{"id":"4401bbe6-3e09-49b9-aecc-033e8aeb413b","arxiv_id":"2412.13971","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For gentle algebras, almost-tilting modules always complete to tilting modules with at most 2n complements, matching a modified version of Happel's conjecture.","lead":"This paper proves that every almost-tilting module over a gentle algebra can be extended to a full tilting module, and that there are at most 2n such extensions. It also builds counterexamples showing that smaller pre-tilting modules cannot always be completed.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 2n complement bound rests on Lemma 3.19's unproven 'straightforward check' case count; an omitted configuration could break Theorem 3.20.","rationale":"The paper's main theorem is clearly stated and the surface-cutting reduction is a coherent strategy. Corollary 3.13 and Proposition 3.15, if accepted, would indeed reduce the theorem to Lemma 3.19, and the paper gives real geometric content: arcs as modules, intersections as extensions, and a concrete cutting construction. The reader's weakest-assumption identification is accurate: Lemma 3.19 is the exact place where the proof of the 2n bound is generated, and it is also where the argument switches from a derivation to a 'straightforward check'. I did not find an internal inconsistency in the rest of the proof, and I am not claiming the lemma is false. The concern is that the central numerical claim is only as secure as the completeness of that case check, which is not demonstrated in the text. The proposed enumeration would settle the question for small ranks and would likely expose any systematic missing family; if no violation appears, the conditional verdict can be upgraded. Thus the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":27505,"tokens_out":10603,"duration_ms":100676,"concrete_test":"Exhaustively enumerate, for n≤8, all simple coordinates on the three base surfaces of Lemma 3.19 (disk, once-punctured disk, annulus) that satisfy the bigon and zero-weight hypotheses, using the quiver data |Q0|=n and |Q1|=|M◦|−2χ to bound the search; for each configuration count the candidate completing ◦-arcs that are zigzag and have zero-weight intersections with all existing arcs. If every configuration has 1≤r≤n+1, the lemma's count is supported; if any configuration has r>n+1, Theorem 3.20's bound is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.20's conclusion that an almost-tilting module has at most 2n complements is obtained by cutting along the n−1 given arcs and applying Lemma 3.19, whose bound m ≤ n~+1 is then converted to m ≤ 2n via n~ ≤ 2n−1. The entire upper bound therefore depends on Lemma 3.19's assertion that the exceptional subsurface is one of three pictured shapes and that the number r of completing arcs satisfies 1 ≤ r ≤ n+1. That assertion is not actually proved: the disk case is dismissed as 'The proof of the claim is a straightforward check' with four representative pictures, and the once-punctured disk and annulus are treated by 'a similar argument' plus one exception. The text does not show that all possible simple coordinates satisfying the bigon conditions are covered, nor does it prove that the pictured dashed arcs exhaust all zigzag ◦-arcs with zero-weight intersections. Proposition 3.15 similarly delegates a case-by-case count to Figure 21, but the decisive issue is still Lemma 3.19. Since the existential half of the theorem only needs r≥1, a missed case with r>n+1 would leave 'almost-tilting implies partial-tilting' intact but destroy the quantitative claim that makes the paper answer Happel's modified conjecture. This is not a disagreement with the geometric framework; it is a precise missing verification at the point where the 2n bound is generated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that every almost-tilting module over a gentle algebra of rank n is partial-tilting and has at most 2n complements, confirming a modified Happel conjecture for this class. The proof uses the surface model for module categories of gentle algebras, introduces a cutting construction for marked surfaces and simple coordinates, and reduces the problem to three base cases: disks, once-punctured disks, and annuli. The paper also constructs, for every n ≥ 3 and 1 ≤ m ≤ n−2, a connected gentle algebra with a pre-tilting module of rank m that is not partial-tilting, extending the Rickard–Schofield counterexample.","tokens_in":27770,"tokens_out":4970,"duration_ms":43424,"significance":"If the main theorem is correct, it affirmatively answers question (C_{n−1}) for all gentle algebras and supplies a finite, explicit bound 2n for the number of complements, matching the asymptotic bound suggested by Mantese's example. The surface-cutting reduction is a promising technique with potential applications beyond tilting completion. The paper is well-structured and carefully integrates existing geometric models from [BC21, OPS18, C23]. However, the quantitative bound 2n depends on a case count in Lemma 3.19 that is delegated to a 'straightforward check' with illustrative pictures rather than a complete proof, and this is the central load-bearing step of the paper.","major_comments":[{"comment":"The bound 1 ≤ r ≤ n+1 in Lemma 3.19 is the exact input that Theorem 3.20 converts into the final bound m ≤ 2n. The proof of this lemma is not complete: the disk case is dismissed with 'The proof of the claim is a straightforward check' and four representative pictures in Figure 15, while the once-punctured disk and annulus are treated by 'a similar argument' with four representatives in Figure 16 plus one exception in Figure 17. The manuscript does not prove that these pictures exhaust all simple coordinates satisfying the bigon conditions, nor does it show that the dashed ◦-arcs are the only zigzag arcs with zero-weight intersections. A missed configuration with r > n+1 would leave the statement 'almost-tilting implies partial-tilting' intact but would invalidate the quantitative claim that gives the paper its main theorem. This is a missing verification, not a disagreement with the geometric framework; the proof should supply a complete, exhaustive case analysis or a verification procedure that covers all possible configurations.","section":"§3.3, Lemma 3.19"},{"comment":"The rank increase rank(Sγ, Mγ) = rank(S, M) + 1 is used in Theorem 3.20 to conclude that the induced algebra AΓ has rank 2n−1, and the equality |Q̂1| = |Q1| is used implicitly in the same reduction. The proof of Proposition 3.15 derives the rank formula from the assertions that the number of marked points increases by two and the Euler characteristic changes by one, with the marked-point count said to be 'proved case-by-case, seeing the pictures in Figure 21'. Since this numerical formula is load-bearing for the main theorem, the proof should include the explicit five-case verification (or a uniform argument) rather than relying solely on a figure. The statement also contains a typographical error: the equality should read |Q̂1| = |Q1|, not |Q1| = |Q1|.","section":"§3.2, Proposition 3.15"}],"minor_comments":[{"comment":"The sentence 'the Euler character of S is the same as the Euler character of the topological quotient of S, that is, equals χ − 1' is confusing and appears to contain a typo; χ was defined as the Euler characteristic of the original surface S, so the statement should refer to the Euler characteristic of Sγ, not of S.","section":"§3.2, Proposition 3.15"},{"comment":"In the proof of the claim for the torus example, the phrase 'if the endpoints of γ are pi, 2 ≤ i ≤ p − 1' uses p without definition; it should presumably be 2 ≤ i ≤ n−1, with n the number of marked ◦-points on the boundary.","section":"§3.4, Theorem 3.22"},{"comment":"The three possibilities for the exceptional subsurface (disk, once-punctured disk, annulus) are said to be 'depicted in Figure 13', but the text also notes that the annulus is homotopic to a once-punctured disk when the boundary formed by ◦-arcs is viewed as a puncture; the criterion for distinguishing these cases in the subsequent count should be stated explicitly.","section":"§3.3, Lemma 3.19"},{"comment":"The appendix explicitly constructs the algebra associated with the cutting surface only under the assumption that the cutting arc γ intersects each •-arc at most once; it would be helpful to state whether the cases appearing in Lemma 3.19 satisfy this assumption, or to what extent the general construction remains open.","section":"Appendix"},{"comment":"The URL for [S23] ends in 'FD-Atlas.htmpl', which looks like a typo for 'FD-Atlas.html'.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is significant and the overall strategy is convincing, but the paper's central quantitative claim currently rests on the unproved case analysis in Lemma 3.19. I recommend requiring a complete proof of that lemma—ideally a systematic enumeration of all possible simple coordinates and a verification that the listed dashed arcs are exhaustive—before publication. The paper also leans heavily on the author's own [C23] for the Ext/weight correspondence; while the published sources [BC21, OPS18] provide independent grounding, the referee report should ask the author to make the dependence precise. The self-citation pattern is not inappropriate for a self-contained research area, but the exposition in §3.3 would benefit from more detail in the case count."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Wen Chang proves that almost-tilting modules over gentle algebras are partial-tilting and have at most 2n complements, confirming a modified Happel conjecture for this class. The surface-cutting technique is the real contribution: it turns a module-theoretic completion problem into a topological reduction to disks, once-punctured disks, and an annulus. The counterexamples for pre-tilting modules that are not partial-tilting (n≥3, 1≤m≤n−2) are explicit and include the known n=3 example, which is a nice sanity check. The geometric dictionary and the cutting construction are explained thoroughly; the appendix even gives a concrete quiver presentation for the cut algebra. This is a genuine within-field advance, not a repackaging of the silting results in [JSW23, LZ23], which live in the derived category.\n\nThe soft spot is exactly where the reader and stress-test put it. Lemma 3.19 is load-bearing: it classifies the exceptional subsurface after cutting and asserts 1≤r≤n+1. The proof says 'straightforward check' and shows representative pictures, with 'a similar argument' for two of the three cases. That is not enough for a result whose quantitative bound (2n) depends on this count. A missed configuration with r>n+1 would not affect the 'almost-tilting implies partial-tilting' half, but it would invalidate the bound that distinguishes the paper as an answer to Happel's conjecture. The author should either write out the finite case analysis in full, or replace it with an algorithm that enumerates the possible zigzag completions. Proposition 3.15's arrow count |Q1|=|Q1| is asserted more than proved, though Corollary 3.7 plus the Euler characteristic computation makes it plausible; that is minor. Reliance on [C23] for the Ext/weight correspondence is acceptable because the same model is supported by the published constructions of BC21 and OPS18; it is not a circularity.\n\nThe existential claim is probably fine even with the gap, and the counterexample part is concrete. But the paper, as written, is not fully checked. A serious referee should get it and ask for the missing cases. I would send it to peer review rather than desk-reject, and I would expect a revised version to come back with a longer proof of Lemma 3.19.","headline":"A real advance for gentle algebras with a reusable surface-cutting method, but the proof of the main bound rests on a 'straightforward check' that a referee should demand to see in full.","tokens_in":28301,"tokens_out":2760,"would_cite":true,"duration_ms":23722,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16D90","16E35","57M50"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every almost-tilting module over a gentle algebra can be completed to a tilting module, with at most 2n complements.","keywords":["gentle algebras","tilting modules","almost-tilting modules","partial-tilting modules","complements","marked surfaces","surface cuts","string modules"],"falsifier":"Find a gentle algebra of rank $n$ with an almost-tilting module having more than $2n$ complements, or exhibit a marked surface whose exceptional subsurface after cutting $n-1$ pre-tilting arcs is not one of the three shapes in Lemma 3.19. Either would break the chain from Lemma 3.19 to Theorem 3.20.","tokens_in":27278,"feed_emoji":"✂️","tokens_out":8575,"duration_ms":71030,"temperature":0.7,"pith_summary":"The paper establishes that over any gentle algebra of rank $n$, every almost-tilting module—a self-orthogonal module of finite projective dimension with exactly $n-1$ indecomposable direct summands—can be completed to a tilting module, and that the number of non-isomorphic complements is at most $2n$. This gives a positive answer to the completion question $(C_{n-1})$ for gentle algebras and confirms a modified complement conjecture in this class. The proof is geometric: modules are viewed as zigzag arcs on a marked surface, and completing a module becomes finding an arc that finishes a dissection. By cutting the surface along the given arcs, the problem reduces to three small base surfaces, where the possible completing arcs can be counted. The paper also constructs, for every $n \\geq 3$ and $1 \\leq m \\leq n-2$, a connected gentle algebra with a pre-tilting module of rank $m$ that is not partial-tilting, showing that completion can fail when more than one summand is missing.","feed_headline":"Almost-tilting modules over gentle algebras always complete","feed_subtitle":"A surface-cutting proof gives at most 2n complements and settles a modified complement conjecture.","key_machinery":"The proof works on the marked surface attached to a gentle algebra: indecomposable string modules are represented by zigzag arcs, extensions and projective dimensions are read from weighted intersections of arcs, and a pre-tilting module becomes a collection of non-intersecting zigzag arcs whose oriented intersections all have weight zero. Cutting the surface along such an arc produces a new marked surface and simple coordinate whose associated gentle algebra has rank $n+1$ but the same number of arrows. Cutting along all $n-1$ arcs of an almost-tilting module leaves a single exceptional subsurface of rank one, which Lemma 3.19 classifies as a disk, a once-punctured disk, or an annulus; counting the possible completing arcs on that small surface gives between $1$ and $n+1$ choices, and lifting these choices back through the cuts yields the bound $2n$ complements.","core_discovery":"The central claim, Theorem 3.20, is that an almost-tilting module over a gentle algebra is always partial-tilting: it has at least one complement, and the total number of complements is bounded by $2n$, where $n$ is the rank of the algebra, meaning the number of non-isomorphic indecomposable projective modules. The same statement verifies the modified complement conjecture for gentle algebras: complements are finite in number, and the bound $2n$ replaces the originally proposed $2n-1$ because a rank-two example already admits four complements. A complementary construction, Theorem 3.22, shows that for any $n \\geq 3$ and $1 \\leq m \\leq n-2$, some connected gentle algebra of rank $n$ has a pre-tilting module of rank $m$ that cannot be completed; in particular, the positive result for almost-tilting modules is close to optimal.","pith_inferences":["The paper leaves open whether the $2n$ bound is sharp: it constructs an annulus example attaining $2n-1$ complements and sketches a gluing route toward $2n$, but does not exhibit an algebra reaching the bound.","The same surface-cutting induction could be applied to silting theory in the derived category of a gentle algebra, where completion questions for pre-silting objects are known to behave differently; the cutting picture may identify exactly where module-level and derived-level completability diverge.","The results suggest an extremal dichotomy for gentle algebras: completion always works for full rank and rank $n-1$, while every rank $\\leq n-2$ admits a failure. One could test whether a similar dichotomy holds for other tame algebras.","The proof reduces complement counting to the topology of a single exceptional subsurface, so a combinatorial model for complements of a maximal partial-tilting module might be extracted purely from the shape of that subsurface."],"forward_implications":["The completion question $(C_{n-1})$ has a positive answer for every gentle algebra of rank $n$: every almost-tilting module is partial-tilting.","A maximal partial-tilting module over a gentle algebra has finitely many complements, at most $2n$, so the finiteness and bounded-complement conjectures hold in this class.","An orthogonal module of full rank $n$ over a gentle algebra is automatically tilting, so the tilting condition (T3) can be replaced by a rank count in this setting.","For ranks $m \\leq n-2$, completion can fail, so the positive result for $n-1$ summands is the strongest possible statement that close to full rank.","The cutting construction gives a new gentle algebra of rank $n+1$ with the same number of arrows, providing a concrete reduction tool for further module-theoretic questions."],"supporting_citations":[{"why":"Introduces the geometric model of the module category of a gentle algebra, providing the bijection between arcs and string modules that the proof uses throughout.","marker":"[BC21]"},{"why":"Refines the model with weighted intersections, so that extensions and projective dimensions of string modules can be read from arc intersections; the cutting construction relies on its notation.","marker":"[C23]"},{"why":"Supplies the rank criterion for pre-silting complexes used to identify tilting dissections and to justify that full-rank orthogonal modules are tilting.","marker":"[APS23]"},{"why":"Raises the completion question $(C_m)$ and gives the rank-three counterexample that the paper generalizes in Theorem 3.22.","marker":"[RS89]"},{"why":"Provides the rank-two algebra with four complements that motivates changing the conjectured bound from $2n-1$ to $2n$.","marker":"[M05]"},{"why":"Develops the complement theory for partial-tilting modules, including the notion of maximal partial-tilting module and conditions for finitely many complements.","marker":"[CHU94]"},{"why":"Classifies indecomposable modules over gentle algebras as string and band modules, allowing the paper to restrict attention to string modules since band modules have nontrivial self-extensions.","marker":"[BR87]"}],"fun_headline_variants":["Gentle algebras: almost-tilting always completes","At most 2n tilting complements for gentle algebras","Surface cuts prove tilting completion for gentle algebras","Almost-tilting modules complete over gentle algebras","Modified Happel conjecture holds for gentle algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 3.19's assertion that after cutting, the single exceptional subsurface is a disk, a once-punctured disk, or an annulus with between $1$ and $n+1$ completing arcs; this classification is recorded as a 'straightforward check' rather than a fully enumerated case analysis.","fun_headline_variants_meta":{"raw":{"variants":["Gentle algebras: almost-tilting always completes","At most 2n tilting complements for gentle algebras","Surface cuts prove tilting completion for gentle algebras","Almost-tilting modules complete over gentle algebras","Modified Happel conjecture holds for gentle algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000275,"raw_usage":{"total_tokens":1597,"prompt_tokens":855,"completion_tokens":742,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":471,"completion_tokens_details":{"reasoning_tokens":669}},"tokens_in":471,"tokens_out":742,"duration_ms":6284,"temperature":1.0,"reasoning_tokens":669,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:35:47.276515+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a gentle algebra of rank $n$ with an almost-tilting module having more than $2n$ complements, or exhibit a marked surface whose exceptional subsurface after cutting $n-1$ pre-tilting arcs is not one of the three shapes in Lemma 3.19. Either would break the chain from Lemma 3.19 to Theorem 3.20.","supporting_citations":[],"review_version":1}