{"id":"beca21e5-e0ee-40da-8b91-6196096061c0","arxiv_id":"2607.13376","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A K-theoretic logarithmic double ramification class is constructed, shown to satisfy a GL_r(Z)-invariant product formula in colimit log K-theory, and computed by a new stack-valued Thom–Porteous formula.","lead":"This paper constructs a K-theory analogue of the logarithmic double ramification cycle—the virtual class of curves where a line bundle becomes trivial—and proves a product formula compatible with GL_r(Z) changes of contact data. It also gives an explicit degeneracy-locus formula via a Grothendieck-polynomial operator, using a new K-theoretic Thom–Porteous formula for algebraic stacks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Colimit logK-theory ring structure relies on Artin-fan functoriality (Prop B.7 via Lemmas B.5–B.6) whose proofs are too weak and were previously mistaken; if these fail, the product formula and the domain of Theorem 1.8 collapse.","rationale":"The reader identified exactly the same load-bearing assumption: the compatibility/uniqueness of the Artin-fan maps in Lemmas B.5–B.6, on which the colimit logK-theory ring structure and the product formula rest. My independent reading of the manuscript confirms that these lemmas are crucial and that their proofs are materially weaker than the surrounding technical apparatus. The acknowledgment of a prior mistake in these lemmas, coupled with the fact that the proofs invoke descent and constructibility claims that are not fully justified, makes this the most exposed point in the paper. I agree with the readers' CONDITIONAL verdict: the central claims may well be true, but the foundation is not yet secure enough to warrant full acceptance. If the concrete test (or an independent verification of Lemmas B.5 and B.6) passes, the verdict could move to ACCEPT; if it fails, the product formula and the colimit ring definition would need substantive revision. No other issue—such as the Eagon-Northcott resolution or the degeneracy-locus computation—seems as fragile, since those parts are backed by concrete citations to existing results and the explicit finite resolution avoids the stack-theoretic convergence problems.","tokens_in":41097,"tokens_out":22498,"duration_ms":216389,"concrete_test":"Independently re-derive Proposition B.7 from first principles. Concretely: (1) In Lemma B.5, take X = A^1 with log structure at 0 and P = N; write out the coequalizer of Artin fans and verify that the resulting map Θ_X→Θ_P is representable, étale, and pulled back from Spec P, checking each step. (2) In Lemma B.6, let F be the extension by zero of the constant sheaf Z on the open stratum of Spec N (so F(Spec N)=0 but F has stalk Z on the generic point) and compute Γ(X, F|_X) and Γ(Θ_X, F|_{Θ_X}) for X = A^1 with log structure at 0. If the claimed bijection between these groups fails, the lemma is false. If both checks pass, the concern is resolved and the conditional can be lifted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction of logK^0(X) (Def 4.3) and hence the product formula (Thm 4.19), GL_r(Z)-invariance, and even the claim that the log DR class lies in logK^0(M_{g,n}) (Thm 4.10, Lemma 4.5) depend on the existence and uniqueness of induced maps between Artin fans, Proposition B.7. Its proof relies on Lemmas B.5 and B.6, which assert that any strict map X→Θ_P induces an étale representable map Θ_X→Θ_P, and that a section of a sheaf pulled back from Spec P over X extends uniquely to Θ_X. The proofs of these lemmas are little more than sketches: Lemma B.5 claims a coequalizer of face inclusions is pulled back from Spec P without checking representability or étaleness in the non-strict case; Lemma B.6 claims global sections of a constructible sheaf on an atomic log scheme are determined by the stalk at the deepest closed stratum, which fails for general constructible sheaves (e.g., a skyscraper with nontrivial monodromy or a sheaf supported on a higher stratum). The acknowledgments state that the proofs of Lemmas B.5 and B.6 were 'refined in collaboration with AI upon noticing a mistake in an earlier version,' so there is concrete reason to worry that the current proofs may still be incomplete. If the induced arrow Θ_{X_1}→Θ_{X_2} is not unique or does not compose properly, the Gysin maps φ^! in Def 4.3 are not well-defined, logK^0(X) is not a ring, and Thm 4.19 (the log product formula and GL_r(Z)-invariance) cannot hold. Even Theorem 1.8 as stated uses aj^!_a on classes in logK^0(\\tilde M_{g,n}), so its meaning is undermined by the same missing foundation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a K-theoretic logarithmic double ramification (log DR) class and studies its formal properties. It defines log perfect obstruction theories for the log DR space and the Abel--Jacobi pullback description, proves their equivalence, introduces a colimit log K-theory ring, and establishes a log product formula together with GL_r(Z)-invariance. It also proves a finite K-theoretic Thom--Porteous formula for degeneracy loci on algebraic stacks, using an Eagon--Northcott resolution, and applies it to compactified Jacobians to express the structure sheaf of the zero section as eG([Rπ'_*F]) and the log DR class as an explicit Gysin pullback of this operator. The main theorems are Theorem 1.8, Theorem 4.19, and Theorem 6.9/Proposition 6.14.","tokens_in":41572,"tokens_out":10399,"duration_ms":103442,"significance":"If correct, the paper is a substantial contribution: it gives a K-theoretic refinement of logarithmic double ramification cycles, a product formula in a genuinely ring-valued log K-theory, and an explicit finite Thom--Porteous formula that remains valid on algebraic stacks where the usual infinite Grothendieck-polynomial series can fail to converge. The degeneracy-locus argument in §5--§6 is independent of the target statement and is not circular; in particular, the Eagon--Northcott resolution is a real methodological strength. However, the framework of colimit log K-theory, and hence Theorems 1.3, 1.4 and 4.19, rests on the Artin-fan functoriality claims in Appendix B, whose proofs are currently not complete. The central formula in Theorem 1.8 also requires clarification of the K-theory group in which the class [Rπ'_*F] is interpreted. For these reasons the paper needs substantial revision before I can recommend acceptance.","major_comments":[{"comment":"The definition of colimit log K-theory and the ring structure on logK^0(X) depend on the existence, uniqueness and compatibility of maps between Artin fans: these are the maps φ: Θ_{X1}→Θ_{X2} used to define the Gysin pullbacks φ^! in Definition 4.3 and in the proof of Theorem 4.4. Proposition B.7 is therefore load-bearing. Its proof is a sketch: Lemma B.6 asserts a uniqueness/extension statement for sheaves on Spec P, but the proof claims that global sections are determined by the stalk at the deepest closed stratum, which is false for general constructible sheaves (e.g. a skyscraper supported on a higher stratum has zero stalk at the deepest stratum but nonzero global sections). The statement of Lemma B.6 also does not assume the constructibility or constancy hypotheses used in the proof. Lemma B.5 similarly asserts an induced étale representable map without a complete verification of","section":"Appendix B, Proposition B.7; Definition 4.3; Theorem 4.19"},{"comment":"The identification of the degeneracy locus with the zero section E^{1/r}→J^{1/r} is imported: the proof says 'which [CH25, Lemma 4.4] then identifies with the zero locus', and Proposition 6.14 uses '[CH25, Lemma 4.5]' to identify a gerbe. These identifications are central to Theorem 1.8. The paper should either state these lemmas with their precise hypotheses and prove them in the required generality, or give a complete reference and explain why the cited statement applies to the present compactified-Jacobian setting. As written, a key step of the main formula is outsourced.","section":"§6, proof of Theorem 6.9 and Proposition 6.14"},{"comment":"The operator eG is defined on differences E−F of vector bundles. In Theorem 6.9 it is applied to [Rπ'_*F]=[π'_*F]−[R^1π'_*F]. The proof derives equality of this class with [π'_*F(D)]−[π'_*F_D(D)], where the latter two are vector bundles, but the equality is obtained in G-theory. Unless the cohomology sheaves are locally free, or K^0 is interpreted as K-theory of perfect complexes and eG is extended to that group, the formula eG([Rπ'_*F]) is not literally well-defined as stated. Please clarify which K-group is used throughout §6 and prove that eG extends to the relevant perfect-complex class.","section":"Definitions 1.5 and Theorem 6.9; Eq. (35)"},{"comment":"The proof of the product formula invokes a pullback square involving the simultaneous triviality locus and then applies '[CHL23, Remark 1.8]' to identify ∆^†_B(⊠_i [DR_{P_i,a_i}]^{ℓvir}) with [DR_{P,A}]^{ℓvir}. This is a substantial compatibility statement between the log Gysin map and the log perfect obstruction theories. The paper should justify this square and the obstruction-theory identification in detail, or state the exact theorem from [CHL23] being used.","section":"Theorem 4.19 and §4.3"}],"minor_comments":[{"comment":"The notation \\(\\tilde M_{g,n}\\) appears in the statement without definition; it is later described as a log alteration. Please define it in the glossary or in the theorem statement.","section":"§1, Theorem 1.8"},{"comment":"The proof cites '[hg]' (a MathOverflow answer) with only an initial; please give the full author name and a stable reference, or replace by a direct proof.","section":"§5, Lemma 5.2"},{"comment":"The paragraph on motivations from quantum K-theory and physics is long and disconnected from the mathematical content. It could be condensed to a few sentences or moved to a remarks subsection.","section":"§1.6"},{"comment":"The statement 'By Remarks 2.18, 2.19 and their analogue in higher rank, we have a pullback square' would be easier to check if the square were written explicitly with arrows and the maps identified.","section":"§4.3, proof of Theorem 4.19"},{"comment":"The corollary states that it suffices that the grade of the ideal sheaf I_{X/Y} is r_2−r_1+1, but the preceding proof assumes a regular immersion. Please include a short explanation of the grade condition and its compatibility with the determinantal complex.","section":"§5, Corollary 5.5"}],"recommendation":"major_revision","confidential_remarks":"I want to be clear that the acknowledgment of AI assistance and the previously mistaken version of Lemmas B.5–B.6 is not by itself grounds for rejection. The manuscript contains substantial independent and correctly reasoned material, especially the Eagon–Northcott resolution in §5 and the degeneracy-locus computation in §6. However, the colimit log K-theory ring used in the main theorems depends on Appendix B, and the current level of detail there is not sufficient for a journal referee to certify the proof. I would send the paper back for a major revision with a request that Proposition B.7 and the related lemmas be either proved in full or replaced by precise citations. I would also ask for a clarification of the K-group conventions in Theorem 6.9. If those points are addressed, the paper could become acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about arXiv:2607.13376. First, it does something genuinely new: it constructs the K-theoretic logarithmic double ramification class, proves the product formula and GL_r(Z) invariance in colimit log K-theory, and derives an explicit degeneracy-locus formula [e] = eG(Rπ'_*F). Second, the product formula's foundation is the weak spot. The stack-level Thom-Porteous formula (Thm 1.6/Cor 5.5) using the finite Eagon-Northcott resolution is a real contribution; it avoids the infinite sums of Buch/Anderson that diverge on stacks, and Section 6's application to the zero section is convincing. The comparison of the two log obstruction theories (Cor 3.1) is also solid and useful.\n\nWhere I'd worry: the colimit logK ring is defined via Gysin pullbacks along maps of Artin fans, and the existence/uniqueness of those maps is Proposition B.7, proved through Lemmas B.5-B.6. Those proofs are sketches. The stress-test's counterexample to B.6 doesn't quite land—the lemma assumes the sheaf is constant on each stratum, and a skyscraper on an open stratum is not—but the lemma's proof is still far too compressed to justify the uniqueness claim over Spec Z. The paper even acknowledges an earlier mistake in those lemmas and an AI-assisted refinement, which makes me want a clean, complete proof or a precise reference. If Prop B.7 fails, Theorem 4.19 and the ring structure on logK^0 collapse. The main degeneracy-locus formula (Thm 1.8) lives in ordinary K-theory of J and appears independent of that, so the paper's centerpiece formula may survive even if Appendix B needs major rework.\n\nThe citation practice is fine: they credit the Chow-level results and extend them. The limitations are honestly stated. I think this deserves a serious referee; the referee should spend their time on Appendix B and on the universal property of Artin fans over Z. I'd take the paper's main claims with moderate confidence until that is fixed.","headline":"K-theoretic log DR class with a genuinely new stack-level Thom-Porteous formula; the formula and product are likely right, but the colimit logK ring rests on a sketched Artin-fan functoriality that needs a real proof.","tokens_in":42133,"tokens_out":9416,"would_cite":true,"duration_ms":95340,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14H10","14C17","14C35","14N35","14D23","19E08"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single explicit operator computes the K-theoretic log double ramification class and also yields a GL_r(Z)-invariant product formula.","keywords":["double ramification cycle","logarithmic geometry","algebraic K-theory","colimit log K-theory","Thom–Porteous formula","degeneracy locus","compactified Jacobian","moduli of curves"],"falsifier":"Using the explicit eG operator, compute both sides of the product formula for g=1, n=2 with interaction matrix A=[[−1,3],[1,−3]] and M=[[−5,2],[−3,1]] (a worked example in the paper); the two products of log DR classes must coincide in logK°(M_{1,2}). Any difference in a computed class would disprove the product formula.","tokens_in":41017,"feed_emoji":"🧮","tokens_out":16944,"duration_ms":140249,"temperature":0.7,"pith_summary":"The paper constructs a K-theoretic analogue of the logarithmic double ramification cycle, the virtual class of curves on which a given line bundle becomes fiberwise trivial. It establishes two main results: the class lives in the colimit log K-theory ring and satisfies a product formula with GL_r(Z)-invariance—products for the columns of A equal those for the columns of AM and both equal the higher-rank log DR class—and, at positive genus, the class is explicitly computed by applying an operator eG, a Grothendieck-type polynomial built from exterior powers and determinants, to the derived pushforward of the universal line bundle over a compactified Jacobian. The operator is finite, thanks to a new K-theoretic Thom–Porteous formula for algebraic stacks that uses an explicit finite free resolution rather than divergent infinite series. Together these give both a computational formula for K-theoretic DR classes and the correct ring structure needed to multiply them.","feed_headline":"Log double ramification classes computed by one explicit operator","feed_subtitle":"A single operator makes log double ramification classes computable, and yields a GL_r(Z)-invariant product law.","key_machinery":"The central object is the operator eG on the K-theory of a stack, defined by eG(E−F) = [O_Y] − [det E]·(λ_t(F∨)/λ_{t^{-1}}(E))_{t^{r1}}, where λ_t is the exterior power series and the subscript means the coefficient of t^{r1}. This operator converts the class of a vector-bundle map to the pushforward of the structure sheaf of the locus where the map does not have maximal rank. Its stack-valid form comes from an explicit finite resolution (the Eagon–Northcott–Buchsbaum–Rim complex) of that structure sheaf, bypassing the infinite λ/divided-power expansions that truncate on schemes but can diverge on algebraic stacks. The surrounding structure is colimit log K-theory logK°(X) = colim over log a","core_discovery":"At positive genus, the structure sheaf of the zero section e: M_{g,n}→J in a compactified Jacobian equals eG([Rπ'_*F]) in K°(J), where eG(E−F) = [O_Y] − [det E]·(λ_t(F∨)/λ_{t^{-1}}(E))_{t^{r1}}. Consequently, after pulling back along the Abel–Jacobi map, the K-theoretic log double ramification class satisfies [DR^log_{g,n,a}]^{ℓvir} = aj^!_a eG(Rπ'_*F). The proof identifies the zero section with the (r1−1)-degeneracy locus of a map of vector bundles whose ranks differ by 1−g, and applies a finite-resolution K-theoretic Thom–Porteous formula valid for algebraic stacks, bypassing the infinite λ/divided-power expansions that diverge there.","pith_inferences":["The finite form of eG suggests that all K-theoretic degeneracy loci on algebraic stacks admit finite alternating-sum formulas, so the apparent need for infinite Grothendieck polynomials is a scheme-specific artifact of nilpotence; analogous finite formulas should hold for rank-drop loci beyond the first.","Because the r-th root variant pushes down to the same eG formula, one can conjecture a full K-theoretic Pixton formula—[DR^{log}]·ψ^u = r^{u+1} ϵ_* c_{g+u}(−Rπ_* L^{1/r})—with explicit r-dependence for each u>0, parallel to the Chow precursor; this is testable by the same degeneracy-locus method.","The reliance on compactified Jacobians is likely temporary; once the logarithmic Picard stack LogPic has sufficiently developed Brill–Noether theory, the same formula should be provable directly on LogPic, eliminating the quasistable-model and admissibility detour."],"forward_implications":["For g>0, the K-theoretic log double ramification class can be computed by evaluating the finite operator eG on Rπ'_*F, without virtual localization or infinite series; the formula is valid on algebraic stacks and in mixed characteristic.","The GL_r(Z)-invariance of products means the class attached to a matrix A is unchanged, after base change, under the action of M; this gives a universal identity in logK°(M_{g,n}) that reduces products of DR classes to a single higher-rank class.","Because the classes lie in colimit log K-theory, which is a ring, intersections of log DR classes with one another and with other K-theoretic classes are well-defined—something that fails for limit log K-theory—so this ring is the natural home for double DR intersections and quantum K-theoretic integration.","The K-theoretic Thom–Porteous formula extends degeneracy-locus computations from schemes to stacks and is valid in mixed characteristic, making the same finite-resolution tool applicable to other moduli stacks where classes of virtual rank zero are not nilpotent."],"fun_headline_variants":["Explicit operator computes K-theoretic log DR classes","Log DR class from a single Grothendieck polynomial operator","K-theoretic log DR via finite-resolution Thom–Porteous","GL_r(Z)-invariant product law for log double ramification","One operator yields GL_r(Z)-invariant log DR class"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The product formula holds only if the pullback maps between different log refinements of the moduli space are compatible enough to form a true ring; a single incompatible pair of refinements would destroy the product structure.","fun_headline_variants_meta":{"raw":{"variants":["Explicit operator computes K-theoretic log DR classes","Log DR class from a single Grothendieck polynomial operator","K-theoretic log DR via finite-resolution Thom–Porteous","GL_r(Z)-invariant product law for log double ramification","One operator yields GL_r(Z)-invariant log DR class"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000201,"raw_usage":{"total_tokens":1169,"prompt_tokens":650,"completion_tokens":519,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":394,"completion_tokens_details":{"reasoning_tokens":444}},"tokens_in":394,"tokens_out":519,"duration_ms":5752,"temperature":1.0,"reasoning_tokens":444,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T05:20:49.183345+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Using the explicit eG operator, compute both sides of the product formula for g=1, n=2 with interaction matrix A=[[−1,3],[1,−3]] and M=[[−5,2],[−3,1]] (a worked example in the paper); the two products of log DR classes must coincide in logK°(M_{1,2}). Any difference in a computed class would disprove the product formula.","supporting_citations":[],"review_version":1}