{"id":"24e97612-75ea-43bb-b20b-196500caa224","arxiv_id":"2501.16903","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The moduli space of total semi-stability conditions on Euclidean quivers is described by partition data and contracts linearly to any non-concentrated stability condition.","lead":"This paper gives an explicit description of the space of total semi-stability conditions for Euclidean quivers, equivalently coherent sheaves on tame weighted projective lines, and shows this space linearly contracts to any non-concentrated point. The result provides a new route toward contractibility of stability spaces and yields an alternative proof of the known contractibility for affine type A.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.4's concentrated-case sufficiency is asserted via 'same argument' rather than proved, and the TSD classification and linear contraction rest on it.","rationale":"The reader identified Proposition 3.4's concentrated-case sufficiency as the weakest assumption, and my reading agrees: it is the single point where the transition from a bare central-charge datum to a genuine stability condition is asserted rather than proved. The rest of the paper is either computational (the D/E inequality derivations, which are finite linear algebra and can be checked from the displayed matrices) or imported with transparent attribution (the affine A contractibility from [Qy4], the HKK result). No internal inconsistency or circular dependency beyond the Prop 3.4 sketch was found. The proposed torsion-pair check is the natural way to settle the concern: Condition ‡ is exactly the cut condition that should produce a section Γ and a tilt heart, but the paper never proves closure of the corresponding torsion pair. Because the concern is about proof completeness rather than a demonstrated falsehood, the appropriate recommendation is to keep the conditional verdict pending that verification, not to reject the paper outright.","tokens_in":20289,"tokens_out":17382,"duration_ms":180766,"concrete_test":"Complete the proof of Proposition 3.4 in the Im z = 0 case by explicitly constructing the torsion pair from the sign function: set Γ = {V indecomposable vector bundle | φ(V) = 1 and φ(τ^{-1}V) = 0}, T = ⟨τ≤0Γ, coh0⟩, F = ⟨τ>0Γ⟩, and prove from the AR-quiver data in §§5–6 that (T, F) is a torsion pair in coh(P1_w). Concretely, verify that every quotient of an object of T lies in T, every subobject of an object of F lies in F, and Hom(T, F) = 0, for at least one representative TSD of each affine type (e.g. the μ = 1/w_i datum with real z). If this check succeeds, the concentrated case of Prop 3.4 is established and the conditional verdict can be upgraded; if it fails for any type, the classification theorems and Theorem 1.1 collapse.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3.4 is the engine of the paper: Theorems 4.1, 5.2 and 6.2 identify PToSS with non-degenerate TSDs satisfying explicit inequalities, and the contraction flow in Corollaries 4.2/5.3/6.3 is just the linear path in those TSD parameters. The 'if' direction for Im z = 0 is dispatched in one sentence: 'one can use Condition ‡ and the same argument of the proof of Case II in Proposition 3.2.' But Case II of Proposition 3.2 is a statement about an existing stability condition σ, where Lemma 2.5 supplies φ(τ V) ≤ φ(V); for a bare TSD neither a slicing nor a heart has been shown to exist. What must be proved is that the sign function φ(V) = 0/1, with all tube objects at phase 1, is a genuine slicing: (i) every nonzero Hom X→Y satisfies φ(X) ≤ φ(Y), not merely arrows in the preprojective component; (ii) the subcategories T = ⟨τ≤0Γ, coh0⟩ and F = ⟨τ>0Γ⟩ form a torsion pair in coh(P1_w), so that H = ⟨T, F[1]⟩ is an abelian heart; and (iii) the resulting stability function satisfies the Harder–Narasimhan property. The paper asserts (ii) via 'same argument' without verifying closure under subobjects/quotients or Hom-orthogonality. The claim is plausible—Condition ‡ is the standard cut condition for a section Γ—but the written proof does not establish it. If the sufficiency fails in any Euclidean type, the parameterization of PToSS and hence the linear contraction in Theorem 1.1 no longer follow.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies total semi-stability conditions on the bounded derived category of a Euclidean quiver, equivalently of coherent sheaves on a tame weighted projective line P^1_w. It defines a totally semi-stable datum (TSD) as a tuple of partitions of the imaginary root together with a complex parameter z, and claims that, subject to explicit linear inequalities, non-degenerate TSDs classify the projective moduli space PToSS(P^1_w) for all affine types. For affine type A every non-degenerate TSD gives a total semi-stability condition, which yields a linear contraction of PToSS to any chosen non-concentrated point; combined with a retraction from the first author's earlier work [Qy4], this gives contractibility of the whole stability space for affine type A. For affine types D and E the paper gives inequality descriptions of the admissible TSDs and uses the same linear flow to contract PToSS.","tokens_in":20632,"tokens_out":11270,"duration_ms":100549,"significance":"If correct, the paper is a substantial contribution toward the contractibility conjecture for stability spaces of non-Calabi-Yau categories: it replaces the stability space by a concrete, checkable set of linear inequalities and gives an explicit linear contraction. The affine A and D cases are argued in detail, and the E6 computation is explicit and verifiable; the resulting inequalities are concrete and falsifiable. The paper is also commendable for reducing the entire classification to a single criterion (Condition ‡ of Proposition 3.4), making the structure of the proof clear. However, two load-bearing gaps—the sufficiency direction of Proposition 3.4 in the concentrated case and the omitted E7/E8 calculations—mean the central claim is not fully demonstrated in the present version.","major_comments":[{"comment":"The 'if' direction for the Im z = 0 (concentrated) case is asserted in one sentence: 'one can use Condition ‡ and the same argument of the proof of Case II in Proposition 3.2.' That is not a proof, because Case II of Proposition 3.2 starts from an existing stability condition σ, where Lemma 2.5 supplies phase monotonicity along all nonzero Homs; a bare TSD has no slicing or heart yet. The missing argument must establish: (i) φ(X) ≤ φ(Y) for every nonzero Hom X→Y, not merely for arrows in the preprojective component; (ii) the subcategories T=⟨τ≤0Γ, coh0⟩ and F=⟨τ>0Γ⟩ form a torsion pair in coh(P^1_w), so that H=⟨T,F[1]⟩ is an abelian heart; and (iii) the resulting stability function satisfies the Harder–Narasimhan property. Since Theorems 4.1, 5.2, 6.2 and hence Theorem 1.1 depend on this sufficiency, the gap is load-bearing and must be repaired.","section":"§3.2, Prop. 3.4"},{"comment":"For affine types E7 and E8, the inequalities (6.3) and (6.4) are stated after the sentence 'one can express the central charges ... and obtain (by eliminating z0) a set of inequalities' (see the text following (6.12) and (6.16)). The intermediate elimination and the reduction from the phase conditions (6.13)/(6.17) to the listed inequalities are not shown, and the claimed existence of a non-degenerate TSD of type E7/E8 (by taking µ^i_j = 1/w_i) is not verified against (6.3)/(6.4). Because these inequalities are the classification of PToSS for these types, Theorem 6.2 and the contraction statement for E7/E8 are not fully demonstrated. The authors should include the detailed derivation or a verifiable computational check.","section":"§6.2, §6.3"}],"minor_comments":[{"comment":"The notation TSD = (µ??, z) is not defined; it should indicate the tuple (µ^i_j) for all i,j explicitly.","section":"§3.2, Def. 3.3"},{"comment":"The slicing P constructed from a TSD is described by assigning each indecomposable V[m] a phase, but it is not stated that P(φ) is the additive closure of those indecomposables; this should be made explicit.","section":"§3.2"},{"comment":"The phrase 'the τ-symmetry that rotates tµ^j_i | j ∈ Z_{w_i}u' contains a typographical artifact ('tµ' instead of 'µ') and the action of the symmetry should be described more clearly.","section":"§6.1"},{"comment":"The symbol 'Stab D8(ĄA_{p,q})' should read 'Stab D^b(ĄA_{p,q})'; the '8' appears to be a typo.","section":"Cor. 4.3"},{"comment":"The AR-quiver figures in §5 and §6 are dense and hard to read in the preprint; marking the chosen section Γ more prominently would help the reader follow the calculations.","section":"Figures 2–4"},{"comment":"Corollary 1.2 and the 'similar to the total stability case' step in Proposition 3.4 rely on [Qy4] and [QZ], both listed as 'to appear'; please state the precise results being imported and, if possible, provide accessible preprints.","section":"References and dependencies"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's main results depend on [Qy4] and [QZ], both by the first author and listed as 'to appear'; the editor may wish to verify that these results are indeed available or accepted. The omitted E7/E8 calculations are likely routine but lengthy, and the Im z = 0 sufficiency in Proposition 3.4 needs a real argument. I do not see grounds for rejection if these gaps are filled; the affine A/D parts and the E6 calculation are convincing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is that this paper gives a concrete, explicit model of the space of total semi-stability conditions for all Euclidean quivers, and the model is very likely right—but the written proof has a hole worth fixing before I'd trust it as stated. The TSD parameterization and the linear contraction for affine D and E are genuinely new; the affine A contractibility is not new (HKK), and the paper says so.\n\nWhat's good: the paper reduces total semi-stability to checking finitely many linear inequalities in the mu's, and for affine A and D it does that in detail. The E6 computation is explicit with matrices. There are no fitted constants, no circularity: the retraction theorem imported from [Qy4] is a separate result, and the HKK citation for affine A is honest.\n\nThe soft spots are real but probably patchable. Proposition 3.4 is the engine: it claims a non-degenerate TSD gives a total semi-stability condition iff the arrow condition holds. The 'if' for Im z > 0 is sketched but plausible. The 'if' for Im z = 0 is dispatched with “same argument as Case II of Prop 3.2.” That does not work as written: Case II starts from an existing stability condition, where Lemma 2.5 gives phase monotonicity along AR paths. For a bare TSD you have no slicing and no heart yet. You need to show the sign function defines a genuine slicing: a torsion pair, closure under subobjects/quotients, and the HN property. The paper does not show this. It is probably true—Condition ‡ is the standard cut condition for a section—but the proof is not there.\n\nThe other soft spot is type E7/E8: the inequalities are listed with “omitting detailed calculations.” The matrices are there, so it's checkable, but for a classification theorem you'd want the elimination steps, or at least a repository with the computation. That's a presentation issue rather than a conceptual gap.\n\nAll in all, this is a solid, useful paper with a load-bearing but likely fixable gap in one proposition and some omitted verification in the largest cases. People working on stability spaces and the contractibility conjecture will want this. It deserves a serious referee; I'd send it out but ask the authors to expand Prop 3.4 and to provide the E7/E8 computations. I would not desk-reject it.","headline":"The TSD parameterization and linear contraction for Euclidean D/E are new and largely convincing, but the proof of Proposition 3.4's concentrated-case sufficiency is a real gap and the E7/E8 inequalities are asserted without derivation; still, the paper deserves a serious referee.","tokens_in":21189,"tokens_out":2408,"would_cite":true,"duration_ms":22592,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F08","16G20","18E30","16G70"],"pacs":[],"model":"deepseek-v4-flash","headline":"Total semi-stability spaces of Euclidean quivers are contractible, and for affine type A the full stability space is contractible as well.","keywords":["stability conditions","total semi-stability","Euclidean quivers","weighted projective lines","contractibility","affine type A","Auslander-Reiten quiver","linear flow"],"falsifier":"A concrete way to test the central claim is to take a non-degenerate TSD for affine type $D_n$ or $E_n$ that satisfies the listed inequalities but has $\\operatorname{Im} z=0$, construct the slicing prescribed by Proposition 3.4, and check whether the heart $H=\\operatorname{mod} C\\Gamma^{\\mathrm{op}}$ is closed under extensions and has the same simple objects as the proposed slices; if some indecomposable object in a $\\tau$-orbit acquires a phase outside $\\{0,1\\}$ or some Hom from a later slice to an earlier slice is nonzero, the sufficiency direction fails. For affine type A, one can instead compute the space of non-degenerate TSDs with $\\operatorname{Im} z>0$ and check directly that the linear path (4.2) between any two points stays non-degenerate and stays inside $\\mathrm{PToSS}$; a counterexample would be a point where $Z([V])=0$ for an indecomposable vector bundle $V$ at some interior $t$.","tokens_in":20074,"feed_emoji":"📉","tokens_out":9671,"duration_ms":85291,"temperature":0.7,"pith_summary":"This paper proves that the moduli space of total semi-stability conditions on the bounded derived category of any Euclidean quiver—equivalently, of coherent sheaves on a tame weighted projective line—is contractible. Total semi-stability means that every indecomposable object is semistable, a boundary case of the global-dimension function used to study when stability spaces are contractible. The proof parameterizes each projective stability condition by a total semi-stability datum: a tuple of partitions of the imaginary root plus one complex parameter, and shows that the condition of being totally semi-stable is exactly a list of linear inequalities involving those parameters plus one monotonicity condition along arrows of the AR quiver. Because the inequalities are linear, the straight line between any two data sets is again a datum whenever the target is non-concentrated, giving a contraction of the whole space to that target. For affine type A this yields an alternative proof that the full space of stability conditions is contractible.","feed_headline":"Every Euclidean quiver has a contractible total semi-stability space","feed_subtitle":"A linear flow contracts the whole moduli space to any non-concentrated point; affine type A follows.","key_machinery":"The total semi-stability datum (TSD) is the working object. It packages the normalizations forced by total semi-stability: after fixing a reference simple $S^{\\lambda_0}$ with $Z(\\delta)=-1$ and phase $1$, the imaginary root $\\delta$ splits as $\\delta=[S^1_i]+\\cdots+[S^{w_i}_i]$ in each tube, so the numbers $\\mu_i^j=-Z([S_i^j])$ form a partition of $1$, and one adds $z=Z([P_0])\\in\\mathbb{C}^*_{\\ge 0}$. The datum determines the central charge on a basis of the Grothendieck group. The paper's criterion says the datum induces a total semi-stability condition iff for every arrow $X\\to Y$ in the preprojective component $\\operatorname{AR}\\operatorname{vect}(\\mathbb{P}^1_w)$, the phases satisfy $\\varphi(X)\\le \\varphi(Y)$. The technical content of the paper is rewriting this arrowwise condition as linear constraints: for affine type A there are no further constraints, for type $\\widetilde{D}_n$ the constraints are (5.2), and for $\\widetilde{E}_6,\\widetilde{E}_7,\\widetilde{E}_8$ they are (6.2),(6.3),(6.4). These linearities are what make the straight-line flow (4.2) land inside $\\mathrm{PToSS}$, so the same data description both classifies the space and proves it contracts.","core_discovery":"The central claim, stated as Theorem 1.1, is that for every tame weighted projective line $\\mathbb{P}^1_w$, the projective space $\\mathrm{PToSS}(\\mathbb{P}^1_w)$ of total semi-stability conditions admits a linear flow contracting it to any chosen non-concentrated point. A stability condition in $\\mathrm{PToSS}$ is encoded by a non-degenerate total semi-stability datum $(\\mu_i^j, z)$ with $\\mu_i^j>0$, $1=\\mu_i^1+\\cdots+\\mu_i^{w_i}$ for each of the $l$ tubes, and $z\\in\\mathbb{C}^*_{\\ge 0}$; the central charge is fixed by $Z([S_i^j])=-\\mu_i^j$ and $Z([P_0])=z$. The datum determines a slicing, and Proposition 3.4 asserts that this slicing is a total semi-stability condition iff phases do not decrease along arrows of the preprojective component of the AR quiver. For affine type A this arrow condition is automatic for non-degenerate data; for affine type D and E it becomes the explicit linear inequalities (5.2), (6.2), (6.3), (6.4). Since the interpolation of two data sets preserves both non-degeneracy (when the target has $\\operatorname{Im} z>0$) and all the linear inequalities, the straight line in the data coordinates descends to a contraction of $\\mathrm{PToSS}$. The non-concentrated/concentrated dichotomy is also established: exactly the non-concentrated points have heart $\\operatorname{coh}(\\mathbb{P}^1_w)$, while concentrated points have heart equivalent to $\\operatorname{mod} C\\Gamma^{\\mathrm{op}}$ for some Euclidean quiver $\\Gamma$. In the affine A case, combined with a retraction from the full stability space to $\\mathrm{ToSS}$ from earlier work, this gives contractibility of the full stability space.","pith_inferences":["If a retraction from the full stability space to $\\mathrm{ToSS}$ can be constructed for affine types D and E analogous to the one used for affine type A, then Theorem 1.1 would imply the full contractibility conjecture for all Euclidean quivers; the paper states this as a plausible next step but does not prove it.","The linear inequality description suggests computational verification: for a fixed small Euclidean quiver one could enumerate all arrow orbits in the AR quiver and check the classification's inequalities against an independent slicing construction, which would test Proposition 3.4's sufficiency.","The same TSD encoding may apply to other hereditary categories with a tube structure, such as orbifold projective lines of wild type, although the finiteness of the AR-quiver arrow checks would need to be re-examined.","The concentration dichotomy suggests a boundary stratification of $\\mathrm{PToSS}$ along $\\operatorname{Im} z=0$ where the heart jumps from $\\operatorname{coh}(\\mathbb{P}^1_w)$ to a Euclidean module category; understanding this wall might clarify how total semi-stability spaces glue to the rest of the stability space."],"forward_implications":["$\\mathrm{PToSS}(\\mathbb{P}^1_w)$ is contractible for every Euclidean quiver, equivalently for every tame weighted projective line.","$\\mathrm{ToSS}(\\mathbb{P}^1_w)$, the unprojectivized total semi-stability space, is contractible as well in each Euclidean type.","For affine type $\\widetilde{A}_{p,q}$, the full stability space $\\mathrm{Stab}\\,D^b(\\widetilde{A}_{p,q})$ is contractible, giving a proof that avoids quadratic differentials.","Every non-concentrated point of $\\mathrm{PToSS}$ is a possible contraction target: the same linear flow works for any choice of $\\sigma_0$ with $\\operatorname{Im} z_0>0$.","For affine types D and E, the explicit inequalities give a finite description of the moduli space as a semialgebraic set, so the space is not just contractible but convex in the TSD coordinates."],"supporting_citations":[{"why":"Supplies the definition of stability conditions and Proposition 5.3, used to pass from a stability function on a heart to a genuine stability condition.","marker":"[Bri1]"},{"why":"Provides the global dimension function and the retraction of the full stability space to ToSS for affine type A, used in Corollary 4.3.","marker":"[Qy4]"},{"why":"Gives the equivalence between total semi-stability and $\\operatorname{gldim}\\le 1$, which underpins the whole study.","marker":"[Qy3]"},{"why":"Introduces the classification strategy for total stability conditions that the TSD method adapts.","marker":"[QZ]"},{"why":"Establishes the derived equivalence $D^b(C_w)\\cong D^b(\\mathbb{P}^1_w)$ and the AR-quiver structure used in the arrow checks.","marker":"[GL]"},{"why":"Provides the canonical algebras and the AR quivers for affine types E that underlie the dimension-vector calculations.","marker":"[Rin]"},{"why":"The previous proof of contractibility of Stab for affine type A via quadratic differentials, which the paper recovers by a different route.","marker":"[HKK]"},{"why":"Proved the first case of Proposition 3.2 (non-concentrated implies heart equals $\\operatorname{coh}$) for orbifold projective lines, as acknowledged in the paper.","marker":"[Ota]"}],"fun_headline_variants":["Total semi-stability moduli contract linearly for all Euclidean quivers","Every Euclidean quiver has contractible total semi-stability moduli","Linear flow contracts stability moduli for Euclidean quivers","Affine type A: full stability space contractible via linear flow","Tame weighted lines: total semi-stability space contracts linearly"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that phase monotonicity along every arrow in the preprojective component of the AR quiver is enough to force the whole slicing to be a valid total semi-stability condition; in the borderline case where all central charges are real, the paper only sketches the proof of this sufficiency, and if that step fails, the classification of total semi-stability spaces collapses.","fun_headline_variants_meta":{"raw":{"variants":["Total semi-stability moduli contract linearly for all Euclidean quivers","Every Euclidean quiver has contractible total semi-stability moduli","Linear flow contracts stability moduli for Euclidean quivers","Affine type A: full stability space contractible via linear flow","Tame weighted lines: total semi-stability space contracts linearly"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001173,"raw_usage":{"total_tokens":4893,"prompt_tokens":1034,"completion_tokens":3859,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":650,"completion_tokens_details":{"reasoning_tokens":3772}},"tokens_in":650,"tokens_out":3859,"duration_ms":27363,"temperature":1.0,"reasoning_tokens":3772,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T05:44:17.985824+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete way to test the central claim is to take a non-degenerate TSD for affine type $D_n$ or $E_n$ that satisfies the listed inequalities but has $\\operatorname{Im} z=0$, construct the slicing prescribed by Proposition 3.4, and check whether the heart $H=\\operatorname{mod} C\\Gamma^{\\mathrm{op}}$ is closed under extensions and has the same simple objects as the proposed slices; if some indecomposable object in a $\\tau$-orbit acquires a phase outside $\\{0,1\\}$ or some Hom from a later slice to an earlier slice is nonzero, the sufficiency direction fails. For affine type A, one can instead compute the space of non-degenerate TSDs with $\\operatorname{Im} z>0$ and check directly that the linear path (4.2) between any two points stays non-degenerate and stays inside $\\mathrm{PToSS}$; a counterexample would be a point where $Z([V])=0$ for an indecomposable vector bundle $V$ at some interior $t$.","supporting_citations":[],"review_version":1}