{"id":"757cbac6-a096-471b-b055-e4b7d1805f93","arxiv_id":"2607.19945","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For gentle algebras from surface tilings, the endomorphism algebra of a tilting module attached to a faithful dissection is isomorphic to an explicitly defined quiver algebra, and a new tiling realizes that endomorphism algebra.","lead":"Using curves on marked surfaces, this pure-math paper gives an explicit quiver-with-relations description of the endomorphism algebra of any tilting module over a gentle algebra. This turns a piece of tilting theory into surface combinatorics and yields new derived-equivalent algebras by iterating the construction.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.7's figure-based case analysis is the load-bearing step: if any geometric configuration is omitted, the basis of rad Hom collapses and B_R ≅ End_{A_P} M(R) fails.","rationale":"The reader's weakest-assumption identification is exactly the right one: Theorem 3.7's geometric case enumeration is the point where the proof is least secure, and it is load-bearing because every later claim about the arrow set and relations of B_R presupposes that the radical morphism basis is given by P-free angles. I agree with the CONDITIONAL verdict: the paper contains a plausible but non-formalized exhaustive analysis, and the main theorem should be accepted only after the case list is verified or replaced by a formal argument. I did not find a separate fatal flaw; the loop-relation concern I initially considered dissolves under closer analysis because faithfulness forces separating arcs between the two copies of a loop in R, so the definition of I_R is likely consistent. No independent, machine-checked verification exists, and the figure-based proof is exactly the sort of thing that can hide a missed case. The proposed test—comparing bases in concrete examples or exhaustively enumerating fan orders—would settle whether the concern lands.","tokens_in":22742,"tokens_out":42839,"duration_ms":394075,"concrete_test":"For a small tiling with a faithful dissection (e.g., the annulus example in §4.2 or a disk with 4-5 marked points), compute for every pair γ1,γ2∈R with Int(γ1,γ2)=0 the dimension of rad Hom_{A_P}(M(γ1),M(γ2)) using Proposition 3.2 / string combinatorics, and count the number of P-free negatively oriented angles ∠^-_{P-free}(γ1,γ2). If they agree for all pairs, the classification passes this test. More analytically, enumerate all possible complete fan orders at a shared endpoint and at the two endpoints, and check that each of the 16 type combinations (eγ1 head/tail × eγ2 head/tail) is either shown in Figures 10–14 or has Int(γ1,γ2)>0. Pay particular attention to eγ2∈eγ2(++) with eγ1∈eγ1(−−).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central isomorphism B_R ≅ End_{A_P} M(R) (Theorem 3.18) rests on Theorem 3.7, which claims that radical morphisms between M(γ1) and M(γ2) are in bijection with P-free negatively oriented angles. The proof classifies admissible segment pairs into four head/tail types and asserts, with reference to Figures 10–14, that all geometrically possible configurations when Int(γ1,γ2)=0 are accounted for. This is a finite but intricate case analysis; it is not formalized, and no lemma states precisely which fan sequences at the two endpoints are compatible with non-intersection. If any configuration is missing, some basis element f∈B (from Proposition 3.2) would not be realized as f_∠α, so the arrow set of B_R would not span the radical morphisms, and the quiver B_R would be too small to be isomorphic to End. The external input Proposition 3.2 is standard, so the risk concentrates in the enumeration. A secondary weakness is Proposition 4.6, whose proof is a one-line assertion; however, Theorem 3.18 itself depends on Theorem 3.7.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops geometric models for endomorphism algebras of tilting modules over gentle algebras. Working with a tiling (S,M,P) and its associated gentle algebra A_P, the authors use Baur–Simões' correspondence between permissible arcs and string modules, together with the τ-tilting/dissection bijection, to show that faithful dissections R correspond to tilting A_P-modules M(R). The central result (Theorem 3.18) states that an explicitly defined quiver-with-relations algebra B_R, whose arrows are indexed by PR-free negatively oriented angles between arcs of R, is isomorphic to End_{A_P}M(R). The paper also constructs a tiling (S_R,M_R,P_R) whose tiling algebra is isomorphic to End_{A_P}M(R), and studies a tilting flip operation that preserves the property of being a tilting module.","tokens_in":23076,"tokens_out":3170,"duration_ms":36680,"significance":"If the main theorem is correct, it gives a concrete, explicit presentation of the endomorphism algebra of a tilting module over a gentle algebra purely in terms of the surface geometry. This is valuable: it yields explicit derived equivalent gentle algebras, enables iteration of the construction, and connects tilting theory for gentle algebras with surface combinatorics. The paper also reproves and slightly extends known correspondences between faithful dissections and tilting modules. The main strengths are the explicitness of the proposed algebra B_R and the clear overall strategy. However, the central isomorphism depends on a case analysis in Theorem 3.7 that is presented through figures rather than a formal enumeration, and the identification of the relation ideal in Theorem 3.18 is asserted rather tersely. These points are load-bearing and need to be made fully rigorous.","major_comments":[{"comment":"The proof that radical morphisms are spanned by morphisms induced by P-free negatively oriented angles rests on a four-type classification of the heads and tails of the admissible segments, with the geometric possibilities displayed only in Figures 10–14. The text says that, using Int(γ1,γ2)=0, 'the only geometrically possible cases' are as listed, but no formal enumeration of the possible fan sequences at the two endpoints is given. If a configuration is omitted, some basis element from Proposition 3.2 would not be realized as f_∠α, and the equality B=B′ would fail, undermining Theorem 3.18. This case analysis is the load-bearing step of the paper and should be replaced by a complete and checkable enumeration, or by a more conceptual argument.","section":"Section 3.3, Theorem 3.7"},{"comment":"The identification of the relation ideal is too compressed. After Lemma 3.16 it is asserted that eI_R equals the ideal generated by the loop squares and the two-arrow products satisfying the conditions of Lemma 3.10, and in Theorem 3.18 the equality eI_R = I_R is stated without a detailed proof. Lemma 3.15 shows that every element of eI_R contains a zero adjacent pair, and Lemma 3.16 excludes nontrivial linear combinations of distinct paths, but the passage from these statements to an equality of ideals needs to be written out: one must verify closure under right and left multiplication, show that the specified generators indeed lie in eI_R, and rule out any other relations in the path algebra. This is directly needed for the main isomorphism.","section":"Section 3.6–3.7, Lemmas 3.15–3.16 and Theorem 3.18"},{"comment":"The proof of Proposition 4.6 is a single sentence: 'By Definition 3.17 and the definition of tiling algebras, the result follows directly from the construction.' This claims both that (S_R,M_R,P_R) is a tiling and that its tiling algebra is isomorphic to End_{A_P}(M(R)) ≅ B_R. The preceding surgery (inserting marked points, sliding endpoints, merging boundary segments) is described informally and could introduce tiles of unexpected types. The isomorphism A_{P_R} ≅ B_R is not demonstrated. Since this is the basis for the claim that the endomorphism algebra is itself realized by a tiling, the proof needs to be substantially expanded.","section":"Section 4.2, Proposition 4.6"}],"minor_comments":[{"comment":"The title contains a typo: 'TIL TING' should be 'TILTING'.","section":"Title"},{"comment":"'Tiling for a tilting endmorphism algebra' should read 'endomorphism algebra'.","section":"Section 4.2 heading"},{"comment":"The text refers to 'Figure 20' when displaying the example; the actual figure is numbered Figure 8. Figure numbering should be checked throughout.","section":"Example 3.4"},{"comment":"There is a typo 'we hve' for 'we have' in the proof.","section":"Lemma 3.10 proof"},{"comment":"The name 'Simões' appears as 'Sim˜oes' in several places due to LaTeX escaping; this should be fixed.","section":"Abstract / Section 1"}],"recommendation":"major_revision","confidential_remarks":"The paper's central result is plausible and potentially useful, but the refereeing process should focus on the rigor of Theorem 3.7 and the relation-ideal identification. The authors also rely on an unpublished Master's thesis [21] for the faithful-dissection/tilting correspondence and the flip criterion; they reprove Theorem 2.18, but the flip criterion in Theorem 4.4 uses [21] as well. The editor may wish to check that [21] is publicly available or that the argument in the present paper is self-contained enough for the claims made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real contribution, not just a repackaging. The genuinely new content is Theorem 3.18, giving B_R ≅ End_{A_P} M(R) for faithful dissections, and the tiling realization in Section 4.2. They are transparent about the fact that Theorem 2.18 (faithful dissections ↔ tilting modules) is due to Nie and reprove it. The comparison with [7]'s admissible segments is careful, and the algebra B_R is defined cleanly from PR-free angles. I think the central isomorphism is very likely true.\n\nWhere I'd push back: the proof rests heavily on Theorem 3.7, which claims every radical morphism comes from a P-free negatively oriented angle. The proof is a four-type head/tail case analysis, with Figures 10–14 carrying the geometric exclusion argument. That is exactly the kind of step that needs a written enumeration of which fan sequences at the endpoints are compatible with Int(γ1,γ2)=0. If a configuration is missing, the basis B' would be too small and B_R would not be isomorphic to End. Nothing in the paper suggests such an omission, and the figures look comprehensive, but this is load-bearing and not fully formalized. I would want the authors to state the combinatorial content explicitly.\n\nSecondary soft spots: the equality eI_R = I_R after Lemma 3.16 is asserted in a sentence; the proof that the constructed triple (S_R,M_R,P_R) is a tiling (Proposition 4.6) is basically one line, and the tiling's type restrictions are not checked tile-by-tile. Those are presentation gaps rather than suspected errors. The use of existing results—Baur–Simões Proposition 3.2, the τ-tilting bijection from He–Zhou–Zhu—is legitimate, and the self-citation in Lemma 2.15 is not doing any hidden work.\n\nOverall: the mathematics is sound in outline, the writing is clear, and the attribution is honest. The paper belongs in a specialist venue. I'd recommend sending it to peer review, with referees asked to scrutinize Theorem 3.7's case analysis and the construction in 4.2. Not a desk reject; not a publish-without-revision either.","headline":"A new and largely convincing geometric description of tilting endomorphism algebras, held up by a figure-based case analysis that should be tightened before I'd call it airtight.","tokens_in":23527,"tokens_out":1777,"would_cite":true,"duration_ms":19396,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","16G70","16E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Surface dissections give explicit tilting endomorphism algebras.","keywords":["gentle algebra","tiling algebra","faithful dissection","tilting module","endomorphism algebra","P-free angle","PR-free angle","derived equivalence"],"falsifier":"Pick a small tiling with a faithful dissection and compute both sides independently: the quiver with relations of $B_R$ from the geometric angle rules, and the actual endomorphism algebra $\\operatorname{End}_{A_P} M(R)$ by direct representation-theoretic calculation. Any mismatch in the number of arrows or in a relation (for instance, a zero composition not predicted by the angle configuration) would disprove the isomorphism. More sharply, search for two permissible arcs with zero intersection that admit a nonzero radical morphism but have no common endpoint; such a pair would contradict the configuration enumer","tokens_in":22646,"feed_emoji":"📐","tokens_out":3313,"duration_ms":30817,"temperature":0.7,"texified_at":"2026-08-05T21:35:30.948934+00:00","pith_summary":"The paper proves that for a gentle algebra presented as a tiling algebra on a marked surface, every faithful dissection of the surface yields a tilting module, and the endomorphism algebra of that tilting module is isomorphic to an explicit auxiliary algebra built from the dissection's geometry. The auxiliary algebra's quiver has one vertex per arc in the dissection, with arrows and relations read off from the PR-free negatively oriented angles between arcs. If correct, this gives a purely geometric way to construct endomorphism algebras of tilting modules and therefore new gentle algebras derived equivalent to the original. The paper also constructs a new tiling realizing this endomorphism algebra and introduces a tilting flip that preserves tilting modules.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":5532,"prompt_tokens":719,"completion_tokens":4813,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":719,"completion_tokens_details":{"reasoning_tokens":4147}},"feed_headline":"Surface dissections model tilting endomorphism algebras","feed_subtitle":"A faithful dissection's angles give the quiver and relations of End(M), exposing new derived-equivalent gentle algebras.","key_machinery":"The central object is the PR-free negatively oriented angle: an angle between two arcs of a dissection whose fan contains no interior arcs from the original tiling $P$ or the dissection $R$. These angles index the arrows of $B_R$; the relations come from whether two such angles share a marked point or a loop arc forces zero composition. The machinery also uses the admissible-segment description of $\\operatorname{Hom}$-spaces between string modules to identify radical morphisms with P-free negatively oriented angles.","core_discovery":"The central claim is Theorem 3.18: for a tiling $(S,M,P)$ and a faithful dissection $R$, the algebra $B_R$ — defined by taking the arcs of $R$ as vertices and the PR-free negatively oriented angles as arrows, with relations determined by local angle configurations — is isomorphic to $\\operatorname{End}_{A_P} M(R)$. Since $M(R)$ is a tilting $A_P$-module, $B_R$ is derived equivalent to $A_P$. The proof relies on a geometric basis theorem: radical morphisms between the modules $M(\\gamma_1)$ and $M(\\gamma_2)$ are in bijection with P-free negatively oriented angles from $\\gamma_1$ to $\\gamma_2$, and compositions of these morphisms are governed by adjacency of angles. The paper further constructs a new marked surface tiling whose tiling algebra is exactly End_","pith_inferences":["The geometric basis theorem likely extends to arbitrary partial dissections, suggesting the angle formalism describes endomorphism algebras of τ-rigid modules, not just tilting modules.","Iterating tilting flips and dissection constructions may yield explicit derived autoequivalences, potentially linking to a geometric model of the derived category.","A parallel formulation using skew-tiling algebras could test whether PR-free angles give endomorphism algebras of tilting objects over skew-gentle algebras."],"forward_implications":["For every faithful dissection, End_{A_P} M(R) is a gentle algebra with a fully explicit quiver and relations.","B_R is derived equivalent to A_P, and the construction can be iterated to produce an infinite family of derived-equivalent gentle algebras.","The tilting flip criterion — M(μ_γ(R)) is tilting if and only if R\\{γ} is a faithful partial dissection — gives a combinatorial operation that preserves tilting modules.","The newly constructed tiling realizes the endomorphism algebra as a tiling algebra, placing such endomorphism algebras back into the geometric model.","The same angle formalism applies to partial dissections, so τ-rigid endomorphism algebras also admit geometric descriptions."],"fun_headline_variants":["Surface angles capture tilting endomorphism algebras","Dissections and angles reveal tilting endomorphism algebras","New tilings encode derived-equivalent gentle algebras","Faithful dissections model tilting endomorphism algebras","Flip moves preserve tilting modules in surface tilings"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof hinges on the case-by-case geometric claim in Theorem 3.7 that every radical morphism between two non-crossing permissible arcs arises from a P-free negatively oriented angle; if the figures omit a possible configuration, the basis theorem and the isomorphism $B_R \\cong \\operatorname{End} M(R)$ would fail.","fun_headline_variants_meta":{"raw":{"variants":["Surface angles capture tilting endomorphism algebras","Dissections and angles reveal tilting endomorphism algebras","New tilings encode derived-equivalent gentle algebras","Faithful dissections model tilting endomorphism algebras","Flip moves preserve tilting modules in surface tilings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000172,"raw_usage":{"total_tokens":1050,"prompt_tokens":622,"completion_tokens":428,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":366,"completion_tokens_details":{"reasoning_tokens":355}},"tokens_in":366,"tokens_out":428,"duration_ms":4564,"temperature":1.0,"reasoning_tokens":355,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T11:13:25.022083+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick a small tiling with a faithful dissection and compute both sides independently: the quiver with relations of $B_R$ from the geometric angle rules, and the actual endomorphism algebra $\\operatorname{End}_{A_P} M(R)$ by direct representation-theoretic calculation. Any mismatch in the number of arrows or in a relation (for instance, a zero composition not predicted by the angle configuration) would disprove the isomorphism. More sharply, search for two permissible arcs with zero intersection that admit a nonzero radical morphism but have no common endpoint; such a pair would contradict the configuration enumer","supporting_citations":[],"review_version":1}