{"id":"d90eb471-6bae-434b-801a-3305e4405ee5","arxiv_id":"1908.09977","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For coprime rank r and first Chern class aH, the 2N-th Betti number of the moduli space of sheaves on P^2 stabilizes once c2 >= N + floor((r-1)/(2r)a^2 + (r^2+1)/2).","lead":"This paper proves an explicit lower bound on the second Chern class beyond which the Betti numbers of the moduli space of sheaves on the projective plane stop changing. It turns a qualitative stabilization theorem into a computable threshold, giving a practical stopping rule for computing these cohomology groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The threshold in Theorem 26 rests on Lemma 18's unproved S1 lower bound; the assertion that the non-negative summands in (27) are bounded below by 2r−4 is justified by a single sentence with no derivation, so the explicit constant C0 is not yet established.","rationale":"Agree with the reader's weakest_assumption. The central new content is the explicit stabilization threshold, obtained by combining Proposition 17's coefficient vanishing with the blow-up formula and the Mozgovoy/Joyce inputs. The only step that produces the numerical constant is Lemma 18's bound on (24). The S1 estimate is where the argument most needs a proof, and the text provides only an assertion. This is not a challenge to the qualitative stabilization result, which already follows from Coskun–Woolf; the paper's contribution is the explicit C0, so the unproved bound is load-bearing. I found no independent internal inconsistency: the compressed handling of the special case l=2, μ_F-difference=−1 can be understood via (1−q) telescoping of the q^{b2} tail, and the case analyses in Proposition 28 are consistent in the cases computed. The main weakness is a missing proof rather than a demonstrated falsehood; hence the reader's CONDITIONAL verdict is appropriate.","tokens_in":31775,"tokens_out":34199,"duration_ms":346409,"concrete_test":"Write a script that, for each r from 2 to 10, enumerates all l, compositions r_1+...+r_l=r, and all integer tuples (a_i,b_i) satisfying the hypotheses of Lemma 18 (sum r_i a_i = a, sum b_i = b, 0≤a,b≤r−1, and the Case A/B conditions implied by S_μ≠0), computes the expression in (24) and the S1 part, and checks the claimed lower bounds κ≥−(r−1) and S1≥−r+3−4/r. If any admissible tuple violates the bounds, the proof of Theorem 26 is refuted; if none does, re-derive the inequality after (27) as a finite combinatorial lemma and check it symbolically for all index partitions with those r_i.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 17's constant C0=(r^2+1)/2 requires the lower bound κ≥−(r−1) for the expression in (24). In Lemma 18 this is reduced to bounding the quadratic S1. After deriving the weighted sum on the right of (27), the proof says: 'By further examining the summands with non-negative coefficient, we see that together they must be bounded below by (2r−4) because all the inequalities in the summations cannot be simultaneously compatible.' This sentence is the entire proof of the S1 bound. The non-negative summands involve products r_i r_j with position-dependent coefficients, and whether they are simultaneously compatible depends on the order of the partition and the values of the r_i; no compatibility argument is supplied. If the claimed 2r−4 bound fails for some composition r_i, the bound κ≥−(r−1) fails, and the maximum defining C0 in Proposition 17 can exceed (r^2+1)/2. Theorem 26 then has no proven explicit threshold, although qualitative stabilization may still hold. Proposition 28 inherits the same gap through its S1+S2 case analysis. The dependence on the unpublished Coskun–Woolf preprint is a secondary but real obstacle to independent verification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies stabilization of the Betti numbers of the moduli space M_{P^2,H}(r,aH,c_2). Its main result, Theorem 26, claims that for r >= 2 and gcd(r,a)=1, the 2N-th Betti number stabilizes once c_2 >= N + floor( (r-1)/(2r) a^2 + (r^2+1)/2 ). The method follows the Coskun--Woolf framework: it encodes the moduli spaces in the ring A^- through generating functions, uses Mozgovoy's formula for ruled surfaces and Joyce's wall-crossing formula on F_1, proves a coefficient-vanishing estimate for (1-q)G_{r,c}(q), and returns to P^2 via the blow-up formula. The key new ingredient is a combinatorial estimate, Lemma 18, that bounds an expression S_1+S_2 and yields the constant C_0=(r^2+1)/2 in Proposition 17. The paper also gives improved bounds for ranks 2 and 4 in Propositions 27 and 28.","tokens_in":31976,"tokens_out":13537,"duration_ms":141922,"significance":"If the main theorem is established, it is a genuinely useful result: it gives an explicit linear-in-N stabilization threshold for all ranks and first Chern classes, rather than only the qualitative statement that stabilization occurs. The approach is systematic, the architecture of the proof is coherent, and the final bound is not fitted but derived from a chain of known structural results. The paper also demonstrates the method by recovering the known rank-2 threshold after a separate calculation and by improving the general bound in a rank-4 example. However, the correctness of the explicit threshold currently rests on an unproved combinatorial assertion in Lemma 18, so the central claim is not yet fully justified. The heavy reliance on the unpublished Coskun--Woolf preprint is a secondary obstacle to independent verification.","major_comments":[{"comment":"The lower bound for S_1 is not established. After deriving Eq. (27), the proof states: 'By further examining the summands with non-negative coefficient, we see that together they must be bounded below by (2r-4) because all the inequalities in the summations cannot be simultaneously compatible.' No argument is supplied for this compatibility claim, and it is not apparent from the displayed sums: whether the positive terms are forced to contribute depends on the order and sizes of the r_i and on the a_i values. Moreover, even accepting the separate bounds on the positive summand (2r-4) and on the negative summand (-(r^2-r)) as stated, the conclusion would only give S_1 >= (2r-4) - (r^2-r), which is not the claimed bound -r+3-4/r; an additional interaction between the positive and negative parts is needed. Because this estimate feeds directly into kappa in Eq. (32), then into C_0 in Proposition 17, and finally into the explicit threshold in Theorems 25 and 26, the main quantitative claim is not yet proven.","section":"Lemma 18, Eq. (24)-(27)"},{"comment":"The improved rank-4 bound inherits the same gap. The proof asserts a sequence of statements of the form 'we see that S_1 >= ...' and 'S_1+S_2 may attain the least possible value ...' for the partitions (r_1,...,r_l) = (3,1), (1,3), (2,1,1), (1,2,1), (1,1,2), and (1,1,1,1), without showing the computations that rule out all remaining integer vectors satisfying the linear constraints. This is a finite check that could be made explicit, but as written it is not verifiable and it depends on the same unproved S_1 estimate from Lemma 18. The text also contains a typo: the second displayed formula for 'S_1' in this proof is in fact the definition of S_2, which makes the case analysis harder to follow.","section":"Proposition 28"}],"minor_comments":[{"comment":"The reduction to a_i in {-1,0,1,2} should be justified. The Lagrange multiplier calculation gives a/r - 1/2 <= a_i <= a/r + 1/2, and with 0 <= a <= r-1 these intervals are contained in (-1/2, 3/2); the text should explain why checking { -1,0,1,2 } suffices.","section":"Lemma 18, partition argument"},{"comment":"The same symbol M is used for both the moduli stack and the coarse moduli space; Proposition 10 relates them by a factor (L-1), so the two objects must be typographically distinguished throughout.","section":"Section 2 and Proposition 10"},{"comment":"Several foundational statements are quoted from the unpublished preprint [CW] (Propositions 2 and 10, Theorem 9, Remark 16, and the sign conditions used in Lemma 18). The author should state the status of this preprint and, if possible, include the necessary statements or proofs so that the main argument can be independently checked.","section":"References"},{"comment":"There is a typographical error in the citation: '[Mo][Theorem 1.1)' should be '[Mo][Theorem 1.1]'.","section":"Theorem 13 reference"}],"recommendation":"major_revision","confidential_remarks":"The main issue is concentrated in Lemma 18: an honest gap in the proof of a load-bearing estimate, not an obvious contradiction. A revised version that supplies a complete proof of the S_1 bound (or replaces it with a weaker but proven bound and adjusts C_0 accordingly) could be publishable. I found no circularity: the quoted external results are used as black boxes and the target theorem is not assumed. The dependence on the unpublished Coskun--Woolf preprint is worth flagging to the editors, as several key structural inputs come from it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives the first explicit stabilization threshold for the Betti numbers of MP2,H(r,aH,c2) for all ranks r≥2 with gcd(r,a)=1: if c2 ≥ N + floor((r−1)/2r a^2 + (r^2+1)/2), then b_{2N} stabilizes, with the stable value determined by Coskun–Woolf. That is a real new result; prior work had qualitative stabilization and low-rank examples. The strategy is coherent: motivic generating functions, Mozgovoy's stack formula, Joyce's wall-crossing, and the blow-up formula are assembled in the expected way, and the rank-1 section cleanly recovers the known Hilbert-scheme stabilization. The worked examples are also honest: Proposition 27 reduces the rank-2 bound to the known c2 ≥ N+1, and Proposition 28 gets a better bound for rank 4. These checks give me confidence that the machinery is fundamentally right.\n\nThe problem is real, though. Lemma 18 is load-bearing for the constant C0=(r^2+1)/2, and its proof of the lower bound for S1 is a single sentence: 'By further examining the summands with non-negative coefficient, we see that together they must be bounded below by (2r−4) because all the inequalities in the summations cannot be simultaneously compatible.' That is not a proof. The expression in (27) has position-dependent coefficients and the compatibility claim depends on the order of the partition and the sizes of the r_i; the reader and the stress-test note are right to flag it. If that bound fails, κ≥−(r−1) fails and the stated threshold is unjustified, though a weaker one might still hold. Proposition 28 inherits the same gap through its S1+S2 case analysis, and its claims like 'we see that S1≥−3/4' are asserted without the supporting computation.\n\nI don't think this is fatal: the architecture is sound, the bound matches all known data, and the missing piece is a combinatorial estimate that may be fixable with a real argument. But it is exactly the kind of assertion that a referee should not let through as is. The reliance on the unpublished Coskun–Woolf preprint is secondary but a stable reference or an appendix stating the needed results would help.\n\nWho is this for? People working on moduli spaces of sheaves on surfaces, stable cohomology, and Donaldson-type invariants. A serious referee will get value from engaging, and after a revision that proves Lemma 18 and clarifies the dependence on [CW], this would be a solid paper.\n\nMy recommendation: send it to peer review rather than desk-reject; the referee should require a real proof of the S1 bound and a stable reference for the input results. With those fixed, I'd be happy to cite it.","headline":"Explicit stabilization threshold is new and plausible, but the load-bearing S1 bound in Lemma 18 is asserted rather than proved; should go to referees with a requirement to fix it.","tokens_in":32554,"tokens_out":2980,"would_cite":true,"duration_ms":29166,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D20","14J60","14J26","14F45"],"pacs":[],"model":"deepseek-v4-flash","headline":"For sheaves on the projective plane with coprime rank and first Chern class, the paper proves that the 2N-th Betti number of the moduli space becomes independent of the second Chern class once $c_2 \\geq N + \\left\\lfloor \\frac{r-1}{2r}a^2…","keywords":["moduli spaces of sheaves","Betti numbers","stabilization","projective plane","motivic generating functions","wall-crossing","Hilbert scheme of points","slope stability"],"falsifier":"A finite computer search over integer tuples $(l, r_i, a_i)$ satisfying $\\sum r_i = r$, $\\sum r_i a_i = a$, and the slope inequalities of the wall-crossing sign rule, evaluating the expression $S_1$ in equation (25), would settle the bound. If any tuple gives $S_1 < -r + 3 - \\frac{4}{r}$, the claimed $\\kappa \\geq -(r-1)$ and hence $C_0=\\frac{1}{2}(r^2+1)$ would fail; if no such tuple exists for all $r$ and $a$, the bound is confirmed.","tokens_in":31511,"feed_emoji":"📐","tokens_out":7471,"duration_ms":68120,"temperature":0.7,"pith_summary":"This paper establishes an explicit numerical threshold beyond which the even Betti numbers of the moduli space of slope-semistable sheaves on the projective plane stop changing. For rank $r \\geq 2$ and first Chern class $aH$ with $\\gcd(r,a)=1$, the theorem says that the $2N$-th Betti number depends only on $N$ once $c_2 \\geq N + \\left\\lfloor \\frac{r-1}{2r}a^2 + \\frac{1}{2}(r^2+1) \\right\\rfloor$. Because the stable values are already known through a universal generating function, the result turns infinitely many previously unknown Betti numbers into explicit numbers. The argument works by showing that all terms of a motivic generating function that could contribute to variation vanish above the threshold.","feed_headline":"Even Betti numbers stabilize past an explicit c2 threshold","feed_subtitle":"For coprime rank and first Chern class, the 2N-th Betti number stops changing once c2 reaches the stated linear threshold.","key_machinery":"The engine is the motivic generating function $G_{r,c}(q)=\\sum_{\\Delta \\geq 0}[\\mathcal{M}_{\\mathbb{P}^2,H}(r,c,\\Delta)]L^{r^2(1-2\\Delta)}q^{r\\Delta}$, together with its blow-up counterpart on the one-point blow-up of $\\mathbb{P}^2$. A criterion from earlier work identifies stabilization of Betti numbers with the vanishing of the coefficient of $L^{-N}q^{\\Delta}$ in $(1-q)G_{r,c}(q)$ for $\\Delta$ large relative to $N$. The paper obtains such a vanishing bound by combining a formula for stack classes on ruled surfaces, a wall-crossing identity that relates two polarizations on the blow-up, and a blow-up formula connecting $\\mathbb{P}^2$ to its blow-up; the final constant $C_0=\\frac{1}{2}(r^2+1)$ comes from bounding a quadratic expression in the first Chern classes of the direct-summand characters.","core_discovery":"The paper's central claim is Theorem 26: for $r \\geq 2$ and $a$ coprime to $r$, the $2N$-th Betti number of $M_{\\mathbb{P}^2,H}(r,aH,c_2)$, the moduli space of slope-$H$-semistable torsion-free sheaves with those Chern classes, stabilizes in $c_2$ once $c_2 \\geq N + \\left\\lfloor \\frac{r-1}{2r}a^2 + \\frac{1}{2}(r^2+1) \\right\\rfloor$. Stabilization means the value of $b_{2N}$ is independent of $c_2$ above the threshold, and it coincides with the coefficient predicted by the stable generating function for sheaves on $\\mathbb{P}^2$. The paper proves this by reducing the question to a vanishing statement for coefficients of $L^{-N} q^{\\Delta}$ in $(1-q)G_{r,aH}(q)$, a motivic generating function built from stack classes.","pith_inferences":["The general threshold is almost certainly not optimal: the worked examples show case-by-case improvements, so a sharper uniform constant could exist, possibly closer to $\\frac{r-1}{2r}a^2$ plus a smaller rank-dependent term.","The same coefficient-vanishing strategy should extend to other rational surfaces or to stabilization statements for higher-degree Betti numbers; the only surface-specific inputs are the blow-up formula and the wall-crossing data.","The unproved combinatorial assertion in Lemma 18 could be tested by a finite minimization over integer tuples; if it fails for some $r$, a weaker explicit threshold would still follow from the surrounding method."],"forward_implications":["For every pair $(r,a)$ with $\\gcd(r,a)=1$, all Betti numbers $b_{2N}$ of the corresponding moduli spaces become explicitly computable once $c_2$ crosses the stated line, since the stable generating function is known.","The rank-one case is recovered as a special case, with stabilization for $c_2 \\geq 2N$.","For the known rank-two example $r=2$, $a=-1$, the method yields the improved threshold $c_2 \\geq N+1$, matching the previously tabulated Betti numbers.","For $r=4$, $a=1$, the bound is improved to $c_2 \\geq N+5$, showing the general threshold is not sharp in every example."],"supporting_citations":[{"why":"Supplies the stable generating function for the Betti numbers of these moduli spaces and the criterion that identifies stabilization with convergence of $(1-q)G$ at $q=1$; the paper's goal is to make that stabilization effective.","marker":"[CW]"},{"why":"Gives the closed formula for the ruled-surface stack generating function $H_{r,c}(q)$ and the blow-up formula that connects $\\mathbb{P}^2$ to its blow-up; both are used to bound the vanishing of coefficients.","marker":"[Mo]"},{"why":"Provides the wall-crossing identity comparing classes of semistable stacks for two polarizations on the blow-up, which is the step that introduces the summands to be bounded.","marker":"[J08]"},{"why":"Supplies the discriminant relation used to eliminate $\\Delta$ from the wall-crossing summands and to express exponents in terms of Chern classes.","marker":"[Y96b]"},{"why":"Provides the algebraic identities for $S_1$ and $S_2$ and the case-by-case bounding style that the paper adapts for its improved examples.","marker":"[Ma14]"}],"fun_headline_variants":["Stable even Betti numbers for P^2 sheaves past explicit c2 cutoff","Explicit c2 threshold freezes even Betti numbers on P^2 moduli","Betti numbers stabilize on P^2 sheaf moduli once c2 passes bound","Even Betti numbers constant after c2 hits explicit formula"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole explicit bound rests on an unproved combinatorial assertion inside Lemma 18: that a certain collection of non-negative summands is jointly bounded below by $2r-4$ because the relevant inequalities cannot be satisfied simultaneously; if this fails, the stated constant $\\frac{1}{2}(r^2+1)$ is not justified, though a weaker threshold could still hold.","fun_headline_variants_meta":{"raw":{"variants":["Stable even Betti numbers for P^2 sheaves past explicit c2 cutoff","Explicit c2 threshold freezes even Betti numbers on P^2 moduli","Betti numbers stabilize on P^2 sheaf moduli once c2 passes bound","Even Betti numbers constant after c2 hits explicit formula"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000471,"raw_usage":{"total_tokens":2303,"prompt_tokens":867,"completion_tokens":1436,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":1351}},"tokens_in":483,"tokens_out":1436,"duration_ms":10296,"temperature":1.0,"reasoning_tokens":1351,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:57:10.622777+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A finite computer search over integer tuples $(l, r_i, a_i)$ satisfying $\\sum r_i = r$, $\\sum r_i a_i = a$, and the slope inequalities of the wall-crossing sign rule, evaluating the expression $S_1$ in equation (25), would settle the bound. If any tuple gives $S_1 < -r + 3 - \\frac{4}{r}$, the claimed $\\kappa \\geq -(r-1)$ and hence $C_0=\\frac{1}{2}(r^2+1)$ would fail; if no such tuple exists for all $r$ and $a$, the bound is confirmed.","supporting_citations":[],"review_version":1}