{"id":"347eaaa5-96cf-4277-862e-00a8ac94a9a2","arxiv_id":"2604.04505","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Bicompact torsion classes equal functorially finite ones for hereditary algebras and semistable cases, so Demonet Conjecture implies Enomoto Conjecture.","lead":"The paper defines bicompact torsion classes as those that are both compact (generated by a single module) and dually compact in the module category of a finite-dimensional algebra. It conjectures these coincide with functorially finite torsion classes, proves the claim for hereditary algebras and semistable torsion classes, and shows this implies Demonet Conjecture entails Enomoto Conjecture on brick-infinite algebras.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Equivalence of bicompact and functorially finite torsion classes established only for hereditary algebras and semistable classes; general case open","rationale":"The reader's weakest-assumption statement matches the precise point at which the argument is least secure: the special-case proofs exploit properties (hereditary closure, semistable Hom-vanishing) that are not available in general, and the paper correctly presents the general statement as a conjecture rather than a theorem. No internal inconsistency or overclaim is visible in the abstract or the described results; the load-bearing risk is simply the open status of the general case. This does not alter the UNVERDICTED verdict.","tokens_in":1626,"tokens_out":419,"duration_ms":44989,"concrete_test":"Take a small non-hereditary algebra (e.g., the 3-arrow Kronecker quiver with a single relation or the algebra KQ/I for a bound quiver of dimension 4–6), enumerate all torsion classes by computing their generators and cogenerators, and check whether every bicompact class is functorially finite and conversely; a single mismatch falsifies the conjecture.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the conjecture that bicompact torsion classes (compact and dually compact) coincide with functorially finite torsion classes in mod A for arbitrary finite-dimensional A, together with the deduction that this would imply Demonet Conjecture entails Enomoto Conjecture. The paper proves the equivalence when A is hereditary (using that every torsion class is generated by a single module via the absence of relations and direct-sum decompositions) and when the torsion class is semistable (using Hom-vanishing between stable and unstable summands). In a general algebra these closure and approximation properties need not hold: a torsion class generated by one module may fail to admit the left or right approximations required for functorial finiteness once relations are present. Consequently the implication between the two brick-infiniteness conjectures rests on an unproven general identification.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript defines a torsion class T in mod A (A finite-dimensional over a field) to be compact if it is the smallest torsion class containing some module M, and bicompact if it is both compact and dually compact. It conjectures that bicompact torsion classes coincide exactly with functorially finite torsion classes, proves the equivalence when A is hereditary (using the absence of relations and direct-sum decompositions) and when T is semistable (using Hom-vanishing between stable and unstable summands), and deduces that this identification would imply the Demonet conjecture entails the Enomoto conjecture on brick-infinite algebras.","tokens_in":1809,"tokens_out":441,"duration_ms":35070,"significance":"If the conjecture holds in generality, it would relate two central open questions on brick infiniteness by identifying compactness properties of torsion classes with the existence of left and right approximations. The verified cases supply rigorous, self-contained arguments that exploit standard closure properties of hereditary categories and Hom-vanishing for semistable classes; these constitute concrete progress and may guide attempts to remove the restrictions.","major_comments":[{"comment":"In the section deriving the implication between the Demonet and Enomoto conjectures, the claim that the bicompact-functorially-finite equivalence yields Demonet implies Enomoto is not fully justified for arbitrary finite-dimensional algebras, because the equivalence is established only for hereditary algebras and semistable torsion classes; the manuscript should clarify whether the two conjectures are understood to apply only in those restricted settings or whether an additional reduction argument covers the general case.","section":"section deriving the implication between Demonet and Enomoto conjectures"}],"minor_comments":[{"comment":"The abstract would benefit from a one-sentence reminder of the statements of the Demonet and Enomoto conjectures to make the implication immediately intelligible to readers outside the immediate subfield.","section":"Abstract"},{"comment":"Notation for the smallest torsion class generated by a module (e.g., the symbol used for the closure operation) should be introduced explicitly at the first occurrence rather than relying on context.","section":"Introduction"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and valuable comments on our manuscript. We address the major comment below and have revised the text for clarity.","responses":[{"response":"We thank the referee for this observation. The equivalence between bicompact and functorially finite torsion classes is conjectural in general and is proven only for hereditary algebras and semistable torsion classes. The section derives the logical implication Demonet implies Enomoto under the assumption that the bicompact-functorially finite identification holds for arbitrary finite-dimensional algebras. The Demonet and Enomoto conjectures are general statements about brick-infinite algebras; our argument shows that the general conjecture would entail the desired implication between them. The special-case proofs provide supporting evidence for the conjecture but are not used to cover the general derivation. We have revised the manuscript to state this distinction explicitly and to clarify that no reduction argument to the special cases is claimed for the implication itself.","revision_made":"yes","referee_comment":"[section deriving the implication between Demonet and Enomoto conjectures] In the section deriving the implication between the Demonet and Enomoto conjectures, the claim that the bicompact-functorially-finite equivalence yields Demonet implies Enomoto is not fully justified for arbitrary finite-dimensional algebras, because the equivalence is established only for hereditary algebras and semistable torsion classes; the manuscript should clarify whether the two conjectures are understood to apply only in those restricted settings or whether an additional reduction argument covers the general case."}],"tokens_in":1274,"tokens_out":333,"duration_ms":30901,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The one thing to know is that Asai introduces bicompact torsion classes and proves they match functorially finite torsion classes when the algebra is hereditary or the class is semistable. This also shows that if the Demonet conjecture holds then so does the Enomoto one. The new part is the definition of bicompact as both compact and dually compact, the conjecture itself, and the two proofs. The hereditary case uses that every torsion class comes from a single module without extra relations getting in the way. The semistable case uses vanishing of Homs between stable and unstable summands. These build on standard facts about torsion classes and seem to hold up. The limitation is that nothing is shown for general finite-dimensional algebras. The properties used might not extend when there are relations, so a singly generated torsion class may not have the approximations needed to be functorially finite. The implication between conjectures therefore depends on the unproven general statement. This paper is for people deep in the representation theory of algebras, especially those tracking torsion classes and questions about brick-infinite algebras. It gives concrete progress on special cases and a new way to link two open problems. I would mention it in a reading group focused on rep theory conjectures. It should go to peer review because the special proofs are useful and the conjecture is worth testing, even with the general case open.","headline":"Asai defines bicompact torsion classes and proves they equal functorially finite ones for hereditary algebras and semistable classes, which yields Demonet implies Enomoto.","tokens_in":2292,"tokens_out":355,"would_cite":false,"duration_ms":35216,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Representation-theoretic conjecture on bicompact torsion classes; no overlap with RS cost or forcing chain","alignment":"orthogonal","rationale":"The paper's core objects (bicompact torsion classes T=T(M)∩T⊥=F(N), semistable Tθ/Tθ via Euler form on K0(proj A)R, rigid θ in g-fan, Demonet/Enomoto implications) live entirely in mod A and silting theory. RS framework derives J-cost, φ-ladders, 8-tick periodicity and D=3 from a single distinction (AbsoluteFloorClosure, Cost.FunctionalEquation, AlexanderDuality). No shared machinery, no ratio-symmetric cost, no parameter-free constants, no contradiction with any RS theorem.","tokens_in":49901,"confidence":"high","tokens_out":174,"duration_ms":10864,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Bicompact torsion classes are conjectured to coincide exactly with functorially finite torsion classes over finite-dimensional algebras.","keywords":["torsion classes","bicompact torsion classes","functorially finite","hereditary algebras","brick infinite algebras","Demonet conjecture","Enomoto conjecture"],"falsifier":"An explicit finite-dimensional algebra together with a torsion class that is bicompact but not functorially finite, or functorially finite but not bicompact.","tokens_in":2524,"feed_emoji":"","tokens_out":612,"duration_ms":21730,"temperature":0.7,"pith_summary":"The paper defines a torsion class in mod A as bicompact when it is the smallest torsion class containing a single module M and the dual condition holds in the opposite category. It conjectures that these bicompact classes are precisely the functorially finite ones. The claim is established for hereditary algebras by using their homological properties and separately for semistable torsion classes. This yields the meta-result that the Demonet conjecture on brick infiniteness implies the Enomoto conjecture.","feed_headline":"Bicompact torsion classes match functorially finite ones for hereditary algebras","feed_subtitle":"The equality also holds for semistable cases, so Demonet conjecture implies Enomoto conjecture on brick infiniteness.","key_machinery":"Bicompact torsion class: a torsion class that is the smallest one containing some module M and satisfies the dual generation condition.","core_discovery":"Bicompact torsion classes, those generated by a single module both directly and dually, are conjectured to be exactly the functorially finite torsion classes. The equality holds when A is hereditary or when the torsion class is semistable, which in turn shows that Demonet’s conjecture implies Enomoto’s conjecture on algebras with infinitely many bricks.","pith_inferences":["The conjecture, if true in full generality, would give a single-module test for functorial finiteness of torsion classes.","The link between the two brick-infinite conjectures may extend to other classes of algebras closed under derived equivalence.","Counterexamples, if they exist, are likely to appear first among non-hereditary algebras with complicated torsion lattices."],"forward_implications":["For every hereditary algebra, bicompact torsion classes are functorially finite.","Semistable torsion classes that are bicompact are functorially finite.","Demonet’s conjecture on brick-infinite algebras implies Enomoto’s conjecture.","Compactness of a torsion class implies functorial finiteness in the settings where the equality is proven."],"fun_headline_variants":["Bicompact torsion classes equal functorially finite for hereditary algebras","Bicompact functorially finite equality holds in semistable cases","Demonet conjecture implies Enomoto via bicompact torsion class equality","Bicompact torsion classes link Demonet and Enomoto conjectures on brick infiniteness"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The closure properties and Hom-vanishing conditions that make the proofs work for hereditary algebras and semistable torsion classes extend to arbitrary finite-dimensional algebras.","fun_headline_variants_meta":{"raw":{"variants":["Bicompact torsion classes equal functorially finite for hereditary algebras","Bicompact functorially finite equality holds in semistable cases","Demonet conjecture implies Enomoto via bicompact torsion class equality","Bicompact torsion classes link Demonet and Enomoto conjectures on brick infiniteness"]},"model":"grok-4.3","cost_usd":0.01471,"raw_usage":{"total_tokens":6279,"prompt_tokens":575,"num_sources_used":0,"completion_tokens":74,"cost_in_usd_ticks":147099500,"prompt_tokens_details":{"text_tokens":575,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5630,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":575,"tokens_out":74,"duration_ms":86167,"temperature":1.0,"reasoning_tokens":5630,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-10T20:09:21.269635+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit finite-dimensional algebra together with a torsion class that is bicompact but not functorially finite, or functorially finite but not bicompact.","supporting_citations":[],"review_version":1}