{"id":"3c1d2033-e09a-41b2-bcd6-e69b9b96cb38","arxiv_id":"2608.04233","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every Artin algebra admits a minimal consistent scattering diagram built from bounded-length module categories and their picture groups, matching Bridgeland's stability diagram over C.","lead":"This paper proves that every Artin algebra has a minimal consistent scattering diagram, a geometric object encoding stability conditions on its modules. The construction uses bounded-length module categories as finite approximations and recovers Bridgeland's stability scattering diagrams over C.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.11 defines g_gamma via an inverse limit but never proves it equals the ordered wall-crossing product required by Definition 7.5.","rationale":"The reader identified the inverse-limit step in Theorem 7.11 as the weakest assumption, and I agree that this is the most load-bearing concern. The paper's construction of g_gamma as a compatible tuple is natural and almost certainly correct, but the proof does not connect this tuple to the ordered wall-crossing product that defines a scattering diagram. For a D(A)-generic path crossing infinitely many walls, the ordered product is only defined coordinatewise in the inverse limit after proving that for each ell only finitely many factors contribute; this is true because D_ell(A) has finitely many walls, but the paper never says so. The missing lemma would be a short verification using Lemma 7.8, and the Kronecker example provides a concrete setting where infinitely many walls accumulate. Since the gap is real but clearly addressable, the appropriate verdict remains conditional: the main theorem is plausible and the finite-level machinery appears sound, but the proof as written does not fully establish that the inverse limit is a scattering diagram in the sense of Section 7.1. The paper should either prove the ordered-product identity or explicitly redefine g_gamma as the coordinatewise inverse limit and state that this is the intended notion of well-definedness for infinite-wall scattering diagrams. I also note that Theorem 8.3 is compressed into a few lines, but the inverse-limit issue is more fundamental because it affects the definition of the object being compared.","tokens_in":45679,"tokens_out":15507,"duration_ms":147532,"concrete_test":"In Theorem 7.11, add a verification that for every D(A)-generic path gamma the ordered product over the (possibly infinite) set S_gamma of crossed walls converges coordinatewise in G(A)=lim G_ell(A) to the tuple (g^ell_gamma). Concretely, for each ell, form the finite product over the walls d in S_gamma whose projection Phi_ell(d) is non-trivial, ordered by the path parameter, and prove it equals g^ell_gamma using Lemma 7.8. A good test case is the Kronecker algebra over C: a path crossing the wall of S(1) at a nonzero x-coordinate with transverse slope crosses infinitely many walls (1,n) accumulating at that crossing; verify that the coordinatewise products are finite and equal the tuple. If the identity fails for such a path, the object constructed in Theorem 7.11 is not a scattering diagram under Definition 7.5.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence theorem, Theorem 7.11, defines the element g_gamma for a D(A)-generic path as the compatible inverse-limit tuple (g^ell_gamma)_ell, where g^ell_gamma = rho([t0,(T_gamma(0))_ell]^{-1}[t0,(T_gamma(1))_ell]) in G_ell(A). However, Definition 7.5 defines a scattering diagram as well defined only when g_gamma is the ordered product over the walls crossed, g_gamma = prod_{d in S_gamma} Phi(d)^{epsilon_d}, and the introduction states that g_gamma should be 'calculated as the ordered product' of the wall labels. The proof never shows that the inverse-limit tuple is this ordered product. For each fixed ell, Lemma 7.8 does identify the finite product in D_ell(A) with g^ell_gamma, so the coordinatewise ordered product in the inverse limit (where only finitely many factors survive modulo ell) would coincide with the tuple. But this identification is exactly the step that makes the inverse-limit object a scattering diagram in the usual sense, and it is neither stated nor proved. Because G(A) is an inverse limit of arbitrary groups with no pro-nilpotent structure and no convergence criterion for infinite products is given, the existence of an ordered product in G(A) is not automatic. Without this step, consistency of the tuple is not the same as consistency of a scattering diagram, and the comparison with Bridgeland's stability scattering diagram in Theorem 8.3 lacks its target notion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for every Artin algebra A, a minimal consistent scattering diagram whose support is the wall-and-chamber structure of A and whose group is an inverse limit of picture groups associated with the bounded-length categories (mod A)_ell. The finite-level diagrams are built from lattices of torsion classes, a finite wall-and-chamber structure D_ell(A), and a categorical presentation of the picture group G_ell(A); consistency is proved via the groupoid of intervals in the lattice of numerical torsion classes. Passing to the inverse limit over ell, the author claims a scattering diagram for mod A and, for A = CQ/I, an isomorphism with Bridgeland's stability scattering diagram. The finite-level parts are developed in detail; the main gap is the passage from the compatible inverse-limit tuples in Theorem 7.11 to the ordered wall-crossing products required by the paper's own Definition 7.5.","tokens_in":45915,"tokens_out":8121,"duration_ms":83620,"significance":"If the construction is fully justified, the paper settles a natural open problem: it gives every Artin algebra a canonical wall-and-chamber scattering diagram, without motivic Hall algebra techniques, and recovers Bridgeland's stability scattering diagram as a special case. The manuscript contains several independently useful contributions: the lattice-theoretic analysis of tors_ell(A) (Theorem 3.1, Proposition 3.10, Theorem 3.12), the characterization of ell-TF-equivalence in Theorem 4.11, the categorical presentation of the picture group in Theorem 6.21, and the finite-level consistency theorem (Theorem 7.9). These parts are largely rigorous and provide a transparent categorical framework. The inverse-limit step and the comparison with Bridgeland's diagram, however, are not yet established at the level of detail required for the central claims.","major_comments":[{"comment":"The proof defines g_gamma as the compatible inverse-limit tuple (g^ell_gamma)_{ell in N}, but Definition 7.5 defines g_gamma as the ordered product of the wall labels along gamma, and the introduction (p. 2) states that g_gamma should be 'calculated as the ordered product' of the elements associated to the walls crossed. The proof never shows that the inverse-limit tuple is this ordered product in G(A). For each fixed ell, Lemma 7.8 identifies the finite-level product with g^ell_gamma, so the tuple is a compatible family of finite ordered products; however, G(A) is an inverse limit of arbitrary groups with no pro-nilpotent structure and no convergence criterion is given for infinite products. Lemma 7.3 only shows that each individual wall is crossed finitely many times, not that a D(A)-generic path crosses only finitely many walls in total. Remark 7.12 explicitly notes that the green-path argument uses Proposition 5.8, while arbitrary generic paths require the missing identification. Thus the well-definedness of the infinite-level scattering diagram in the sense of Definition 7.5 is not established. The paper should either prove that the tuple satisfies the ordered-product property with respect to a defined notion of infinite product in G(A), or explicitly amend Definition 7.5 to define well-definedness via inverse-limit compatibility; the latter would still require a comparison with the standard notion used in Theorem 8.3.","section":"§7.3, Theorem 7.11"},{"comment":"The claimed isomorphism between the torsion scattering diagram and Bridgeland's stability scattering diagram is underproved. The map I is defined only on the images Im(Phi_A) and Im(Phi-hat_A), not on the ambient groups G(A) and H-hat(A); to speak of an isomorphism of scattering diagrams one must show these images generate the respective groups and that I extends to a group isomorphism compatible with all wall labels and with the inverse-limit structures. The proof merely cites [Bri17, Lemma 6.6], Lemma 7.8 and Proposition 5.4, but it does not verify compatibility with the infinite products/inverse limits that are essential at the A-level. In particular, Bridgeland's group is pro-unipotent with a convergence mechanism, while G(A) is an inverse limit of picture groups without an analogous mechanism; the comparison therefore needs a precise statement of how the two inverse systems are identified. Corollary 8.5 inherits this issue. This is load-bearing for the second half of Theorem 1.1.","section":"§8, Theorem 8.3"}],"minor_comments":[{"comment":"In the first paragraph, 'Krull–Schimidt' should be 'Krull–Schmidt'.","section":"§2.1"},{"comment":"In the proof, the notation 'rT' is used for the torsion class associated with the uniquely determined torsion-free class F(F_ell) before it is introduced; please define this notation explicitly.","section":"§3.2, Proposition 3.6"},{"comment":"The relation is written as 'g_d2 = g_d1 g_d1', where the two occurrences of d1 denote cones of different codimension; this is confusing. Use distinct symbols, for example g_{d_2} = g_{d_1} g_{d^1} with d_1 in D^0 and d^1 in D^1.","section":"§6.2, Definition 6.13"},{"comment":"In the statement and proof, 'Phi^l_A' should be 'Phi^ell_A' for consistency with the notation D_ell(A) and G_ell(A).","section":"§7.2, Theorem 7.9"},{"comment":"The walls in Figure 5(c)–(e) are labelled by modules such as 1/22 and 11/22, but the labels are not explained in the caption; please add a sentence explaining the notation or refer explicitly to Example 4.22.","section":"§4.2, Figure 5"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper constructs a \"torsion scattering diagram\" for every Artin algebra by first building finite diagrams on the categories of modules of length at most ℓ and then taking an inverse limit. This is a genuinely new approach, and the finite-level part looks like solid mathematics. The author develops a lattice of torsion classes for these bounded-length categories, shows it is completely semidistributive with brick labels, builds wall-and-chamber structures and picture groups, and proves consistency for the finite diagrams. The categorical description of the picture groups via subcategories and the ∗-product is a nice contribution in its own right. I have no real complaints about Sections 3–6; the arguments are detailed and mostly convincing.\n\nThe main result, Theorem 7.11, is where I part ways. The paper defines g_γ for a D(A)-generic path as the compatible family (g^ℓ_γ) in the inverse limit group G(A), where each g^ℓ_γ comes from the finite diagram. But the definition of a scattering diagram in Section 7.1 requires g_γ to be the ordered product of the wall labels along γ. The paper never proves the inverse-limit tuple is that ordered product. This is not a cosmetic issue: D(A) can have infinitely many walls, and G(A) is an arbitrary inverse limit of groups, so infinite products do not automatically exist. The stress-test note has this exactly right. The gap is probably fixable — for each fixed ℓ only finitely many walls have non-trivial label in G_ℓ(A), so one can define the ordered product coordinatewise and then show the tuple is that product — but as written the argument is missing. Without this step, consistency of the tuple is not consistency of a scattering diagram in the sense the paper itself advertises.\n\nThe comparison with Bridgeland's stability scattering diagram (Theorem 8.3) is also too compressed. The map I is only defined on the images of the wall labels; no isomorphism between the ambient groups G(A) and the pro-unipotent group from the motivic Hall algebra is constructed. That may be enough for a weak notion of \"isomorphic diagram\", but it needs to be stated and argued more carefully.\n\nThere are minor issues: Proposition 3.7 is left to the reader, and some \"straightforward\" steps in Section 5 deserve more detail. These are not serious.\n\nBottom line: this is a promising paper with a real new construction, but the infinite-level theorem has a load-bearing gap. I would recommend sending it to a serious referee and asking for a revision that either proves the inverse-limit tuple equals the ordered product, or changes the definition of consistency for inverse-limit diagrams. The target audience is representation theorists working on stability conditions, τ-tilting theory, and scattering diagrams; they will find the finite-level machinery useful regardless.","headline":"Solid finite-level construction; the inverse-limit step in Theorem 7.11 needs a real argument before the infinite diagram is justified.","tokens_in":46471,"tokens_out":3592,"would_cite":true,"duration_ms":33813,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G10","16G20","16G70","18E40","13F60"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any Artin algebra, this paper constructs a minimal consistent scattering diagram from the lattice of torsion classes, and shows that for path algebras over \\mathbb{C} it agrees with the established stability scattering diagram.","keywords":["Artin algebras","scattering diagrams","torsion classes","wall-and-chamber structures","stability conditions","picture groups","inverse limits","bounded-length modules"],"falsifier":"For an algebra with infinitely many walls, such as the Kronecker algebra over $\\mathbb{C}$, take a generic path $\\gamma$ that crosses infinitely many walls. For each $\\ell$, compare the element $g^\\ell_\\gamma$ given by the compatible family with the image in $G_\\ell(A)$ of the finite ordered product over the walls of $\\mathcal{D}_\\ell(A)$ crossed by $\\gamma$; a single level where they differ, or a path where the finite products have no limit, would falsify Theorem 7.11.","tokens_in":45417,"feed_emoji":"🧱","tokens_out":9044,"duration_ms":75822,"temperature":0.7,"pith_summary":"This paper proves that every Artin algebra—a finite module category over an artinian ring—has a minimal consistent scattering diagram, a wall-and-chamber picture that records how stability conditions change as a path crosses walls. This matters because scattering diagrams encode representation-theoretic data such as torsion classes and $\\tau$-tilting modules, and until now they were known only for finite-dimensional algebras over the complex numbers. The proof cuts the module category into layers of modules of length at most $\\ell$, builds a finite scattering diagram for each layer, and assembles them by inverse limit. For algebras of the form $\\mathbb{C}Q/I$, the newly built diagram is isomorphic to the already-known stability scattering diagram.","feed_headline":"Every Artin algebra carries a minimal scattering diagram","feed_subtitle":"An inverse-limit construction over bounded-length modules extends wall-crossing diagrams beyond complex path algebras.","key_machinery":"The argument is carried by the bounded-length truncation $(\\operatorname{mod} A)_\\ell$, the full subcategory of modules of length at most $\\ell$. Each truncation has only finitely many walls, so every generic path crosses finitely many of them; its torsion classes $T_\\ell = T \\cap (\\operatorname{mod} A)_\\ell$ form a complete semidistributive lattice with a brick labelling, and its $\\ell$-TF-equivalence classes form the finite cone complex $\\mathcal{D}_\\ell(A)$. To each $\\mathcal{D}_\\ell(A)$ the paper attaches a picture group $G_\\ell(A)$, generated by semistable subcategories under the extension product $\\ast$, so that the wall label $\\Phi_\\ell(d) = (\\operatorname{mod}^{\\mathrm{ss}}_d A)_\\ell$ makes $(\\mathcal{D}_\\ell(A), \\Phi_\\ell)$ a minimal consistent scattering diagram. A compatible family of path elements $g^\\ell_\\gamma$ then defines $g_\\gamma$ in the inverse limit $G(A) = \\varprojlim G_\\ell(A)$.","core_discovery":"The central claim is that the data of any Artin algebra $A$ determine a minimal consistent scattering diagram $(\\mathcal{D}(A), \\Phi \\colon \\mathcal{D}^1(A) \\to G(A))$. Here $\\mathcal{D}(A)$ is the cone complex formed by TF-equivalence classes of stability conditions on $\\operatorname{mod} A$, $G(A)$ is the inverse limit of picture groups $G_\\ell(A)$ attached to the subcategories of modules of length at most $\\ell$, and the wall label $\\Phi(d)$ is the semistable subcategory $\\operatorname{mod}^{\\mathrm{ss}}_d A$. The diagram is minimal because no wall receives the identity label, and consistent because every generic path $\\gamma$ has an element $g_\\gamma$ that depends only on its endpoints. When $A = \\mathbb{C}Q/I$ with $Q$ a finite quiver and $I$ an admissible ideal, this torsion scattering diagram is isomorphic to the stability scattering diagram previously constructed for such algebras.","pith_inferences":["The inverse-limit definition of $g_\\gamma$ as a compatible family may not literally equal the ordered product of infinitely many wall labels in $G(A)$; if that equality fails, the group would need an ordering or completion to match the classical scattering-diagram convention. This can be checked on Kronecker-type examples with infinitely many walls.","The same length-truncation and inverse-limit recipe should work for any abelian length category with finitely many simple objects, not only module categories of Artin algebras.","If the paper's conjectures hold—that each $\\mathcal{D}_\\ell(A)$ is connected and that every $T_\\ell$ is determined by its semibricks—the finite diagrams become algorithmically tractable, making $G(A)$ and its wall-crossing products computable in concrete examples.","The isomorphism with the stability scattering diagram suggests that $G(A)$ is a purely combinatorial model for the pro-unipotent group built from Hall algebras, so homological information about $G(A)$ may carry stability-space information."],"forward_implications":["Every Artin algebra, even one with infinitely many walls, now has a wall-crossing product for every generic path in its stability space, assembled from finite length-level data.","For $\\tau$-tilting finite algebras the construction stabilizes: $\\mathcal{D}(A) = \\mathcal{D}_\\ell(A)$ and $G(A) = G_\\ell(A)$ for some $\\ell$, recovering the finite picture-group scattering diagram.","For finite-dimensional $\\mathbb{C}$-algebras of the form $\\mathbb{C}Q/I$, the torsion scattering diagram is isomorphic to the stability scattering diagram, so the classical motivic Hall algebra construction is not needed to obtain it.","The wall-and-chamber structure $\\mathcal{D}(A)$ is recovered as an inverse limit of finite cone complexes $\\mathcal{D}_\\ell(A)$, giving finite approximations to the stability space that can be computed level by level.","Minimality is built in: no wall is labelled by the identity, so every wall contributes an actual crossing effect to the scattering process."],"supporting_citations":[{"why":"defines the stability scattering diagram for finite-dimensional C-algebras and proves its consistency, the object compared in Theorem 8.3.","marker":"[Bri17]"},{"why":"introduces scattering diagrams, consistency, and the wall-crossing product the paper adapts.","marker":"[KS06]"},{"why":"provides the wall-and-chamber structure of an algebra and its relation to support tau-tilting modules, the basis of D(A).","marker":"[BST19]"},{"why":"establishes TF-equivalence classes as open convex cones and wall-crossing by bricks, generalized here to length-bounded modules.","marker":"[Asa21]"},{"why":"supplies the partitioned-fan construction of tau-cluster morphism categories and picture groups used for D_l(A) and G_l(A).","marker":"[Kai25]"},{"why":"characterizes tau-tilting finiteness by bounded brick lengths, used to show the construction stabilizes in that case.","marker":"[ST22]"},{"why":"shows each stability condition induces two torsion pairs, the categorical input for the groupoid consistency proof.","marker":"[BKT14]"},{"why":"provides the motivic Hall algebra structure, including invertibility of semistable classes, that underlies the comparison in Section 8.","marker":"[Joy07]"}],"fun_headline_variants":["Every Artin algebra gets a minimal scattering diagram","Inverse limits yield scattering diagrams for all Artin algebras","Torsion scattering diagrams: now universal for Artin algebras","Minimal consistent scattering diagrams for every Artin algebra","From bounded-length modules to scattering diagrams on all Artin algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction depends on identifying the inverse-limit element assigned to a generic path with the ordered product of the wall labels it crosses; if that identification fails, the infinite object is not a scattering diagram in the usual sense.","fun_headline_variants_meta":{"raw":{"variants":["Every Artin algebra gets a minimal scattering diagram","Inverse limits yield scattering diagrams for all Artin algebras","Torsion scattering diagrams: now universal for Artin algebras","Minimal consistent scattering diagrams for every Artin algebra","From bounded-length modules to scattering diagrams on all Artin algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000155,"raw_usage":{"total_tokens":1195,"prompt_tokens":907,"completion_tokens":288,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":208}},"tokens_in":523,"tokens_out":288,"duration_ms":3057,"temperature":1.0,"reasoning_tokens":208,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:41:24.325526+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For an algebra with infinitely many walls, such as the Kronecker algebra over $\\mathbb{C}$, take a generic path $\\gamma$ that crosses infinitely many walls. For each $\\ell$, compare the element $g^\\ell_\\gamma$ given by the compatible family with the image in $G_\\ell(A)$ of the finite ordered product over the walls of $\\mathcal{D}_\\ell(A)$ crossed by $\\gamma$; a single level where they differ, or a path where the finite products have no limit, would falsify Theorem 7.11.","supporting_citations":[],"review_version":1}