{"id":"9c4c28e6-4905-433e-97cd-fa6889e50295","arxiv_id":"2506.21372","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Mutation of complete τ-exceptional sequences is transitive for every τ-tilting finite algebra.","lead":"This paper proves that for any finite-dimensional algebra with finitely many support tau-rigid modules, all complete tau-exceptional sequences lie in one orbit under mutation. It settles the transitivity question for the tau-tilting finite case, unifying earlier partial results.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proof is internally coherent, with the only substantive risk being reliance on the external gen-minimality criterion [8, Thm. 2.14].","rationale":"The reader's weakest-assumption analysis correctly identifies [8, Thm. 2.14] as the pivotal external input. My pass confirms that the internal chain of reductions is coherent: Lemma 2.1 supplies the gen-minimal pair, Lemma 2.2 realizes adjacent swaps, Proposition 2.3 lifts these to TF-orderings, Corollary 2.4 handles gen-minimal representatives, and Lemma 2.5 provides the descent. I paid special attention to Lemma 2.5's maximality argument: the replacement by a permuted TF-ordering stays in the same mutation orbit because all intermediate permutations are TF by the maximality of s, and the contradiction T(X') superset T(X) is valid. The only unresolved risk is the external theorem, which the paper does not prove; this is a normal mathematical dependency, but it is not an internal inconsistency. Minor typos (e.g., 'hopitality', 'U/ell+1') do not affect correctness. Therefore the ACCEPT verdict with moderate confidence remains appropriate.","tokens_in":995,"tokens_out":749,"duration_ms":301502,"concrete_test":"Independently verify [8, Thm. 2.14]: prove from Theorem 1.1(a) that for every basic tau-rigid module M, gen-minimality is equivalent to M = Ps(perp J(M)), and check the identity computationally for all tau-tilting finite algebras of rank at most 3 (for example, all orientations of the A3 quiver). If any gen-minimal M fails the identity, the proof of Theorem 2.6 collapses at Corollary 2.4 and Lemma 2.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. I traced the main argument: Theorem 2.6 follows from Lemma 2.5 (every mutation orbit contains a sequence whose inverse TF-ordering has gen-minimal direct sum) and Corollary 2.4 (gen-minimal representatives with the same J are mutation-equivalent). Corollary 2.4 uses Proposition 2.3, which reduces adjacent swaps to Lemma 2.2; Lemma 2.5 uses Proposition 2.3 both to permute TF-orderings and to justify the maximality contradiction. I found no internal gap: the chain of TF-orderings in Lemma 2.5 is valid by maximality of s, the descending chain in Lemma 2.2 Case 3 is finite by tau-tilting finiteness, and the use of Lemma 1.2 in Lemma 2.5 is sound. The genuinely load-bearing external input is Theorem 1.20, quoted from [8, Thm. 2.14] and not proved here. If that characterization failed, both the identification of gen-minimal modules in Proposition 2.3 and the reduction in Corollary 2.4 and Lemma 2.5 would break. This is a dependency risk, not a detected error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proves that for a τ-tilting finite algebra Λ, mutation of complete τ-exceptional sequences is transitive. The precise statement is Theorem 2.6: for any wide subcategory W of rank r, any two τ-exceptional sequences in τ-es(W) are in the same orbit under the mutations φ_{r+1}, ψ_{r+1}, ..., φ_{n-1}, ψ_{n-1}; taking W = 0 gives transitivity on complete τ-exceptional sequences. The proof uses the bijection ω between TF-ordered τ-rigid modules and τ-exceptional sequences, the E-map, and gen-minimal τ-rigid modules. After establishing that adjacent swaps in a TF-ordering can be realized by mutations (Proposition 2.3, via Lemma 2.2), the authors reduce to the case of gen-minimal modules (Corollary 2.4) and prove every mutation orbit contains a sequence whose associated TF-ordered module is gen-minimal (Lemma 2.5). The argument is detailed and internally coherent, and the main theorem is not assumed anywhere in the proof.","tokens_in":13165,"tokens_out":12093,"duration_ms":122109,"significance":"If correct, this paper settles a natural question left open in [8] and generalizes the previously known rank-two and Nakayama cases. The proof is well structured: the reduction to TF-orderings, the treatment of adjacent swaps, and the descent to gen-minimal representatives are clear. A particular strength is that the paper gives a complete chain of implications from Lemma 2.2 through Proposition 2.3, Corollary 2.4, and Lemma 2.5 to Theorem 2.6, with the only substantive external input being the gen-minimality criterion quoted as Theorem 1.20 from [8]. I found no circularity and no internal gap in the main argument.","major_comments":[],"minor_comments":[{"comment":"The line 'we must have U/ell+1 = U' should read 'U_{\\ell+1} = U'; the LaTeX has lost the subscript braces, and the same typo appears later in the same paragraph.","section":"Lemma 2.2, Case 3"},{"comment":"The maximality condition is stated as 'we do not have T(X ′) ⊊ T(Y) for any X ′ in O', but Y is undefined; it should presumably be 'T(X) ⊊ T(X′)' or an equivalent formulation. As written, the sentence is not meaningful.","section":"Lemma 2.5"},{"comment":"The references to 'Propositions 1.8 and 1.14' should be to 'Lemma 1.8 and Proposition 1.14'; there is no Proposition 1.8, and the displayed factorization of E_{M_i} uses Lemma 1.8.","section":"Proposition 2.3"},{"comment":"The proof of the main theorem depends essentially on this quoted theorem; the authors should state explicitly that it is proved in [8] and, if [8] has not yet appeared, indicate its availability or status. This is a completeness concern rather than a mathematical objection.","section":"Section 1.4, Theorem 1.20"},{"comment":"The word 'hopitality' should be 'hospitality'.","section":"Acknowledgments"}],"recommendation":"minor_revision","confidential_remarks":"The paper is closely tied to the companion preprint [8], and Theorem 1.20 is load-bearing for the main result. I recommend that the editor verify whether [8] is published or accepted, or require the authors to include a proof or a precise reference to [8, Thm. 2.14]. With that caveat, the mathematics in this manuscript appears sound and suitable for publication after the minor typographical and referencing corrections noted in the report."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: this is a solid, genuinely new result — transitivity of mutation of complete τ-exceptional sequences for all τ-tilting finite algebras. The reader's ACCEPT verdict is right. The proof strategy is the real contribution: instead of case-by-case for rank two or Nakayama, they reduce to finding a gen-minimal representative in every orbit and then use a swap lemma (Prop 2.3) to connect all gen-minimal representatives. The maximal-torsion-class argument in Lemma 2.5 is the heart and it works. I traced the chain and found no gap. The use of the same authors' earlier gen-minimality criterion [8, Thm 2.14] as Theorem 1.20 is the only load-bearing external input. That's a dependency risk, not an error; the criterion is stated with hypotheses and used non-circularly. If you want to referee, that's the first thing to check. There are a few typos (e.g., 'U/ell+1' in Lemma 2.2 Case 3, 'hopitality' in acknowledgments). Also the proof of Lemma 2.2 Case 3 is intricate and worth a careful check. The introduction and background are efficient and well organized. This paper deserves a serious referee and should be published after minor revision. It will be cited by anyone working on τ-exceptional sequences. Reading group: yes, it's a good example of a clean structural proof.","headline":"A solid, genuinely new transitivity theorem for τ-exceptional sequences; referee it.","tokens_in":13753,"tokens_out":1169,"would_cite":true,"duration_ms":12295,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16D90","16G10","16G20","16S90"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a τ-tilting finite algebra, any two complete τ-exceptional sequences are connected by a finite chain of left and right mutations.","keywords":["τ-tilting theory","τ-exceptional sequences","mutation transitivity","gen-minimal modules","wide subcategories","τ-tilting finite algebras","torsion classes","τ-perpendicular categories"],"falsifier":"Enumerate, for a small $\\tau$-tilting finite algebra of rank 3 (found, say, by computer search), all complete $\\tau$-exceptional sequences and all allowed left/right mutations; the theorem predicts one connected component of the mutation graph, so two sequences in different components would be a counterexample.","tokens_in":12736,"feed_emoji":"🔄","tokens_out":12296,"duration_ms":120883,"temperature":0.7,"pith_summary":"The paper proves that over a $\\tau$-tilting finite algebra—one in which only finitely many basic support $\\tau$-rigid modules exist—left and right mutation can transform any complete $\\tau$-exceptional sequence into any other. $\\tau$-exceptional sequences are ordered lists of indecomposable modules that generalize the exceptional sequences familiar from hereditary algebras, and mutation is the local operation that swaps adjacent entries in a controlled way. Transitivity was already known for algebras of rank two and for Nakayama algebras; this paper establishes it in full generality under the $\\tau$-tilting-finiteness assumption. The proof works by showing each mutation orbit contains a distinguished sequence coming from a gen-minimal module, and that any two such distinguished sequences are related by adjacent swaps that mutations realize.","feed_headline":"Mutation is transitive for τ-tilting finite algebras","feed_subtitle":"Every complete τ-exceptional sequence can be mutated into every other, generalizing the known rank-two and Nakayama cases.","key_machinery":"Two structures carry the argument. The first is the bijection $\\omega$ from TF-ordered $\\tau$-rigid modules to $\\tau$-exceptional sequences: a TF-ordering is an ordering of the indecomposable summands of a $\\tau$-rigid module in which no summand lies in the torsion class generated by the summands after it, and $\\omega$ turns such a module into a $\\tau$-exceptional sequence by applying the reduction functors to each entry. The second is gen-minimality of the underlying module: a $\\tau$-rigid module is gen-minimal if removing any summand strictly shrinks the torsion class it generates. The proof uses the reduction maps $E_T$ and the $\\tau$-perpendicular categories $J(\\cdots)$ to track how a mutation of the sequence changes the corresponding ordered module, showing that each mutation orbit contains a gen-minimal representative and that mutations realize adjacent transpositions between any two orderings of the same gen-minimal module.","core_discovery":"The central theorem is that if $\\Lambda$ is $\\tau$-tilting finite of rank $n$ and $W$ is any wide subcategory of rank $r$, then any two $\\tau$-exceptional sequences $X$ with $J(X)=W$ lie in the same orbit under the mutation operators $\\varphi_{r+1}, \\psi_{r+1}, \\ldots, \\varphi_{n-1}, \\psi_{n-1}$; taking $W=0$ gives the headline statement that any two complete $\\tau$-exceptional sequences are connected by a finite chain of left and right mutations. The argument first shows (Lemma 2.5) that every such orbit contains a sequence whose underlying ordered module is gen-minimal—no proper direct summand generates the same torsion class. It then shows (Corollary 2.4) that all gen-minimal representatives for a fixed $W$ have the same underlying module, and (Proposition 2.3) that adjacent swaps of its summands, which generate every ordering, can be carried out by the allowed mutations. The proof thus upgrades the classical transitivity result for hereditary algebras to every $\\tau$-tilting finite algebra.","pith_inferences":["An implicit consequence is that invariants of a mutation orbit can be defined from the unique gen-minimal underlying module, since the orbit's gen-minimal representatives all share that module.","A testable extension is to bound the number of mutations needed to connect two complete sequences: the proof gives a route through adjacent swaps but no explicit complexity estimate.","Because the proof's maximality argument uses finiteness of torsion classes, the theorem neither predicts nor rules out transitivity beyond $\\tau$-tilting finite algebras; the natural next check is a minimal non-$\\tau$-tilting-finite example."],"forward_implications":["Any two complete $\\tau$-exceptional sequences over a $\\tau$-tilting finite algebra are connected by a finite chain of left and right mutations.","For each wide subcategory $W$ of rank $r$, the operators $\\varphi_{r+1}, \\psi_{r+1}, \\ldots, \\varphi_{n-1}, \\psi_{n-1}$ act transitively on the $\\tau$-exceptional sequences $X$ with $J(X)=W$.","Every mutation orbit contains a representative whose underlying $\\tau$-rigid module is gen-minimal, and this underlying module is the same for all gen-minimal representatives in the orbit.","Previous transitivity results for rank-two $\\tau$-tilting finite algebras and for Nakayama algebras follow as special cases of the main theorem."],"supporting_citations":[{"why":"Introduces left and right mutation of $\\tau$-exceptional pairs and sequences, proves all pairs are mutable in the $\\tau$-tilting finite case, and supplies the gen-minimality criterion Theorem 1.20 on which the descent argument rests.","marker":"[8]"},{"why":"Foundations of $\\tau$-tilting theory: defines $\\tau$-rigid, $\\tau$-tilting, and support $\\tau$-rigid modules, and the Bongartz and co-Bongartz completions used throughout.","marker":"[1]"},{"why":"Introduces $\\tau$-exceptional sequences and the reduction maps $E_T$ that translate between sequences and ordered $\\tau$-rigid modules.","marker":"[10]"},{"why":"Provides $\\tau$-tilting reduction and the $\\tau$-perpendicular categories $J(M,P)$, the ambient setting in which mutations are computed step by step.","marker":"[15]"},{"why":"Proves the bijection between TF-ordered $\\tau$-rigid modules and $\\tau$-exceptional sequences (Theorem 1.4), the main translation device of the proof.","marker":"[17]"},{"why":"Establishes that $\\tau$-tilting finite algebras have only finitely many functorially finite torsion classes, used for the maximality contradiction in Lemma 2.5.","marker":"[12]"},{"why":"Shows wide subcategories are $\\tau$-perpendicular with the stated rank formula, so the main theorem applies to every wide subcategory.","marker":"[13]"},{"why":"Supplies the reduction identities $E_{Y\\oplus Z}(X)=E_{J(Z)}^{E_Z(Y)}(E_Z(X))$ and $J(X\\oplus Y)=J_{J(Y)}(E_Y(X))$ used to compose reductions.","marker":"[7]"},{"why":"Gives the bijection between torsion classes and wide subcategories used in Theorem 1.1 to identify gen-minimal modules and split projectives.","marker":"[16]"}],"fun_headline_variants":["τ-exceptional mutation is transitive","All τ-exceptional sequences linked by mutation","Mutation connects every τ-exceptional sequence pair","Transitive mutation for τ-tilting finite algebras","τ-tilting finite: mutation transitivity proven"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is a previously proved criterion, stated as Theorem 1.20 and not reproved here, which says that a $\\tau$-rigid module is gen-minimal (removing any summand strictly shrinks the torsion class it generates) exactly when it equals the split-projective part of the perpendicular of its $\\tau$-perpendicular category; if that criterion failed, the proof's claim that every mutation orbit contains a gen-minimal representative would not follow.","fun_headline_variants_meta":{"raw":{"variants":["τ-exceptional mutation is transitive","All τ-exceptional sequences linked by mutation","Mutation connects every τ-exceptional sequence pair","Transitive mutation for τ-tilting finite algebras","τ-tilting finite: mutation transitivity proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000648,"raw_usage":{"total_tokens":2897,"prompt_tokens":789,"completion_tokens":2108,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":405,"completion_tokens_details":{"reasoning_tokens":2037}},"tokens_in":405,"tokens_out":2108,"duration_ms":14880,"temperature":1.0,"reasoning_tokens":2037,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:25:53.165224+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate, for a small $\\tau$-tilting finite algebra of rank 3 (found, say, by computer search), all complete $\\tau$-exceptional sequences and all allowed left/right mutations; the theorem predicts one connected component of the mutation graph, so two sequences in different components would be a counterexample.","supporting_citations":[{"cited_title":"Adachi, O","cited_arxiv_id":null,"evidence_quote":"Foundations of $\\tau$-tilting theory: defines $\\tau$-rigid, $\\tau$-tilting, and support $\\tau$-rigid modules, and the Bongartz and co-Bongartz completions used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces $\\tau$-exceptional sequences and the reduction maps $E_T$ that translate between sequences and ordered $\\tau$-rigid modules."},{"cited_title":"Jasso, Reduction of τ -tilting modules and torsion pairs","cited_arxiv_id":null,"evidence_quote":"Provides $\\tau$-tilting reduction and the $\\tau$-perpendicular categories $J(M,P)$, the ambient setting in which mutations are computed step by step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the bijection between TF-ordered $\\tau$-rigid modules and $\\tau$-exceptional sequences (Theorem 1.4), the main translation device of the proof."},{"cited_title":"Demonet, O","cited_arxiv_id":null,"evidence_quote":"Establishes that $\\tau$-tilting finite algebras have only finitely many functorially finite torsion classes, used for the maximality contradiction in Lemma 2.5."},{"cited_title":"Demonet, O","cited_arxiv_id":null,"evidence_quote":"Shows wide subcategories are $\\tau$-perpendicular with the stated rank formula, so the main theorem applies to every wide subcategory."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the reduction identities $E_{Y\\oplus Z}(X)=E_{J(Z)}^{E_Z(Y)}(E_Z(X))$ and $J(X\\oplus Y)=J_{J(Y)}(E_Y(X))$ used to compose reductions."},{"cited_title":"Marks and J","cited_arxiv_id":null,"evidence_quote":"Gives the bijection between torsion classes and wide subcategories used in Theorem 1.1 to identify gen-minimal modules and split projectives."}],"review_version":1}