{"id":"9bb001c9-0732-4e2d-88f8-b08bac012213","arxiv_id":"2507.02218","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Colored admissible tagged edges in a twice-punctured disk model the non-tau-rigid modules of type \\widetilde{D}_n and satisfy an intersection-dimension formula.","lead":"This paper constructs a geometric model for the non-tau-rigid modules of the affine Dynkin quiver type \\widetilde{D}_n, representing each module by colored curves in a twice-punctured disk. If correct, it gives a combinatorial way to compute extension dimensions by counting curve intersections, but two load-bearing steps in the proof are sketched rather than fully demonstrated.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The quiver isomorphism φ in Definition 6.1 is asserted rather than proven; the matching condition dim[M] = Int(·) is not shown to be injective or to propagate uniquely, so Theorem A's module-to-curve correspondence is unestablished.","rationale":"The paper's project is sound and potentially valuable: extending geometric models to the regular modules of type ~D_n is a natural completion of existing surface models. The intersection-dimension formula is the payoff, but it only has content if the edge-to-module correspondence is a genuine bijection. The reader identified Definition 6.1 as the weakest assumption, and I agree: this is exactly where an unproven assertion ('it should be clear') carries the whole theorem. The secondary issue in Lemma 7.2 (same-tube intersection by visual inspection) is also real but can be repaired by a direct calculation inside a uniserial tube; the φ problem is more fundamental because, without φ, none of the lemmas can even state their module labels. A concrete small-case census with QPA would settle the issue: it would either exhibit a collision or missing edge, or provide strong evidence that the recursive matching is coherent. The verdict should remain conditional; the concern does not warrant rejection, because the construction is plausible and the gap is fillable, but the theorem should not be accepted as fully proven until the bijection is established. No independent support (machine-checked proofs, reproducible code) is present to outweigh the missing argument.","tokens_in":16985,"tokens_out":11377,"duration_ms":137748,"concrete_test":"Take the smallest case n=4 (twice-punctured disk with two boundary marked points). Enumerate all indecomposable kQ_T-modules with QPA and all endpoint-relative homotopy classes of colored admissible tagged edges. Compute the vector (Int((γ,κ,λ), i))_{i∈T} for every edge and compare with dimension vectors of modules. (a) Verify whether the mouth assignment of Definition 6.1 is injective: if two rank-1 tubes with distinct colors contain modules with the same dimension vector, the vector condition alone does not determine λ, exposing the ambiguity. (b) Run the full recursive assignment and check that it yields a bijection on vertices and arrows, with no module assigned zero edges and no edge assigned two modules. Any collision or missing vertex falsifies the claimed quiver isomorphism and hence Theorem A.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on φ: Γ(mod kQ_T) → Γ(E(T)) being a well-defined quiver isomorphism. Definition 6.1 defines φ on projective, injective, and mouth vertices by requiring dim[M] = (Int((γ,κ,λ), i))_{i∈T}, then extends recursively: if there is an arrow [M]→[N] in the module quiver, an arrow φ([M])→(γ,κ,λ) in Γ(E(T)), and dim[N] equals the intersection vector, set φ([N]) = (γ,κ,λ). This is not a proof of well-definedness. The text says only 'It should be clear from the constructions ...' and asserts uniqueness of components. Two concrete gaps: (1) Existence: it is not shown that for every module [N] there is an edge (γ,κ,λ) with dim[N] = (Int(...)) and the required arrow from the already-mapped predecessor. (2) Uniqueness: the intersection vector is independent of the coloring λ because triangulation edges all have color −∞ (Definition 4.7); hence infinitely many colorings of the same underlying tagged edge share one vector. The recursion carries λ along geometric arrows, but no argument shows that different paths from the projective/mouth sources cannot assign two different edges to the same [N]. If φ is not bijective, the 'corresponding modules' in Lemmas 7.1 and 7.2 are not well-defined and Theorem A does not attach to the module category. Lemma 7.2's same-tube case is left to 'visual inspection', but the load-bearing gap is the definition of φ itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a geometric model for the indecomposable modules, in particular the non-τ-rigid regular modules, over path algebras of acyclic quivers of type \\widetilde{D}_n. The model consists of colored admissible tagged edges in a twice-punctured disk with (n−2) boundary marked points, together with a quiver Γ(E(T)) of such edges. The central result, Theorem A (Corollary 7.3), claims an intersection-dimension formula: for two colored admissible edges (γ1,κ1,λ1) and (γ2,κ2,λ2), the intersection number Int((γ1,κ1,λ1),(γ2,κ2,λ2)) equals dim_k Ext^1(M1,M2)+dim_k Ext^1(M2,M1), where M_i are the modules corresponding under a quiver isomorphism φ: Γ(mod kQ_T) → Γ(E(T)) defined in Section 6. The proof of Theorem A is split into Lemma 7.1 (preprojective versus arbitrary modules) and Lemma 7.2 (regular-regular modules), with the remaining cases handled by duality and known vanishing statements.","tokens_in":17326,"tokens_out":4858,"duration_ms":59056,"significance":"If established, the model would fill a genuine gap in the geometric modeling of module categories over Euclidean quivers: previous work covered types A_n, D_n, \\widetilde{A}_n, and the τ-rigid modules of type \\widetilde{D}_n, but not the non-τ-rigid regular modules. The idea of adding a color λ to admissible edges in order to separate infinitely many stable tubes is natural and is the right kind of move for the problem. However, the main theorem is not proven as written: the quiver isomorphism φ of Definition 6.1 is asserted rather than proved, and the regular-regular case in Lemma 7.2 relies on 'visual inspection'. Both points are load-bearing for the claimed correspondence between geometry and homological algebra, so the paper needs substantial revision before the central claim can be accepted.","major_comments":[{"comment":"The quiver isomorphism φ is load-bearing but is not proved. The recursive definition assumes that for each arrow [M]→[N] there is a unique admissible edge (γ,κ,λ) such that dim[N] equals (Int((γ,κ,λ),i))_{i∈T} and such that the required arrow from φ([M]) exists. No argument is given for existence, uniqueness, or bijectivity; the text says only 'It should be clear from the constructions' and then asserts that φ 'must define an isomorphism of quivers'. The injectivity concern is concrete: by Definition 4.7, all triangulation edges have color −∞ and intersections with them are independent of the color λ, so the condition dim[M]=(Int(...)) cannot by itself distinguish among infinitely many colorings of the same underlying tagged edge. The recursion carries λ along geometric arrows, but no proof rules out different paths assigning different colored edges to the same module [M]. Since Lemmas 7.1 and 7.2 use φ^{-1} and Theorem A is stated in terms of 'corresponding modules', this gap undermines the central claim.","section":"Definition 6.1"},{"comment":"The same-tube case is not proved. After correctly noting that distinct colors give zero on both sides, the proof says that since modules in a stable tube are uniserial, dim_k Ext^1 can be calculated combinatorially, and then 'by construction (and visual inspection)' the intersection number between two admissible edges follows exactly the same pattern. This is not a proof of the claimed equality. What is needed is an explicit verification that the intersection numbers of admissible edges in each of the three types of regular tubes (rank 2, rank n−2, and rank 1) reproduce the extension dimensions dictated by the uniserial composition series; such a verification should include the exceptional tubes T_0, T_1, T_∞ and the homogeneous tubes. As written, the regular-regular case of Theorem A is left to inspection rather than demonstrated.","section":"Lemma 7.2"},{"comment":"The proof of the preprojective case has an unhandled subcase. The argument applies τ^{n+1} to both modules and asserts that if τ^{n+1}M2≠0 then the chain of equalities follows. If M2 is preinjective, τ^{n+1}M2 may be 0, and the proof does not explain why the equality Int(ρ^{n+1}(γ1),ρ^{n+1}(γ2)) = dim_k Hom(P(j),τ^{n+1}M2) is still valid when the right-hand side is interpreted with τ^{n+1}M2=0. In addition, the claim that 'by our assumption that M1 is to the left of M2, Ext^1(M1,M2)=0' needs a justification in terms of the Auslander-Reiten structure of the preprojective component; this is plausible but not automatic for arbitrary preprojective pairs. These steps are part of the proof of Theorem A and should be completed.","section":"Lemma 7.1"}],"minor_comments":[{"comment":"In the final line of the proof, 'the last inequality follows from the Auslander-Reiten formulas' should read 'the last equality follows from the Auslander-Reiten formulas'.","section":"Lemma 7.1"},{"comment":"In case (6), the condition 'ρ^1(γ)=γ' is better written as 'ρ(γ)=γ' for consistency with the notation introduced in Definition 4.14.","section":"Definition 5.4"},{"comment":"The sentence 'ϑγϑ is homotopic to γ' mixes the left and right poliwhirl notations without a definition of their composition; please clarify whether this is intended to be (ϑγ)ϑ or γϑ and spell out the homotopy.","section":"Remark 4.13"},{"comment":"Figure 1 is a schematic diagram of the Auslander-Reiten quiver but is not labeled with the components P(T), Q(T), and R(T); adding such labels would help the reader connect the figure to Definitions 3.7 and 5.2.","section":"Figure 1"},{"comment":"The citation 'Proposition 2.10 of [9]' should be checked against the published version of Fomin-Shapiro-Thurston, since the numbering of propositions differs between the preprint and the journal version.","section":"Theorem 5.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a coherent and quite plausible plan, and the two gaps identified above seem fixable in principle: one would add an inductive proof of well-definedness and bijectivity of φ along the Auslander-Reiten quiver, and a case-by-case verification for Lemma 7.2 using the uniserial structure of stable tubes. I therefore recommend major revision rather than rejection. I would also encourage the author to state explicitly how φ is defined on the exceptional tubes and to provide at least one nontrivial worked regular-regular example beyond the current Example 7.4."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuinely new geometric model for the non-tau-rigid modules of type ~D_n, and the color decoration idea is the right fix for separating the infinitely many homogeneous tubes. The paper is worth taking seriously, but the proof of the central quiver isomorphism is incomplete, and one lemma leans on visual inspection.\n\nWhat is actually new: previous geometric models covered A_n, D_n, ~A_n, skew-gentle algebras, and the tau-rigid modules of ~D_n. This paper adds the regular (non-tau-rigid) modules of ~D_n by coloring admissible tagged edges with lambda in P^1 union {-infinity} and declaring intersections between different non-trivial colors to be zero. That is a natural and plausible way to encode the fact that Ext vanishes between distinct homogeneous tubes, and the main theorem (intersection number equals summed Ext^1-dimensions) is exactly what the AR theory predicts. The paper is clearly written, the elementary moves and mesh relations are laid out carefully, and the running example is helpful.\n\nWhere it gets soft. Definition 6.1 asserts the quiver isomorphism phi without proof. The phrase 'it should be clear' is doing real work: the recursion from projective/mouth vertices to all vertices needs a well-definedness argument. In particular, intersection vectors cannot see the color lambda, since every triangulation edge has color -infinity, so the recursion has to carry the color through arrows; nothing in the text shows that two different paths cannot assign different colored edges to the same module. If phi is not bijective, then Lemmas 7.1 and 7.2 are not statements about the module category, and Theorem A loses its content. This is the load-bearing gap. The same-tube case in Lemma 7.2 is also justified by 'visual inspection' rather than a calculation; that is a smaller but real gap. I do not think these are fatal - the construction is very likely correct and the gaps look fixable - but they are not cosmetic.\n\nBottom line: this is a solid contribution with an unproven central claim. The target audience is representation theorists working on geometric models for Euclidean quivers and cluster theory adjacent people. It deserves a serious referee; the referee should ask for a complete proof of Definition 6.1 and a real proof of Lemma 7.2.","headline":"A plausible and genuinely new geometric model for non-tau-rigid modules of type ~D_n, but the central quiver isomorphism is asserted, not proven.","tokens_in":17880,"tokens_out":2493,"would_cite":true,"duration_ms":29268,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G70","13F60","16G20","05E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For affine type D-tilde-n, every indecomposable module is represented by a colored curve in a twice-punctured disk, and the intersection number of two curves equals the two-way dimension of the extension spaces between the modules.","keywords":["geometric model","non-tau-rigid modules","affine type \\widetilde{D}_n","intersection-dimension formula","colored admissible tagged edges","stable tubes","Auslander-Reiten quiver","twice-punctured disk"],"falsifier":"Fix an acyclic triangulation $T$ of the twice-punctured disk with $(n-2)$ boundary marked points. For each indecomposable module $M$ in a stable tube, compute the vector $(\\operatorname{Int}((\\gamma,\\kappa,\\lambda), i))_{i\\in T}$ for the curve the paper assigns to $M$ and compare it with $\\dim M$. If any module's dimension vector is not realized by an admissible edge, or if two non-isomorphic modules are assigned the same curve, then the quiver isomorphism $\\varphi$ fails and the intersection-dimension formula of Theorem A does not hold for those modules.","tokens_in":2186,"feed_emoji":"🌀","tokens_out":2116,"duration_ms":133648,"temperature":0.7,"pith_summary":"This paper claims that the module category of an acyclic path algebra of affine type $\\widetilde D_n$ can be drawn on a twice-punctured disk with $(n-2)$ marked boundary points. Previous geometric models covered the rigid modules and the types $A_n$, $D_n$, and $\\widetilde A_n$, but left out the regular modules, which are exactly the non-$\\tau$-rigid modules that tilting theory tends to ignore. The new ingredient is a color $\\lambda\\in \\mathbb P^1\\cup\\{-\\infty\\}$ on each admissible tagged edge, one color per stable tube, so that curves living in different tubes do not intersect. The payoff is the intersection-dimension formula: for any two modules $M_1,M_2$ represented by colored edges, the geometric intersection number of the two curves equals $\\dim_k \\mathrm{Ext}^1(M_1,M_2)+\\dim_k \\mathrm{Ext}^1(M_2,M_1)$, so extension dimensions become a counting problem.","feed_headline":"Curves in a twice-punctured disk count extension dimensions","feed_subtitle":"Colored curves represent every module of affine type D-tilde-n, so homological data becomes counting intersections.","key_machinery":"The machinery is the category $\\mathcal E(T)$ of colored admissible tagged edges in the twice-punctured disk, together with the tagged rotation $\\rho$ and the six classes of elementary moves that generate the arrows of the quiver $\\Gamma(\\mathcal E(T))$. An admissible tagged edge is a curve that is properly embedded, not null-homotopic, not homotopic to a boundary segment, and not allowed to cut out a once-punctured monogon; the color $\\lambda\\in \\mathbb P^1\\cup\\{-\\infty\\}$ records which stable tube a regular module belongs to, and edges of distinct non-$-\\infty$ colors are declared to have intersection number zero. The tagged rotation $\\rho$ is the geometric analogue of the Auslander-Reiten translation, and the elementary moves are the geometric analogues of the middle terms of almost split sequences. The quiver isomorphism $\\varphi$ is defined on the mouth vertices of stable tubes by matching a module's dimension vector to the intersection numbers of its curve with the triangulation, then extended recursively along arrows; the mesh relations in $\\mathcal E(T)$ play the role of the Auslander-Reiten mesh relations.","core_discovery":"The central claim is Theorem A. For a triangulation $T$ of a twice-punctured disk with $(n-2)$ boundary marked points whose quiver $Q_T$ is an acyclic orientation of type $\\widetilde D_n$, every indecomposable finite-dimensional module over $kQ_T$ is represented by a colored admissible tagged edge $(\\gamma,\\kappa,\\lambda)$, and for any two such edges, $\\operatorname{Int}((\\gamma_1,\\kappa_1,\\lambda_1),(\\gamma_2,\\kappa_2,\\lambda_2))=\\dim_k\\operatorname{Ext}^1(M_1,M_2)+\\dim_k\\operatorname{Ext}^1(M_2,M_1)$. The paper further claims that the correspondence $\\varphi\\colon \\Gamma(\\operatorname{mod} kQ_T)\\to \\Gamma(\\mathcal E(T))$ is an isomorphism of quivers that sends the Auslander-Reiten translation $\\tau$ to the tagged rotation $\\rho$, so the entire combinatorial skeleton of the module category, including the infinitely many homogeneous stable tubes of the regular component, is mirrored in the disk.","pith_inferences":["An implication the paper leaves implicit is that if the recursive definition of $\\varphi$ can be made fully explicit, the same 'dimension vector equals intersection vector' check would give a direct bijective proof of the quiver isomorphism for the whole Auslander-Reiten quiver, not just for the tube mouths.","The coloring idea should transfer to other affine types whose regular components split into infinitely many disjoint tubes: assign each tube a distinct point of $\\mathbb P^1$ and declare different colors non-intersecting, and the same intersection-dimension formula would hold whenever a surface model exists.","A testable extension is to use the model as a combinatorial filter for $\\tau$-rigidity: since $\\mathrm{Ext}^1$ dimensions are intersection counts, a module is $\\tau$-rigid exactly when its curve has controlled self- and mutual intersections, so one could read off the support $\\tau$-tilting data of the regular component from the disk.","The paper leaves open whether the quiver isomorphism can be upgraded to an equivalence of additive categories, with elementary moves and mesh relations matching irreducible morphisms and Auslander-Reiten sequences; such a categorification would make the geometric model a full replacement for the module category rather than only its quiver."],"forward_implications":["For any two modules, the number $\\dim_k\\operatorname{Ext}^1(M_1,M_2)+\\dim_k\\operatorname{Ext}^1(M_2,M_1)$ can be computed by drawing two curves and counting intersections; no extension calculation is needed.","The Auslander-Reiten quiver of $kQ_T$ is isomorphic, as a quiver, to the quiver of admissible edges, so the irreducible-morphism combinatorics of the module category is displayed directly in the disk.","Modules in different stable tubes have zero intersection and zero extension dimensions, because their curves carry different colors and are declared non-intersecting.","The regular modules, which are not $\\tau$-rigid and are invisible to tilting theory, are included in the model, so homological questions about the entire module category of type $\\widetilde D_n$ become geometric questions.","Because intersection numbers are invariant under the tagged rotation, computations for non-projective modules can be moved along the Auslander-Reiten quiver to simpler representatives, matching the $\\tau$-invariance of $\\mathrm{Ext}^1$."],"supporting_citations":[{"why":"Supplies the surface-and-triangulation framework: tagged edges, intersections, triangulations, and the quiver $Q_T$ associated to a triangulation.","marker":"[9]"},{"why":"Describes the stable tubes of Euclidean type $\\widetilde D_n$, including the ranks $2,2,n-2$ and the homogeneous rank-$1$ tubes, and gives the combinatorial Ext computation inside a tube used in Lemma 7.2.","marker":"[23]"},{"why":"Provides the bijection between representations of an acyclic quiver and modules over its path algebra, together with the standard facts on projective and injective modules used throughout.","marker":"[22]"},{"why":"Supplies the background on irreducible morphisms, almost split sequences, and the Auslander-Reiten translation that the geometric model is meant to mirror.","marker":"[1]"},{"why":"Provides the Auslander-Reiten formulas used in Lemma 7.1 to convert Ext spaces into Hom spaces and thereby into dimension-vector entries.","marker":"[2]"},{"why":"Gives the earlier geometric model for tame categories of type $\\widetilde A_n$, the closest predecessor that the present model extends to the non-tau-rigid modules of type $\\widetilde D_n$.","marker":"[3]"},{"why":"Supplies the geometric model for cluster categories of type $D_n$ via triangulations of a once-punctured polygon, a main predecessor for surface models of type $D$.","marker":"[21]"},{"why":"Introduces the completion of a tagged edge, a technical device needed to define elementary moves for curves ending at a puncture.","marker":"[17]"}],"fun_headline_variants":["Counting curve crossings gives Ext dimensions for all D~n modules","Twice-punctured disk hosts all D~n modules as tagged curves","Ext spaces from intersection numbers in a pierced-disk model","Non-τ-rigid modules join the geometric disk model","Curves on a twice-punctured disk calculate homological dimensions"],"cache_read_input_tokens":19840,"weakest_assumption_plain":"The result rests on the claim that every indecomposable module is matched to exactly one curve, with the curve's intersection counts against the fixed triangulation equal to the module's dimension vector at every position in the Auslander-Reiten quiver, not only at the mouths of the stable tubes; if that matching ever identifies two different modules or loses a module, Theorem A stops being a statement about modules.","fun_headline_variants_meta":{"raw":{"variants":["Counting curve crossings gives Ext dimensions for all D~n modules","Twice-punctured disk hosts all D~n modules as tagged curves","Ext spaces from intersection numbers in a pierced-disk model","Non-τ-rigid modules join the geometric disk model","Curves on a twice-punctured disk calculate homological dimensions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000784,"raw_usage":{"total_tokens":3468,"prompt_tokens":958,"completion_tokens":2510,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":574,"completion_tokens_details":{"reasoning_tokens":2431}},"tokens_in":574,"tokens_out":2510,"duration_ms":20076,"temperature":1.0,"reasoning_tokens":2431,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:34:26.225368+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix an acyclic triangulation $T$ of the twice-punctured disk with $(n-2)$ boundary marked points. For each indecomposable module $M$ in a stable tube, compute the vector $(\\operatorname{Int}((\\gamma,\\kappa,\\lambda), i))_{i\\in T}$ for the curve the paper assigns to $M$ and compare it with $\\dim M$. If any module's dimension vector is not realized by an admissible edge, or if two non-isomorphic modules are assigned the same curve, then the quiver isomorphism $\\varphi$ fails and the intersection-dimension formula of Theorem A does not hold for those modules.","supporting_citations":[{"cited_title":"Part I: Cluster complexes, Acta Mathematica201(2008), no","cited_arxiv_id":null,"evidence_quote":"Supplies the surface-and-triangulation framework: tagged edges, intersections, triangulations, and the quiver $Q_T$ associated to a triangulation."},{"cited_title":"2, Cambridge University Press, Cambridge, 2007","cited_arxiv_id":null,"evidence_quote":"Describes the stable tubes of Euclidean type $\\widetilde D_n$, including the ranks $2,2,n-2$ and the homogeneous rank-$1$ tubes, and gives the combinatorial Ext computation inside a tube used in Lemma 7.2."},{"cited_title":"1, Springer, 2014","cited_arxiv_id":null,"evidence_quote":"Provides the bijection between representations of an acyclic quiver and modules over its path algebra, together with the standard facts on projective and injective modules used throughout."},{"cited_title":"1, Cambridge University Press, Cambridge, 2006","cited_arxiv_id":null,"evidence_quote":"Supplies the background on irreducible morphisms, almost split sequences, and the Auslander-Reiten translation that the geometric model is meant to mirror."},{"cited_title":"Smalø,Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995","cited_arxiv_id":null,"evidence_quote":"Provides the Auslander-Reiten formulas used in Lemma 7.1 to convert Ext spaces into Hom spaces and thereby into dimension-vector entries."},{"cited_title":"A geometric realization of tame categories","cited_arxiv_id":"1502.06489","evidence_quote":"Gives the earlier geometric model for tame categories of type $\\widetilde A_n$, the closest predecessor that the present model extends to the non-tau-rigid modules of type $\\widetilde D_n$."},{"cited_title":"1, 1–21, Publisher: Springer","cited_arxiv_id":null,"evidence_quote":"Supplies the geometric model for cluster categories of type $D_n$ via triangulations of a once-punctured polygon, a main predecessor for surface models of type $D$."},{"cited_title":"A geometric model for the module category of a skew-gentle algebra","cited_arxiv_id":"2004.11136","evidence_quote":"Introduces the completion of a tagged edge, a technical device needed to define elementary moves for curves ending at a puncture."}],"review_version":1}