{"id":"1ec81f08-414d-49b0-8df1-7bb3caf4f38c","arxiv_id":"2501.13694","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Over Nakayama algebras, the mutation operation on τ-exceptional sequences is transitive, with explicit combinatorial formulas for each mutation step.","lead":"This pure math paper gives explicit formulas for a 'mutation' operation on ordered module lists over Nakayama algebras, and proves that this operation can connect any two complete lists. The result settles a natural question in representation theory: transitivity of mutation, previously known only for hereditary algebras and rank-2 algebras, now holds for all Nakayama algebras.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.1's transposition claim relies on an unproved bijectivity/finiteness step; if it fails, Lemma 5.2 and hence Theorem 5.15 collapse.","rationale":"The reader identified Lemma 4.7 as the weakest assumption, but Lemma 4.7 has a short and essentially correct proof: every object of J(M,P) is an object of modΛ, hence uniserial, so the equivalent algebra Γ has only uniserial modules and is Nakayama. The genuinely under-justified step is in Lemma 5.1, which the reader flagged as a red flag but did not make the central concern. Lemma 5.1 is directly used in Lemma 5.2, the first main component of the transitivity proof, so a gap there is at least as load-bearing as Lemma 4.7. The paper has substantial independent support: explicit formulas, worked examples, and a geometric model, all of which make the main conclusion plausible. The recommended verdict remains CONDITIONAL (equivalently UNCHANGED here), because the concern is a proof gap that should be fixed or explicitly justified, not a demonstrated counterexample to the theorem. A computational verification as proposed would either confirm the lemma or produce a counterexample, thereby settling the issue.","tokens_in":48127,"tokens_out":7951,"duration_ms":66859,"concrete_test":"Enumerate all Nakayama algebras of rank n≤6; for each, list all ordered pairs (B,C) of indecomposable τ-rigid modules such that both B⊕C and C⊕B are TF-ordered. Compute φ using the explicit formulas of Theorem 1.3 and check that the forward φ-orbit of B⊕C always contains C⊕B. In the same computation, test whether φ is injective (equivalently bijective) on this finite set. A single counterexample would refute Lemma 5.1; if no counterexample is found, the missing step can be completed by a case analysis of the possible φ-orbits.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Lemma 5.1 must show that any two TF-orders B⊕C and C⊕B of the same two-summand τ-rigid module are connected by mutations of TF-orders. The proof splits into cases, but in the 'left regular and at least one summand non-projective' case it argues: if φ(C⊕B)=B⊕C, then 'there must be an i such that φ^i(B⊕C)=C⊕B ... since Λ is τ-tilting finite.' This is not justified. Finiteness of the set of TF-ordered τ-rigid pairs does not by itself force the forward orbit of B⊕C to hit C⊕B; one also needs that φ is injective (or bijective) on this finite set, or an explicit analysis of the orbit. No such argument or citation is provided. The concern is load-bearing because Lemma 5.2 uses Lemma 5.1 to connect arbitrary TF-orders of the same τ-tilting module; if Lemma 5.1 fails, the first half of the transitivity argument in Theorem 5.15 fails. The later cases (TF-1b, TF-4) are handled by explicit two-step sequences, but the generic non-projective regular case is left to this terse finiteness assertion, and the surrounding text does not rule out that φ(B⊕C) lands in a different cycle of the finite mutation graph.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the mutation of τ-exceptional sequences introduced in [BHM24] through the equivalent language of TF-ordered τ-rigid modules. The authors translate the general mutation rules into operations on TF-ordered modules using the V-map and E-map, then specialize to Nakayama algebras, where they obtain explicit formulas for left mutation in four mutually exclusive cases (TF-1a, TF-1b, TF-2a, TF-2b, TF-3, TF-4). The main application is Theorem 5.15, which states that mutation of TF-ordered τ-tilting modules, and hence of complete τ-exceptional sequences, is transitive for Nakayama algebras. The proof strategy is to show (Lemma 5.2) that any two TF-orders of the same τ-tilting module are connected by mutations, and (Proposition 5.14) that a single τ-tilting mutation lifts to a mutation between suitable TF-orders; connectivity of the τ-tilting mutation graph then gives transitivity.","tokens_in":48301,"tokens_out":12551,"duration_ms":108212,"significance":"If the main theorem is correct, it provides the first known non-hereditary class of algebras of arbitrary rank for which mutation of complete τ-exceptional sequences is transitive, complementing the rank-2 result of [BHM24]. The paper is also valuable for its explicit combinatorial description of mutation for Nakayama algebras, including the irregular case, and for its use of Adachi's disk model to visualize the formulas. The authors build on published results rather than introducing ad-hoc assumptions, and the paper contains many worked examples that illustrate all six mutation cases. The central derivation is detailed and the overall strategy is coherent; however, one load-bearing proof step in Lemma 5.1 is under-justified, and one structural lemma (Lemma 4.7) is stated with a proof that is too terse for the role it plays in the induction.","major_comments":[{"comment":"The proof of the first case relies on the assertion: 'If the second equality is true then there must be an i such that φ^i(B⊕C)=C⊕B as wanted, since Λ is τ-tilting finite.' This is not justified by finiteness alone. From φ(C⊕B)=B⊕C, one cannot conclude that the forward orbit of B⊕C under φ ever reaches C⊕B unless φ is known to be injective (or bijective) on a finite invariant set containing both elements. The paper does not cite or prove such injectivity at this point. Since Lemma 5.2 and hence Theorem 5.15 depend on this step, the argument must be repaired. One possible repair is to cite the invertibility of the mutation operation from [BHM24] and explain that φ is a bijection on the relevant finite set. Alternatively, the step can be avoided entirely: because C⊕B is TF-ordered, C∉GenB by Definition 2.13, so in the stated left-regular case Theorem 3.26 already yields φ(B⊕C)=C⊕B directly, with the projective subcase handled by Lemma 3.4(c).","section":"§5, Lemma 5.1"},{"comment":"Lemma 4.7 is load-bearing because Proposition 5.14 applies Lemma 5.1 inside τ-perpendicular categories J(M,P), which must themselves be module categories of Nakayama algebras. The proof is a single sentence: since every indecomposable Λ-module is uniserial, so is every indecomposable object of J(M,P), hence the Morita equivalent algebra Γ is Nakayama. This needs expansion: J(M,P) is a wide subcategory, and it is not immediate that the subobject lattice of an object of J(M,P) in the ambient category is inherited by the wide subcategory in a way that preserves uniseriality. The claim is true, but the proof should spell out that the subobjects of an object in the wide subcategory are exactly the ambient submodules that also lie in the subcategory, so the submodule lattice of each object remains a chain.","section":"§4, Lemma 4.7"}],"minor_comments":[{"comment":"There is a typo: 'Nakayama algebas' should read 'Nakayama algebras'.","section":"§5, first paragraph"},{"comment":"In the paragraph beginning 'Next we claim that it follows that Hom(X, Z) ≠ 0', the same statement appears twice; the second occurrence should likely refer to Hom(X, U1) ≠ 0, based on the surrounding argument.","section":"§5, proof of Lemma 5.11"},{"comment":"The notation B(M) for the Bongartz completion is used before its formal definition in Section 2.1; while the definition appears earlier, it would help the reader to include a forward reference at first use in the introduction of the Nakayama-specific results.","section":"§4, Proposition 4.9"},{"comment":"There is a typo: 'Nakayma algebra' should be 'Nakayama algebra'.","section":"§6, Example 6.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution to the representation theory of Nakayama algebras and the theory of τ-exceptional sequences, and the main theorem is likely correct. The only serious obstacle to acceptance is the unjustified finiteness step in Lemma 5.1, which is load-bearing for the transitivity theorem. This is fixable either by citing the invertibility of mutation from [BHM24] or by the direct case analysis suggested above. I recommend major revision rather than rejection; after the proof of Lemma 5.1 is repaired and Lemma 4.7 is expanded, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, what you should know: this paper gives explicit combinatorial formulas for mutating TF-ordered τ-rigid modules over Nakayama algebras (Theorem 1.3) and uses them to prove that mutation of complete τ-exceptional sequences is transitive (Theorem 5.15). That is a real, new result; previous work only covered rank-2 algebras. The general framework in Section 3, translating mutation into V- and E-maps, is genuinely useful and looks correct.\n\nThe proofs are mostly careful. Section 4's explicit Bongartz and co-Bongartz completions for Nakayama algebras are a nice contribution, and the case analysis for mutation, including the irregular case, is concrete and checkable. The V-map computations in Corollary 4.16 are elegant. The overall strategy for transitivity—connect TF-orders of the same module, then connect neighboring τ-tilting modules via a single TF-mutation—is sound in principle.\n\nNow the soft spots. Lemma 5.1 is load-bearing and under-proved. The proof says: if φ(C⊕B) = B⊕C, then there must be an i with φ^i(B⊕C) = C⊕B because Λ is τ-tilting finite. Finiteness alone does not rule out tails in a non-injective map. You need either to prove φ is injective (hence a permutation) on the finite set of TF-ordered pairs, or to exhibit the sequence explicitly. The later case analysis does this for the projective and irregular cases, but the generic non-projective regular case is left to that finiteness assertion. This is not a fatal flaw—I expect it can be repaired—but it is a genuine gap in the proof of Theorem 5.15.\n\nTwo smaller things. Lemma 4.7 (τ-perpendicular categories are again Nakayama) is a one-sentence uniseriality argument; it is plausible but deserves a real proof, since the induction in Proposition 5.14 depends on it. And the homological arguments in Lemmas 5.9, 5.12, and 5.13 are dense enough that a referee should ask for expansion; I did not find a contradiction, but they are not easy to verify.\n\nBottom line: the main theorem is likely true, the paper is a solid contribution to τ-tilting theory, and the explicit formulas will be cited by people working on mutation and exceptional sequences. But it is not ready as is. A serious referee should be sent it, with instructions to require a fix for Lemma 5.1 and more detail on Lemma 4.7. My own verdict: conditional accept, if those are addressed.","headline":"A genuinely new transitivity theorem for Nakayama algebras with a mostly sound proof, but Lemma 5.1 has a gap that needs a real fix before the main theorem is fully established.","tokens_in":48935,"tokens_out":5809,"would_cite":true,"duration_ms":50401,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","16G70","18E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Nakayama algebras, mutation connects every complete τ-exceptional sequence to every other.","keywords":["τ-tilting theory","τ-exceptional sequences","TF-ordered τ-rigid modules","Nakayama algebras","uniserial algebras","mutation transitivity","Bongartz completion","τ-perpendicular categories"],"falsifier":"Build the directed graph whose vertices are the TF-ordered τ-tilting modules of the radical-cube-zero Nakayama algebra with cyclic quiver on four vertices and whose edges are the four-case left mutations of Theorem 1.3; if any two vertices in the same connected component of the τ-tilting exchange graph lie in different components of this directed graph, transitivity fails.","tokens_in":47839,"feed_emoji":"🔁","tokens_out":9700,"duration_ms":75502,"temperature":0.7,"pith_summary":"The paper proves that over any Nakayama algebra, mutation is transitive: any complete τ-exceptional sequence can be turned into any other by a finite chain of left mutations. The proof works by translating mutation into an operation on TF-ordered τ-rigid modules, which are ordered collections of τ-rigid modules that encode such sequences, and then giving an explicit four-case formula for the operation. A key step is showing that the relevant τ-perpendicular subcategories are again Nakayama algebras, so the two-term mutation rules can be applied inside reductions. If the result stands, it extends the previously known rank-two case to all uniserial algebras and provides a concrete combinatorial algorithm for moving between complete sequences.","feed_headline":"Mutation connects every τ-exceptional sequence over Nakayama algebras","feed_subtitle":"Explicit four-case mutation rules show any complete τ-exceptional sequence over a Nakayama algebra can reach any other.","key_machinery":"The central objects are TF-ordered τ-rigid modules, ordered τ-rigid modules M1 ⊕ ... ⊕ Mt with Mi not generated by the later summands, which are in bijection with τ-exceptional sequences. The argument is carried by the V-map and E-map associated to a τ-rigid pair: the V-map bijects summands of the co-Bongartz completion to summands of the Bongartz completion, and the E-map reduces τ-rigid pairs to relative τ-rigid pairs in the τ-perpendicular subcategory J(M,P). For Nakayama algebras the paper computes these maps explicitly using the maximal and minimal completions described in Propositions 4.9 and 4.11, then proves the four-case mutation formula of Theorem 1.3. The reduction step is justified by Lemma 4.7, which says J(M,P) is again Morita equivalent to a Nakayama algebra, allowing longer mutations to be handled inside smaller categories of the same type.","core_discovery":"The central claim is Theorem 5.15: for a Nakayama algebra, mutation of TF-ordered τ-tilting modules, and therefore of complete τ-exceptional sequences, is transitive. The paper establishes this by proving Theorem 1.3, which describes left mutation of a TF-ordered module B ⊕ C in four exhaustive and mutually exclusive cases: if C is projective with no homomorphisms to B, the pair simply swaps; if C is projective and Hom(C,B) ≠ 0, the second entry becomes the torsion-free functor f_C(B); if C is generated by B, the first entry becomes a radical of B (or its projective cover when B is projective); and in the remaining irregular case, the second entry becomes the quotient B/C. Transitivity follows by combining a lemma that any two TF-orders of the same τ-tilting module are connected by mutations with a proposition that lifts each ordinary τ-tilting mutation to a sequence of TF-ordered mutations; connectivity of the τ-tilting exchange graph for τ-tilting finite algebras then connects everything.","pith_inferences":["Editorial inference: the only Nakayama-specific input in the transitivity argument is Lemma 4.7, so an analogous statement may hold for any class of τ-tilting finite algebras whose τ-perpendicular reductions stay inside the class.","Editorial inference: the explicit V-map computations could yield a direct count of φ-orbits and mutation graph components in the geometric disk model for Nakayama algebras, refining the comparison with rank-two wide subcategories that Example 6.8 shows is nontrivial.","Editorial inference: one could test whether the four-case formulas of Theorem 1.3 extend to all ordered τ-rigid pairs, not just TF-ordered ones, by inserting the E-map corrections that the paper uses for longer sequences.","Editorial inference: the failure of braid relations combined with transitivity suggests that the mutation graph is a natural combinatorial invariant for Nakayama algebras, and comparing its diameter across the two quiver shapes (linear and cyclic) would be a concrete next question."],"forward_implications":["Any two TF-orders of the same τ-tilting module over a Nakayama algebra are connected by a sequence of left mutations (Lemma 5.2).","Each ordinary τ-tilting mutation between τ-tilting modules can be lifted to a sequence of mutations of TF-ordered modules, so the graph of TF-orders lies over the connected τ-tilting exchange graph (Proposition 5.14).","Combining these, the mutation graph of complete τ-exceptional sequences over a Nakayama algebra is connected (Theorem 5.15).","The four-case formula gives an explicit algorithm for left mutation in terms of radicals, projective covers, and torsion-free functors, so the transitivity proof is constructive.","Transitivity holds even though the mutation does not satisfy braid relations in this setting, as shown by Example 6.9."],"supporting_citations":[{"why":"Defines the mutation of τ-exceptional pairs and sequences that this paper translates into TF-ordered module language.","marker":"[BHM24]"},{"why":"Establishes the bijection between ordered τ-rigid pairs and signed τ-exceptional sequences and introduces the E-map.","marker":"[BM21]"},{"why":"Characterizes TF-ordered τ-rigid modules as exactly the ordered pairs inducing unsigned τ-exceptional sequences.","marker":"[MT20]"},{"why":"Supplies the foundational τ-tilting theory, Bongartz completions, τ-tilting mutation, and the connected exchange graph for τ-tilting finite algebras.","marker":"[AIR14]"},{"why":"Introduces the τ-perpendicular category and τ-tilting reduction used to reduce longer sequences to length-two mutations.","marker":"[Jas15]"},{"why":"Extends the reduction to τ-rigid pairs and proves the order-preserving bijection used in Theorem 3.8 and Lemma 4.7.","marker":"[Dem+23]"},{"why":"Classifies τ-rigid modules over Nakayama algebras and provides the disk model used for examples and Proposition 4.2.","marker":"[Ada16]"},{"why":"Supplies the standard facts about uniserial modules and Nakayama algebras that underlie Propositions 4.6 and 4.9.","marker":"[ASS06]"}],"fun_headline_variants":["Nakayama τ-mutation: any two sequences are connected","τ-rigid mutation: all τ-exceptional sequences connect on Nakayama","Explicit four-case mutation rule proves transitivity on Nakayama","Nakayama τ-mutation graph is fully connected","One mutation path suffices between any τ-sequences on Nakayama"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that every τ-perpendicular subcategory J(M,P) of a Nakayama algebra is again Morita equivalent to a Nakayama algebra, so that the same two-term mutation formulas apply inside each reduction step.","fun_headline_variants_meta":{"raw":{"variants":["Nakayama τ-mutation: any two sequences are connected","τ-rigid mutation: all τ-exceptional sequences connect on Nakayama","Explicit four-case mutation rule proves transitivity on Nakayama","Nakayama τ-mutation graph is fully connected","One mutation path suffices between any τ-sequences on Nakayama"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001194,"raw_usage":{"total_tokens":4868,"prompt_tokens":831,"completion_tokens":4037,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":3948}},"tokens_in":447,"tokens_out":4037,"duration_ms":27485,"temperature":1.0,"reasoning_tokens":3948,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:42:48.479391+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build the directed graph whose vertices are the TF-ordered τ-tilting modules of the radical-cube-zero Nakayama algebra with cyclic quiver on four vertices and whose edges are the four-case left mutations of Theorem 1.3; if any two vertices in the same connected component of the τ-tilting exchange graph lie in different components of this directed graph, transitivity fails.","supporting_citations":[],"review_version":1}