that, over these tensor-product algebras, $\\tau$-exceptional mutation carries no information beyond the underlying quiver: every mutation-theoretic question can be reduced to classical quiver mutation.","The coincidence with $R$-exceptional lattices may make the transitive braid action useful for computing rank-vector invariants along braid orbits of $RQ$-lattices, a direction the paper leaves implicit."],"forward_implications":["Every $\\tau$-exceptional pair over $\\Lambda=R\\otimes_k kQ$ is both left and right mutable, so $\\Lambda$ is mutation complete.","The braid group acts transitively on the set of complete $\\tau$-exceptional sequences in $\\operatorname{mod}\\Lambda$.","Complete $\\tau$-exceptional sequences over $\\Lambda$ are in bijection with complete exceptional sequences over $kQ$, and the mutation graph over $\\Lambda$ is the image of the classical mutation graph under induction.","Crawley-Boevey's $R$-exceptional sequences of $RQ$-lattices coincide with $\\tau$-exceptional sequences, and the two braid actions agree.","Braid relations such as $\\varphi_1\\varphi_2\\varphi_1=\\varphi_2\\varphi_1\\varphi_2$ hold for complete $\\tau$-exceptional sequences over $\\Lambda$."],"supporting_citations":[{"why":"Establishes the induction bijection on $\\tau$-exceptional sequences and the structure of $\\tau$-perpendicular subcategories of $\\operatorname{mod}\\Lambda$; the paper's main comparison rests on these results.","marker":"[Non25]"},{"why":"Defines BHM mutation of $\\tau$-exceptional pairs and sequences and proves the hereditary case in which BHM and classical mutation agree.","marker":"[BHM24]"},{"why":"Introduces $\\tau$-exceptional sequences and the $\\tau$-perpendicular recursion used throughout.","marker":"[BM21b]"},{"why":"Defines classical exceptional sequences and proves the transitive braid group action over hereditary algebras.","marker":"[CB92]"},{"why":"Develops the braid group action on complete exceptional sequences for hereditary Artin algebras.","marker":"[Rin94]"},{"why":"Introduces $RQ$-lattices and $R$-exceptional sequences and proves transitivity of the braid action used in the Section 4 comparison.","marker":"[CB24]"},{"why":"Provides the uniqueness property of complete $\\tau$-exceptional sequences used to identify mutated entries.","marker":"[HT24]"},{"why":"Introduces $\\tau$-perpendicular categories, the setting in which $\\tau$-exceptional pairs are mutated.","marker":"[Jas15]"}],"fun_headline_variants":["Braid group ties all tau-exceptional sequences together","Local algebra quivers: mutation unifies classical and tau","Transitive braid action on tau-exceptional sequences","Same mutation for tau and classical exceptional sequences"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on two imported structural results: the induction functor gives a bijection on $\\tau$-exceptional sequences, and every $\\tau$-perpendicular subcategory of $\\operatorname{mod}\\Lambda$ is again equivalent to $\\operatorname{mod}(R\\otimes_k Q')$ for an acyclic quiver $Q'$; if either failed, the comparison of mutations would lose its ground.","fun_headline_variants_meta":{"raw":{"variants":["Braid group ties all tau-exceptional sequences together","Local algebra quivers: mutation unifies classical and tau","Transitive braid action on tau-exceptional sequences","Same mutation for tau and classical exceptional sequences"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000134,"raw_usage":{"total_tokens":1090,"prompt_tokens":850,"completion_tokens":240,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":176}},"tokens_in":466,"tokens_out":240,"duration_ms":2948,"temperature":1.0,"reasoning_tokens":176,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:01:45.812676+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $\\Lambda=R\\otimes_k kQ$ with $R=k[x]/(x^2)$ and $Q$ a three-vertex wild quiver, list all complete $\\tau$-exceptional sequences by induction from $kQ$, and compute the BHM mutation $\\varphi_1$ on any one of them directly from the pair formulas; if the first entry of $\\varphi_1(\\Lambda\\otimes_{kQ} M)$ is not isomorphic to $\\Lambda\\otimes_{kQ}$ applied to the first entry of $\\sigma_1(M)$, the theorem fails.","supporting_citations":[],"review_version":1}