{"id":"775651bb-a09b-4000-bd27-e6117d3ba1f1","arxiv_id":"2506.13602","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Brick-directed algebras are exactly the algebras whose torsion-class lattice is left modular, extremal, or trim; the wall-and-chamber and Newton-polytope reformulations follow.","lead":"This paper defines a new class of torsion pairs, brick-splitting pairs, and a new class of algebras, brick-directed algebras, which are algebras whose bricks cannot be arranged in a cycle of non-zero non-isomorphisms. It proves that for brick-finite algebras, being brick-directed is exactly the same as the lattice of torsion classes being left modular, extremal, or trim.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.7 appears sound; the wall-chamber and Newton-polytope characterizations rest on an unpublished citation and should remain conditional.","rationale":"The reader's conditional verdict is appropriate, but the precise load-bearing concern is narrower than the reader's weakest_assumption suggests. The imported brick-labeling theorem (Theorem 2.4, from [DI+]) and the chain correspondence (Theorem 2.6, from [KD]) are published and appear to be applied correctly; I do not find them fragile. The genuinely fragile input is Proposition 4.14, cited to the unpublished [As3]. This proposition is essential for Theorem 4.18, Theorem 5.11, and Corollary 5.7, but it is not needed for Theorem 1.7, whose proof can be checked line by line through Proposition 3.4, Theorem 4.1, and Proposition 5.2. The paper's advertised central claim includes the wall-chamber and Newton-polytope characterizations, so the missing public proof is a real support gap. The correct verdict is conditional: the main equivalence and lattice-theoretic spine result are well supported, while the stability-based characterizations should be downgraded or marked as dependent on [As3] until the proof is public or included.","tokens_in":33359,"tokens_out":30024,"duration_ms":300198,"concrete_test":"Ask the authors to either include a self-contained proof of Proposition 4.14 in the paper or replace [As3] with a public preprint containing it. As an independent check of the statement, compute the wall-and-chamber decomposition of the rank-2 wind-wheel algebra Λ_2 from Example 4.11 (which has five bricks) using the published algorithm of [As2]/[BST], and verify for every cover in Hasse(torsΛ_2) labeled by a brick X that there exists θ with T_θ and T_θ equal to the two torsion classes of the cover and X θ-stable. If the check fails, Theorem 4.18 is false; if it passes, the citation gap remains but the statement is plausible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.7 itself is internally coherent: Proposition 3.4 identifies left modular elements with brick-splitting torsion classes using the weakly atomic structure of torsA and the standard brick labeling, and Proposition 5.2 reaches extremality through the length/brick-count argument. I do not see a gap in that chain. The load-bearing weak point is the wall-chamber route. Proposition 4.14 asserts that for every Hasse arrow T→U in torsA labeled by a brick X there is a stability parameter θ with T_θ = T, T_θ = U, and X θ-stable. This input produces the consistent sequence in Theorem 4.18 and, via Lemma 4.15(2), the indivisibility step in Lemma 5.10 that underlies Theorem 5.11 and Corollary 5.7. The paper gives no proof and cites the unpublished manuscript [As3]. Unlike Theorem 2.4 (published [DI+]) or Theorem 2.6 (published [KD]), Proposition 4.14 is a precise compatibility statement between Hasse labels and stability walls, not a standard reference. If the statement or its proof in [As3] fails, the wall-chamber and Newton-polytope characterizations of brick-directed algebras are unsupported. This does not threaten Theorem 1.7, but it means the full central claim advertised in the abstract is conditional on a nonpublic source.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces brick-splitting torsion pairs, i.e. torsion pairs in which every brick lies in the torsion or torsion-free class, and proves that they are exactly the left modular elements of the lattice tors A (Theorem 1.1, via Propositions 3.1 and 3.4). It then defines brick-directed algebras as those with no cycle of non-zero non-isomorphisms between bricks, and characterizes them: by the existence of a maximal chain of brick-splitting torsion classes (Theorem 4.1), and, in the brick-finite case, by left modularity, extremality, and trimness of tors A (Theorem 1.7), by consistency of a sequence in the real Grothendieck group (Theorem 4.18), and by the existence of an indivisible increasing path in a Newton polytope (Theorem 5.11). It also gives an explicit gluing construction producing brick-directed algebras of all representation types (Corollary 4.13).","tokens_in":33701,"tokens_out":15040,"duration_ms":140212,"significance":"If the results hold, the paper gives a satisfying representation-theoretic realization of left modular, extremal, and trim lattices, and a substantial generalization of representation-directed algebras, with explicit examples in the tame and wild worlds. The proof of the main lattice-theoretic theorem (Theorem 1.7) is coherent: it builds on published brick-labeling results [DI+], chain-of-bricks results [KD], and lattice-theoretic facts [TW], [Mu], and the extremality argument via chain length |brickA| is elegant. The paper is also strong on explicit constructions and worked examples. However, the advertised wall-and-chamber and Newton-polytope characterizations are currently conditional on an unpublished source, which is the main obstacle to accepting the full claims in the abstract.","major_comments":[{"comment":"Proposition 4.14 and Lemma 4.15 are load-bearing for the wall-chamber and Newton-polytope results: Theorem 4.18 uses Proposition 4.14 to produce the consistent sequence, and Theorem 5.11 together with Corollary 5.7 uses Lemma 4.15(2) via Lemma 5.10. Both results are cited to the unpublished manuscript [As3]. Proposition 4.14 in particular is a precise compatibility statement between Hasse labels and stability parameters, not a standard reference. As long as [As3] remains unpublished, Theorems 1.10, 1.12, and Corollary 5.7 are conditional. I do not see a gap in the proof of Theorem 1.7 itself, but the abstract advertises these characterizations as theorems, so this dependency needs to be resolved before publication, either by including proofs in an appendix or by citing a publicly available version.","section":"§4.4, Prop. 4.14 and Lemma 4.15"}],"minor_comments":[{"comment":"In the proof of Corollary 3.16 the partition is described as having no arrow from S1 to S2 and then (1−e)Ae = 0 is deduced; this reverses the orientation in Proposition 3.14, which gives no arrow from sim A∩F to sim A∩T. The argument still works after replacing e by 1−e, but the displayed direction should be corrected.","section":"§3.3, proof of Cor. 3.16"},{"comment":"In the extremality implies brick-directed direction, the citation to Theorem 2.6 is not quite the right tool: Theorem 2.6 as stated only produces chains of torsion classes from chains of bricks. The intended conclusion follows directly from Proposition 2.5, since labels along a maximal chain of length |brickA| form a chain of bricks containing all bricks.","section":"§5, proof of Prop. 5.2"},{"comment":"The symbol A is overloaded in the proof of Lemma 3.8, where it denotes both the algebra and the wide subcategory Filt(X⊕Y); using a different letter, such as W, for the latter would avoid confusion.","section":"§3.3, proof of Lemma 3.8"},{"comment":"The phrase 'For any positive integer n>1' would be more naturally rendered as 'For every integer n>1'.","section":"Cor. 1.6"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the manuscript's main lattice-theoretic theorem is in good shape, but the wall-chamber and Newton-polytope sections depend on an unpublished manuscript [As3] by one of the authors. I recommend requesting a preprint of [As3] or a proof of Proposition 4.14 and Lemma 4.15 as part of the revision; otherwise the advertised characterizations remain conditional. There is no concern about circularity: the central proof uses published results as tools."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. This is a good paper. It introduces brick-splitting torsion pairs and brick-directed algebras, and proves the expected characterization: for brick-finite A, brick-directed iff torsA is left modular iff extremal iff trim, with spine equal to the brick-splitting torsion classes. That equivalence is genuinely new, and the proof through Prop 3.4 is clean. The brick quiver material and gluing constructions are also useful, and the examples cover all representation types, which I hadn't expected.\n\nThe main lattice-theoretic chain is sound. Prop 3.4 uses the weakly atomic brick labeling from published sources, and the length/brick-count argument in Prop 5.2 checks out. I do not see a gap in Theorem 1.7.\n\nThe real soft spot is Section 4.4. Prop 4.14, which says every Hasse arrow labeled by a brick is realized by a stability parameter with T_θ = T, T_θ = U, and X θ-stable, is cited to the unpublished manuscript [As3]. That proposition is the bridge from the torsion-class chain to the consistent sequences and then to the Newton polytope characterization. If [As3] doesn't pan out, Theorems 4.18 and 5.11 and Cor 5.7 are unsupported. This is not a flaw in the algebra-lattice part, but it is a load-bearing provenance gap for advertised claims, and the paper should either prove Prop 4.14 or replace the citation with a public reference.\n\nMinor point: the reader's report calls out an orientation discrepancy in the Section 1 idempotent statement. I looked at it; it's a notational swap between e_T in Section 1 and e_F in the proof of Cor 3.16, not a mathematical error. Leave it if you want, but I wouldn't require a correction.\n\nCitations are mostly published and standard; [As3] is the only nonpublic one. No circularity.\n\nWho is this for? Anyone working on torsion classes, τ-tilting theory, or lattices associated to finite-dimensional algebras. It deserves a serious referee. I'd send it out and ask for the Prop 4.14 issue to be fixed before acceptance.","headline":"A solid, genuinely new characterization of brick-directed algebras via left modularity of torsion-class lattices; the only load-bearing gap is a wall-chamber result cited to an unpublished manuscript.","tokens_in":34179,"tokens_out":6330,"would_cite":true,"duration_ms":65400,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G10","06A07","05E10","16S90"],"pacs":[],"model":"deepseek-v4-flash","headline":"Brick-directed algebras are exactly the brick-finite algebras whose lattice of torsion classes is left modular, extremal, and trim, with the brick-splitting torsion classes forming its distributive spine.","keywords":["brick","splitting torsion pair","brick-directed algebra","left modular lattice","trim lattice","torsion classes","wall-and-chamber structure","Newton polytope"],"falsifier":"For the 3-Kronecker quiver, the paper predicts that K0(projA)R admits no consistent or weakly consistent sequence since the algebra is strictly wild; finding such a sequence would falsify Theorem 4.18. Alternatively, exhibiting a brick-finite algebra whose lattice of torsion classes is left modular but whose brick quiver contains a directed cycle would break Theorem 1.7 at its lattice-theoretic core.","tokens_in":33160,"feed_emoji":"🧱","tokens_out":6453,"duration_ms":54303,"temperature":0.7,"pith_summary":"This paper introduces brick-splitting torsion pairs, meaning torsion pairs that split the set of bricks (modules whose endomorphism ring is a division algebra) into torsion and torsion-free parts, and uses them to define brick-directed algebras, which are algebras admitting no cycle of non-zero non-invertible maps between bricks. The main theorem shows that for an algebra with only finitely many bricks, being brick-directed is equivalent to the lattice of torsion classes being left modular, being extremal, and being trim; in that case the brick-splitting torsion classes are exactly the spine of the lattice and form a distributive sublattice. Along the way the paper characterizes brick-splitting torsion classes as left modular elements of the torsion-class lattice, gives a bijection between Hom-orthogonal total orders on bricks and maximal chains of brick-splitting torsion classes, and connects brick-directedness to wall-and-chamber data and to an indivisible increasing path in the Newton polytope of the direct sum of all bricks. The results turn a purely representation-theoretic condition, absence of cycles of bricks, into lattice-theoretic and polyhedral characterizations that generalize classical representation-directed algebras while covering new tame and wild examples.","feed_headline":"Brick cycles decide when a torsion lattice is left modular","feed_subtitle":"For brick-finite algebras, no brick cycles is equivalent to extremality, trimness, and a distributive spine in torsA.","key_machinery":"The brick quiver Qb(A) is the central combinatorial object: its vertices are the bricks of A and it has an arrow X to Y whenever HomA(X,Y) is nonzero, so the absence of directed cycles in Qb(A) is exactly brick-directedness. The lattice-theoretic machinery rests on the brick-labeling theorem for Hasse(torsA): every cover relation in the lattice of torsion classes is uniquely labeled by a brick, labels along a saturated chain are Hom-orthogonal, and every torsion class is determined by the bricks it contains. That labeling, combined with the correspondence between chains of bricks and chains of torsion classes, lets the paper identify brick-splitting torsion classes with left modular elements, with the elements on maximal chains, and with vertices of the Newton polytope of the direct sum of all bricks.","core_discovery":"The central claim, Theorem 1.7, is that for a brick-finite algebra A the following are equivalent: A is brick-directed; the lattice torsA of torsion classes is left modular; torsA is extremal; and torsA is a trim lattice. Moreover, when these hold, the spine of torsA consists exactly of the brick-splitting torsion classes and forms a distributive sublattice. The paper also proves that a torsion class T is brick-splitting precisely when T is a left modular element of torsA, and that A is brick-directed precisely when its brick quiver Qb(A) is acyclic; in that case there is a bijection between total orders on the bricks satisfying Hom-orthogonality and maximal chains of brick-splitting torsion classes. For brick-finite A, brick-directedness is additionally equivalent to the existence of a consistent or weakly consistent sequence in the real Grothendieck group K0(projA)R and to the existence of an indivisible increasing path in the Newton polytope N(M) from 0 to [M], where M is the direct sum of all bricks.","pith_inferences":["If the brick-labeling structure extends to infinite semidistributive lattices, brick-directedness may serve as the right infinite analogue of trimness; the paper itself gestures at this possibility in Remark 1.13 without proving it.","The brick-quiver acyclicity criterion suggests a testable detection strategy for quiver algebras with finite brick data: enumerate bricks, build Qb(A), and check acyclicity, and the wind-wheel family in the paper is a concrete place to run such a check.","The uniqueness of bricks by their dimension vector, if it were known beyond brick-finite algebras, would interact directly with the second brick-Brauer-Thrall expectation that brick-infinite algebras over algebraically closed fields contain infinitely many bricks with a common dimension vector; the paper raises the question but does not settle it."],"forward_implications":["Brick-directed algebras form a strictly larger family than representation-directed algebras: they may be representation-infinite, tame, or wild, and for every rank n>1 the paper constructs explicit examples of all five types (representation-finite, brick-finite tame, brick-infinite tame, brick-finite wild, and brick-infinite wild).","For any brick-finite brick-directed algebra, the lattice torsA has a maximal chain of length |brickA|, and its spine is a distributive sublattice consisting exactly of the brick-splitting torsion classes.","In a brick-finite brick-directed algebra, every brick is uniquely determined by its dimension vector, and each module variety mod(A,d) contains at most one brick component.","Brick-directedness is inherited by quotient algebras A/J, corner algebras eAe, and τ-reductions, so extremality and left modularity of torsA pass to the corresponding torsion-class lattices.","Strictly wild algebras are never brick-directed, and among path algebras exactly the Dynkin quivers and the Kronecker quiver are brick-directed."],"supporting_citations":[{"why":"Supplies the brick-labeling of Hasse(torsA) and the bijection between bricks, join-irreducibles, and meet-irreducibles, on which Propositions 3.4 and 5.2 rest.","marker":"[DI+, Theorem 3.4]"},{"why":"Demonet's map from chains of bricks to chains of torsion classes, used in Theorem 4.1 and in the extremality direction of Theorem 1.7.","marker":"[KD, Appendix]"},{"why":"Gives the equivalence of left modularity, extremality, and trimness for finite semidistributive lattices that Theorem 1.7 realizes in torsA.","marker":"[TW, Theorem 1.4]"},{"why":"Independent proof of the same semidistributive-lattice equivalence, cited as the lattice-theoretic background for the classification.","marker":"[Mu, Theorem 3.2]"},{"why":"Unpublished manuscript invoked as the proof of Proposition 4.14, the existence of stability parameters realizing each arrow of Hasse(torsA), on which the wall-chamber characterizations depend.","marker":"[As3]"},{"why":"Provides the bijection between maximal chains in torsA and increasing paths in the Newton polytope N(M), used in Theorem 5.11.","marker":"[AH+, Subsection 5.3]"},{"why":"Supplies the same Newton-polytope correspondence from another reference, used for Proposition 5.9.","marker":"[Fe2, Subsection 9.2]"},{"why":"Gives the gluing construction used to produce brick-directed algebras of all five representation types in Corollary 4.13.","marker":"[MP1, Section 7]"},{"why":"Classical characterization of splitting torsion pairs that motivates and contrasts with the brick-splitting notion.","marker":"[ASS, VI. Prop. 1.7]"},{"why":"Shows that τ-reduction realizes the module category as a wide subcategory, used to prove brick-directedness is preserved under τ-reduction.","marker":"[Ja, Theorem 3.8]"}],"fun_headline_variants":["No brick cycles: left modular torsion lattice","Brick quiver acyclic iff left modular torsA","Brick-directed iff left modular for brick-finite","Brick-splitting pairs are left modular elements","Left modularity from brick-splitting torsion pairs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the brick-labeling structure of the torsion-class lattice: cover arrows carry unique brick labels, labels along saturated chains are Hom-orthogonal, and each torsion class is determined by the bricks it contains; for the wall-and-chamber results, a further premise is a proof currently cited to an unpublished manuscript rather than supplied in this paper.","fun_headline_variants_meta":{"raw":{"variants":["No brick cycles: left modular torsion lattice","Brick quiver acyclic iff left modular torsA","Brick-directed iff left modular for brick-finite","Brick-splitting pairs are left modular elements","Left modularity from brick-splitting torsion pairs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00161,"raw_usage":{"total_tokens":6427,"prompt_tokens":978,"completion_tokens":5449,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":5374}},"tokens_in":594,"tokens_out":5449,"duration_ms":39485,"temperature":1.0,"reasoning_tokens":5374,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T20:00:04.194424+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the 3-Kronecker quiver, the paper predicts that K0(projA)R admits no consistent or weakly consistent sequence since the algebra is strictly wild; finding such a sequence would falsify Theorem 4.18. Alternatively, exhibiting a brick-finite algebra whose lattice of torsion classes is left modular but whose brick quiver contains a directed cycle would break Theorem 1.7 at its lattice-theoretic core.","supporting_citations":[],"review_version":2}