{"id":"0a4e197d-047a-45d5-be90-b96110dc1775","arxiv_id":"2607.06960","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"An extriangulated functor from (n+1)-term subcategories to (n+1)-term complexes is constructed, with equivalent conditions for fullness yielding an extriangle equivalence and a mutation-compatible silting bijection.","lead":"The paper constructs a functor from (n+1)-term subcategories of algebraic triangulated categories to (n+1)-term complexes, generalizing Yang's n=1 construction. It gives equivalent conditions for this functor to be full, yielding an extriangle equivalence and a mutation-compatible silting bijection.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The proof of Theorem 4.4 hinges on an unverified chain-map composition (step 6: \"a direct check\") that Ω sends extriangles to triangles; a sign or indexing error there would invalidate the extriangulated functor property underlying all main results.","rationale":"The paper's logical structure is sound: Theorem 5.14 is a correct conditional result independent of [13], and Theorem 4.4's construction is detailed enough that the \"direct check\" is a finite, verifiable computation rather than a conceptual gap. The n=1 case provides one data point (matching Yang's prior work). The concern I identify — the unverified chain-map composition — is a legitimate soft spot but is the kind of thing that would be caught by careful line-by-line verification, not a structural flaw. The reader's ACCEPT verdict with MODERATE confidence is appropriate: the arguments are coherent and well-structured, the constructions are parameter-free, and the applications are natural. The paper acknowledges independent work by Silberberg. I agree with the reader's assessment but would note that the load-bearing concern is the Theorem 4.4 computation, not the reliance on [13]. The verdict remains ACCEPT with MODERATE confidence, as the reader stated.","tokens_in":42020,"tokens_out":8018,"duration_ms":598172,"concrete_test":"Independently verify the chain-map composition for n=2. Specifically, take the 2-cluster category of Q: 2→1 (Example 2.17/5.22) with M=add(P1⊕ΣP2), choose a specific extriangle in pr^3_T(M) (e.g., one arising from a conflation in the Frobenius category), and compute: (a) the three chain maps (3.14), (3.12), (4.7) explicitly for this instance, (b) their composition, and (c) the target chain map (4.5). If the composition matches (4.5) entry-by-entry (including signs), the extriangulated functor property is verified for n=2; if not, the main construction has an error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's concern about [13, Theorem 3.7] affects applicability of Theorem 5.14 to general reduced (n−1)-Auslander categories, but not its correctness: Theorem 5.14 is a conditional statement (\"if P exists, then...\") whose proof uses only extriangulated category axioms, not [13]. The more load-bearing concern is internal to the paper: the proof of Theorem 4.4, which establishes that (P,Ω) is an extriangulated functor. The argument reduces to showing that for a conflation ẽu: Y→Z, ẽv: Z→X in the Frobenius category F, the image sequence P(Y)→P(Z)→P(X)→ΣP(Y) is a triangle in K^b(M). After constructing a horseshoe presentation of Z (diagram 4.2) and decomposing Ω(π(ew)) as π(ΣeP(eδ)◦eΓ_{X1}◦eP(eγ_X^0)), the proof states (end of p. 23): \"A direct check shows that this composition is exactly the one shown in (4.5).\" This composition involves three explicitly written chain maps — (3.14), (3.12), and (4.7) — each carrying alternating signs (−1)^i and index shifts. If any sign or degree alignment is off, Ω would fail to realize extriangles as triangles, and (P,Ω) would not be an extriangulated functor. This would invalidate Theorem 1.1, Theorem 5.21, and the applications in Section 6. The n=1 case recovers Yang's [37, Theorem 1.1], providing one check, but the first genuinely new case n=2 has no prior verification. The concern is not conceptual — the maps are all written out — but the final composition is asserted rather than displayed term-by-term.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper constructs an extriangulated functor (P, Ω) from the (n+1)-term subcategory pr_T^{n+1}(M) of an algebraic triangulated category T to the category K[-n,0](M) of (n+1)-term complexes, generalizing Yang's n=1 construction. The functor P restricts to the identity on M. The main result (Theorem 1.2 / Theorems 5.14, 5.17, 5.20) establishes that P is full if and only if Ω is injective if and only if Hom_C(Σ^i M, M) = 0 for 1 ≤ i ≤ n−1, and under these conditions P induces an extriangle equivalence C/[Σ^n M, M] ≃ K[-n,0](M) plus a mutation-compatible bijection on silting subcategories. Applications to n-cluster tilting subcategories and (n+1)-Calabi-Yau categories are given in Section 6.","tokens_in":42343,"tokens_out":1252,"duration_ms":190242,"significance":"The paper makes a substantial contribution by extending the functorial bridge between (n+1)-term subcategories and (n+1)-term complexes from the classical n=1 case to arbitrary n. The construction of P in Section 3 is parameter-free (given M and T, the functor is determined up to natural isomorphism by Proposition 3.9), and the equivalent conditions in Theorem 5.14 are logically derived rather than circularly imposed. The n=1 case recovers Yang's result, providing a consistency check. The applications to silting bijections and cluster tilting in Calabi-Yau categories are concrete and falsifiable. The proofs proceed by careful induction with explicit chain maps.","major_comments":[{"comment":"Proof of Theorem 4.4 (end of p. 23): The final step of the proof asserts that the composition of the three explicitly written chain maps — (3.14), (3.12), and (4.7) — equals the chain map shown in (4.5), verified by 'a direct check.' This composition involves alternating signs (−1)^i and index shifts across three maps, and it is the load-bearing step establishing that (P, Ω) is an extriangulated functor. If any sign or degree alignment is off, Ω fails to send extriangles to triangles, invalidating Theorem 1.1, Theorem 5.21, and the Section 6 applications. The n=1 case provides one check, but the first genuinely new case n=2 has no prior verification. The authors should either display the term-by-term composition or, at minimum, spell out the verification for one representative degree index to confirm the sign convention is consistent across all three maps.","section":null}],"minor_comments":[{"comment":"p. 4, Convention: 'For any two subcategories X and Y of a triangulated category T, we denote by X*Y the subcategory of T consisting of the objects Z such that there exists a triangle X → Z → Y → ΣX with X ∈ X and Y ∈ Y.' This should specify X ∈ X (an object) rather than X ∈ X (the subcategory), for clarity.","section":null},{"comment":"p. 6, line 5: 'natrual' should be 'natural' (appears in the proof of Theorem 4.4: 'since Ω is a natrual transformation').","section":null},{"comment":"p. 23, line 2: 'ΣeP(eδ)◦eΓ_{X1}◦eP(eγ_X^0)' — the notation with the 'e' prefix is used consistently for lifts to F, but a brief reminder at this point that 'e' denotes the lifted version would aid readability, as this is a dense passage.","section":null},{"comment":"Example 2.17 (p. 10): The AR quiver is drawn with specific notation (P_i, S_i) but the relationship between the 2-cluster category structure and the specific triangles listed could benefit from one sentence explaining why M = add(P_1 ⊕ ΣP_2) is 2-rigid.","section":null},{"comment":"p. 37, Corollary 6.6: 'Hom_T(ΣM, M) = 0 for all 1 ≤ i ≤ n−1' should read 'Hom_T(Σ^i M, M) = 0 for all 1 ≤ i ≤ n−1' (missing exponent on Σ).","section":null},{"comment":"The reference [36] to Silberberg's forthcoming work is cited as 'In preparation.' If available by the time of revision, a more complete reference or arXiv link would be appropriate.","section":null},{"comment":"Section 5 is dense; a brief roadmap paragraph at the start of Section 5 outlining the logical flow (Proposition 5.3 → Lemma 5.9 → Lemma 5.11 → Theorem 5.14 → Proposition 5.16 → Theorems 5.17, 5.20) would help the reader navigate the induction.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern about the 'direct check' in Theorem 4.4 is legitimate and is the single most important point to verify. I have flagged it as a major comment because it is load-bearing, but I assess it as a presentation gap (the maps are all written out; the composition is asserted rather than displayed) rather than a likely error, since the n=1 case provides a non-trivial consistency check and the surrounding infrastructure (Lemmas 3.5, 3.6, 3.8, 3.14) is carefully built. The reader's concern about [13, Theorem 3.7] affecting applicability is correctly noted by the reader as not affecting correctness: Theorem 5.14 is a conditional statement whose proof uses only extriangulated category axioms. I agree with this assessment."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for the careful reading and the constructive suggestion regarding the sign verification in the proof of Theorem 4.4. We agree that this step is load-bearing and that the verification should be made explicit rather than left as a 'direct check.' We will revise the manuscript accordingly.","responses":[{"response":"We fully agree with the referee that this step deserves explicit verification. The composition of the three chain maps (3.14), (3.12), and (4.7) is indeed the crux of the proof that (P, Ω) is an extriangulated functor, and the alternating signs (−1)^i together with the degree shifts make a terse 'direct check' inadequate for a load-bearing argument. We will revise the manuscript to include an explicit term-by-term verification of the composition. Specifically, we will display the composition at a representative degree index i (for 0 ≤ i ≤ n−1), showing how the signs (−1)^{i+1} from (4.7), the identity maps and sign (−1) factors from (3.12), and the signs (−1)^n, (−1)^{n−1}, … from (3.14) combine to produce exactly the entries of the chain map (4.5). We will also verify the boundary cases i = 0 and i = n−1 separately, as these involve the terms I(X₁) and the zero objects where the degree alignment is most delicate. This will confirm the sign convention is consistent across all three maps for arbitrary n, not just n = 1.","revision_made":"yes","referee_comment":"Proof of Theorem 4.4 (end of p. 23): The final step of the proof asserts that the composition of the three explicitly written chain maps — (3.14), (3.12), and (4.7) — equals the chain map shown in (4.5), verified by 'a direct check.' This composition involves alternating signs (−1)^i and index shifts across three maps, and it is the load-bearing step establishing that (P, Ω) is an extriangulated functor. If any sign or degree alignment is off, Ω fails to send extriangles to triangles, invalidating Theorem 1.1, Theorem 5.21, and the Section 6 applications. The n=1 case provides one check, but the first genuinely new case n=2 has no prior verification. The authors should either display the term-by-term composition or, at minimum, spell out the verification for one representative degree index to confirm the sign convention is consistent across all three maps."}],"tokens_in":41464,"tokens_out":854,"duration_ms":38999,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"This paper generalizes Yang's recent construction of a functor from 2-term subcategories to 2-term complexes, extending it to (n+1)-term subcategories and (n+1)-term complexes for arbitrary n. The main outputs are: an extriangulated functor (P, Ω) from pr^{n+1}_T(M) to K[-n,0](M) (Theorem 4.4), equivalent conditions for P to be full involving injectivity of Ω and a vanishing condition on Hom(Σ^i M, M) (Theorem 5.14), an induced extriangle equivalence modulo an ideal, and a mutation-compatible silting bijection (Theorem 5.20). Applications to n-cluster tilting in Calabi-Yau categories round things out in Section 6. The n=1 case recovers Yang's result, which is a good consistency check. The construction of P is parameter-free and determined up to natural isomorphism (Proposition 3.9), and the logical structure of the equivalent conditions is genuinely non-circular. The Γ construction (Construction 3.13, Lemma 3.14) connecting P with the suspension is a clean piece of machinery, and the induction arguments in Section 5 are carefully structured with the five-lemma-style diagram chases (Lemma 5.13) doing real work. The paper acknowledges Silberberg's independent forthcoming work, which is the right thing to do. The stress-test concern about the proof of Theorem 4.4 is legitimate but not, I think, fatal. The proof reduces to showing that Ω sends extriangles to triangles, and the final step — verifying that the composition of three explicitly written chain maps (3.14), (3.12), and (4.7) equals the map in (4.5) — is asserted as 'a direct check' rather than displayed term-by-term. Each of the three maps is written out explicitly with its signs and degree shifts, so the verification is in principle mechanical. The concern is that a sign or indexing error in this composition would undermine the extriangulated functor property and hence the main results. The n=1 case provides one check, but n=2 has no prior verification. This is the kind of gap that a referee should ask the authors to fill by either displaying the composition or providing a more conceptual argument for why the signs work out. The reader's concern about the dependency on [13, Theorem 3.7] for applicability to general reduced (n-1)-Auslander categories is well-tated but secondary: Theorem 5.14 is a conditional statement whose proof uses only extriangulated category axioms, not [13]. The algebraic triangulated category assumption is standard for this type of horseshoe-lemma construction. Overall this is a solid paper with a real new construction. The soft spot is the unverified composition in the proof of Theorem 4.4, which deserves scrutiny but is likely fixable. The paper is for specialists in higher homological algebra and silting theory. It deserves a serious referee who can check the diagram chases carefully.","headline":"Generalizes Yang's 2-term functor to (n+1)-term complexes; the construction is sound but one key composition is asserted rather than displayed.","tokens_in":43138,"tokens_out":730,"would_cite":true,"duration_ms":147699,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18G80","16E35","18G05"],"pacs":[],"model":"glm-5.2","headline":"Functor bridges n+1-term subcategories and complexes","keywords":["extriangulated category","n-rigid subcategory","silting subcategory","cluster tilting","Calabi-Yau category","homotopy category","mutation","Auslander category"],"falsifier":"If a reduced (n−1)-Auslander extriangulated category exists that does not arise from an n-rigid subcategory of an algebraic triangulated category, the functor P may not be constructible for that category, limiting the scope of Theorem 5.14.","tokens_in":42041,"feed_emoji":"🔗","tokens_out":1778,"duration_ms":111282,"temperature":0.7,"pith_summary":"The paper constructs an explicit functor P from the (n+1)-term subcategory of a triangulated category generated by an n-rigid subcategory M to the category of (n+1)-term complexes over M, generalizing a previously known construction for the n=1 case. The functor P sends each object, built from n+1 consecutive shifts of M, to a complex of length n+1 built from M, and is shown to respect the extriangulated (generalized exact) structures on both sides. The main theorem proves that P is full if and only if a natural map on extensions is injective, if and only if the groups Hom(Σ^i M, M) vanish for 1 ≤ i ≤ n−1. When any of these equivalent conditions holds, P becomes dense, the extension map becomes a natural isomorphism, and P descends to an extriangle equivalence between the quotient of the source category by a specific ideal and the target complex category. The paper further shows that this equivalence preserves silting subcategories and their mutations, and applies the results to n-cluster tilting subcategories and to n-cluster tilting objects in (n+1)-Calabi-Yau categories, obtaining bijections between cluster tilting objects and silting or tilting complexes.","feed_headline":"Vanishing extensions govern equivalence of subcategories and complexes","feed_subtitle":"A functor between (n+1)-term subcategories and (n+1)-term complexes is an equivalence exactly when intermediate negative extensions vanish,携","key_machinery":"The functor P is built from eM-presentations of length n+1 in a Frobenius exact category F whose stable category is T. Each object X in pr^{n+1}_T(M) is resolved by n+1 conflations using objects from M, producing a complex C•(X) in degrees −n to 0. Morphisms are lifted to chain maps via vanishing of Ext^1(M, Y) for appropriate objects, and null-homotopies are controlled by factoring through projective-injective objects. The extension map Ω is defined by factoring extensions through Σ of objects in pr^n_T(M) and composing with Γ, a natural isomorphism P∘Σ ≅ Σ∘P constructed from the standard triangle structure of the stable category.","core_discovery":"The central discovery is the trifold equivalence: fullness of P, injectivity of the extension map Ω, and the vanishing of Hom(Σ^i M, M) for intermediate shifts 1 ≤ i ≤ n−1. This vanishing condition, which is weaker than full n-rigidity (it excludes only the extreme i = n), is the precise obstruction that determines whether the functor from (n+1)-term subcategories to (n+1)-term complexes is an equivalence. The functor P itself is constructed by lifting objects and morphisms through a Frobenius exact category, using presentations of length n+1 and the horseshoe lemma to build chain maps between complexes, then passing to the stable category. The natural isomorphism Γ between P∘Σ and Σ∘P on a子","pith_inferences":["The vanishing condition Hom(Σ^i M, M) = 0 for 1 ≤ i ≤ n−1 is strictly weaker than n-rigidity (which includes i = n), suggesting that the 'boundary' extension group Hom(Σ^n M, M) plays a different role—it controls the ideal [Σ^n M, M] being quotiented, rather than obstructing the equivalence.","The construction likely extends to non-algebraic triangulated categories if one can find an alternative to the Frobenius lift, since the main theorem is stated for general reduced (n−1)-Auslander extriangulated categories, but the explicit construction of P currently depends on the algebraic hypothesis.","The mutation compatibility of the silting bijection suggests that exchange graphs of silting subcategories on both sides are isomorphic, which could provide a combinatorial tool for computing mutation classes in settings where one side is more tractable."],"forward_implications":["When M is n-cluster tilting and the vanishing condition holds, the entire triangulated category T is equivalent (modulo an ideal) to (n+1)-term complexes over M, providing a higher-dimensional analogue of classical tilting theory.","In (n+1)-Calabi-Yau categories, the functor gives a mutation-compatible bijection between 2-term n-cluster tilting objects and 2-term silting complexes over the endomorphism algebra, extending the support τ-tilting correspondence to arbitrary n.","The equivalence preserves silting subcategories and their left/right mutations, meaning the combinatorial mutation structure on one side is faithfully mirrored on the other.","When Σ^{n+1}M = M, the functor P becomes a genuine equivalence of additive categories, endowing K[-n,0](M) with a triangulated structure inherited from T."],"fun_headline_variants":["Intermediate extension vanishing governs subcategory-complex equivalence","A trifold criterion for extriangulated subcategory-complex equivalence","Mutation-compatible bijections via intermediate extension vanishing","When do (n+1)-term subcategories match (n+1)-term complexes?","Negative extensions dictate equivalence of subcategories and complexes"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The explicit construction of the functor P requires the triangulated category T to be algebraic (the stable category of a Frobenius exact category), so that morphisms can be lifted to the exact category and chain maps can be built using the horseshoe lemma. The main theorem is then stated for any reduced (n−1)-Auslander extriangulated category, relying on a cited classification result that every algebraic such category arises from the construction. If that classification hasg","fun_headline_variants_meta":{"raw":{"variants":["Intermediate extension vanishing governs subcategory-complex equivalence","A trifold criterion for extriangulated subcategory-complex equivalence","Mutation-compatible bijections via intermediate extension vanishing","When do (n+1)-term subcategories match (n+1)-term complexes?","Negative extensions dictate equivalence of subcategories and complexes","Extriangulated equivalence from intermediate extension vanishing"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1304,"prompt_tokens":560,"completion_tokens":744,"prompt_tokens_details":null},"tokens_in":560,"tokens_out":744,"duration_ms":25322,"temperature":1.0,"reasoning_tokens":704,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T01:01:26.103005+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If a reduced (n−1)-Auslander extriangulated category exists that does not arise from an n-rigid subcategory of an algebraic triangulated category, the functor P may not be constructible for that category, limiting the scope of Theorem 5.14.","supporting_citations":[],"review_version":1}