{"id":"0bc2c8c1-57ec-4517-853e-62c4e7a3728f","arxiv_id":"2606.13508","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Establishes poset isomorphisms between (d+1)-term silting subcategories, functorially finite s-torsion pairs in the d-extended heart, and hereditary complete cotorsion pairs, with dg-algebra versions for τ-tilting pairs.","lead":"The paper proves poset isomorphisms linking (d+1)-term silting subcategories with s-torsion pairs in d-extended hearts and hereditary cotorsion pairs. A generalist might read it for new organizational tools in homological algebra that connect silting and torsion theories.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of the d-extended heart's technical conditions as the weakest assumption is accurate and load-bearing for the bijections. No additional internal gap is visible from the given abstract and claim statement; the low-confidence UNVERDICTED verdict therefore remains appropriate until the full proofs are examined.","tokens_in":1615,"tokens_out":317,"duration_ms":15627,"concrete_test":"Verify that the definition of the d-extended heart (presumably §2) produces an abelian category in which every silting subcategory induces a functorially finite s-torsion pair; if the construction is only stated for specific ambient categories (e.g., derived categories of finite-dimensional algebras), test the claim on a non-finite-dimensional example such as the derived category of a hereditary algebra of infinite representation type.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts poset isomorphisms between (d+1)-term silting subcategories, functorially finite s-torsion pairs in the d-extended heart, and hereditary complete cotorsion pairs (plus dg-algebra versions). The reader's weakest assumption correctly flags that these isomorphisms presuppose the d-extended heart exists and carries the required structure (abelian category with functorially finite torsion pairs and hereditary cotorsion pairs). Without the full text, no internal inconsistency, missing hypothesis, or counterexample to the construction can be located; the abstract is consistent with standard techniques in silting theory and extended t-structures.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript establishes poset isomorphisms between (d+1)-term silting subcategories, functorially finite s-torsion pairs in the d-extended heart, and hereditary complete cotorsion pairs in a suitable subcategory of a triangulated category. As an application, it provides dg-algebra versions of these bijections relating τ-tilting pairs, (d+1)-term silting complexes, and functorially finite s-torsion pairs.","tokens_in":1715,"tokens_out":433,"duration_ms":15575,"significance":"If the isomorphisms hold, the results unify aspects of silting theory with torsion and cotorsion pairs in the setting of extended hearts, extending classical correspondences (such as those for 2-term silting and torsion pairs) to higher d. The dg-algebra versions may facilitate applications in derived categories and representation theory of algebras. The paper ships explicit bijections that are functorial in the stated sense, which strengthens the contribution if the technical conditions on extended hearts are verified.","major_comments":[{"comment":"The central claims rely on the existence and properties of the d-extended heart (including that it is abelian and admits functorially finite s-torsion pairs and hereditary complete cotorsion pairs). The abstract and setup assume these without an explicit verification or reference to a prior result establishing the required abelian structure and finiteness conditions for general d; this is load-bearing for all stated isomorphisms.","section":"Introduction / Setup of extended hearts"}],"minor_comments":[{"comment":"Notation for s-torsion pairs and the precise definition of 'hereditary complete cotorsion pairs in a suitable subcategory' should be recalled or cross-referenced in the statement of the main theorems to improve readability.","section":"Main theorems"},{"comment":"The dg-algebra versions are presented as an application; a brief comparison table or diagram relating the classical and dg cases would clarify the functoriality claims.","section":"Application section"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for recognizing the potential of our results to unify aspects of silting theory with torsion and cotorsion pairs via extended hearts. We address the single major comment below.","responses":[{"response":"We agree that the properties of the d-extended heart are foundational to all stated isomorphisms. The construction of the d-extended heart appears in Section 2, where it is introduced as an abelian category following the standard extension procedure from the original heart. However, the introduction and setup do not contain an explicit reference or short verification confirming the abelian structure together with the existence of functorially finite s-torsion pairs and hereditary complete cotorsion pairs for arbitrary d. We will revise the manuscript by adding a reference to the prior result that establishes the abelian structure of the d-extended heart for general d, together with a brief paragraph in the setup section recalling why the required finiteness conditions hold. This change will clarify the load-bearing assumptions without affecting the main theorems or proofs.","revision_made":"yes","referee_comment":"[Introduction / Setup of extended hearts] The central claims rely on the existence and properties of the d-extended heart (including that it is abelian and admits functorially finite s-torsion pairs and hereditary complete cotorsion pairs). The abstract and setup assume these without an explicit verification or reference to a prior result establishing the required abelian structure and finiteness conditions for general d; this is load-bearing for all stated isomorphisms."}],"tokens_in":1201,"tokens_out":324,"duration_ms":17616,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that the authors establish poset isomorphisms linking (d+1)-term silting subcategories, functorially finite s-torsion pairs inside the d-extended heart, and hereditary complete cotorsion pairs in a suitable subcategory. They also record the corresponding statements for dg algebras, relating tau-tilting pairs, (d+1)-term silting complexes, and torsion pairs.\n\nThis extends earlier correspondences between silting objects and torsion pairs by moving to the d-extended setting. The bijections look like a direct, incremental step from the cited literature rather than a wholesale reorganization.\n\nThe work does a reasonable job of collecting these objects into a single framework and stating the isomorphisms cleanly. If the proofs in the body are complete and the definitions of the extended hearts are handled without extra hidden assumptions, the results should be usable by people already working in this corner of representation theory.\n\nThe main limitation visible from the abstract is the lack of any proof outline or concrete example, which makes it hard to judge how much new technical work is required to make the functorially finite conditions hold. The central assumption—that the d-extended heart carries an abelian structure rich enough for these pairs—needs to be checked carefully in the full text, but nothing in the statement itself looks circular or contradictory.\n\nThis is aimed at specialists who already know silting theory and t-structures. A reader who wants explicit bijections for the d-case could extract something usable here.\n\nI would send it to peer review. The claims are specific enough that referees can test them against the existing literature.","headline":"The paper sets up poset isomorphisms between (d+1)-term silting subcategories and torsion/cotorsion pairs in d-extended hearts, plus dg versions.","tokens_in":2170,"tokens_out":406,"would_cite":false,"duration_ms":15937,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Poset isomorphisms connect (d+1)-term silting subcategories to functorially finite s-torsion pairs in d-extended hearts and to hereditary complete cotorsion pairs.","keywords":["silting subcategories","extended hearts","s-torsion pairs","cotorsion pairs","tau-tilting pairs","dg algebras","poset isomorphisms","triangulated categories"],"falsifier":"An explicit triangulated category possessing a d-extended heart in which the map from (d+1)-term silting subcategories to functorially finite s-torsion pairs fails to be bijective or order-preserving.","tokens_in":2514,"feed_emoji":"","tokens_out":735,"duration_ms":15196,"temperature":0.7,"pith_summary":"The paper proves that these three collections of objects stand in order-preserving bijection with one another inside a triangulated category equipped with a d-extended heart. The same pattern is shown to hold, after passage to dg algebras, between tau-tilting pairs, (d+1)-term silting complexes, and s-torsion pairs. A reader following the argument sees that questions about one of these structures can be translated directly into questions about the others while preserving the partial order. The correspondences therefore supply a uniform language for several variants of tilting and silting that appear in representation theory.","feed_headline":"Poset isomorphisms link silting subcategories to s-torsion pairs","feed_subtitle":"The bijections also equate hereditary cotorsion pairs and extend to tau-tilting pairs and silting complexes over dg algebras.","key_machinery":"The d-extended heart, which supplies the ambient abelian category in which s-torsion pairs and hereditary cotorsion pairs are defined and compared with silting subcategories via the stated poset isomorphisms.","core_discovery":"The central claim is that the poset of (d+1)-term silting subcategories is isomorphic to the poset of functorially finite s-torsion pairs inside the d-extended heart, which is in turn isomorphic to the poset of hereditary complete cotorsion pairs in an associated subcategory; the dg-algebra versions replace the first two posets by the poset of tau-tilting pairs and the poset of (d+1)-term silting complexes, respectively.","pith_inferences":["The isomorphisms may be used to transport mutation operations or approximation properties from one structure to the others.","The pattern suggests analogous correspondences could exist for other notions of extended hearts or for n-torsion pairs with n not equal to d.","Concrete computations of these posets for derived categories of gentle algebras or cluster-tilted algebras become interchangeable across the three descriptions."],"forward_implications":["Any classification of (d+1)-term silting subcategories immediately yields a classification of the corresponding s-torsion pairs and cotorsion pairs.","Properties preserved by the poset isomorphisms, such as finiteness or heredity, transfer between the three collections.","In the dg setting the same transfer applies between tau-tilting pairs and silting complexes.","The bijections are compatible with the natural partial orders on each side."],"fun_headline_variants":["Silting subcategories isomorphic to s-torsion pairs in extended hearts","(d+1)-term silting subcategories isomorphic to cotorsion pairs","Tau-tilting pairs isomorphic to (d+1)-term silting complexes","Hereditary cotorsion pairs isomorphic to s-torsion pairs in extended hearts"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The d-extended heart of the triangulated category exists and satisfies the technical conditions that let functorially finite s-torsion pairs and hereditary complete cotorsion pairs be defined inside it.","fun_headline_variants_meta":{"raw":{"variants":["Silting subcategories isomorphic to s-torsion pairs in extended hearts","(d+1)-term silting subcategories isomorphic to cotorsion pairs","Tau-tilting pairs isomorphic to (d+1)-term silting complexes","Hereditary cotorsion pairs isomorphic to s-torsion pairs in extended hearts"]},"model":"grok-4.3","cost_usd":0.008605,"raw_usage":{"total_tokens":3825,"prompt_tokens":551,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":86049500,"prompt_tokens_details":{"text_tokens":551,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3195,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":551,"tokens_out":79,"duration_ms":29270,"temperature":1.0,"reasoning_tokens":3195,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T05:00:54.340086+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit triangulated category possessing a d-extended heart in which the map from (d+1)-term silting subcategories to functorially finite s-torsion pairs fails to be bijective or order-preserving.","supporting_citations":[],"review_version":1}