{"id":"e4b024b1-8326-4501-9d25-f5cf967c112c","arxiv_id":"1908.02932","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Wild motivic McKay correspondence for arbitrary finite groups is proved via motivic integration over formal Deligne-Mumford stacks, yielding stringy motive invariance and a motivic Bhargava mass formula.","lead":"A new theory of motivic integration is built for 'wild' Deligne-Mumford stacks, where stabilizer groups may have order divisible by the field characteristic. It proves a McKay correspondence for linear actions of any finite group and recovers a motivic version of Bhargava's mass formula.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.3's right-hand side is well-defined only by citation to [TY19]; the v-level stratification of Δ_G and convergence of the integral are not proved here, making the central equality conditional on that companion result.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing point: the RHS of the wild McKay correspondence depends on [TY19] for its definition. This is the most serious issue because Theorem 1.3 is an equality between Mst(A^d_k/G) and an integral that, strictly speaking, is not constructed within this paper; a failure of [TY19] would make the central claim vacuous rather than merely unproved. However, the paper is explicit about the citation, the companion paper is said to be completed before this one, and the reduction in §14 (Corollary 14.4) is internally coherent assuming that result. I did not find a more damaging internal gap: the shift-function computation in Lemma 14.2 gives s_X=-v(E), the stratification by v is locally constructible by Lemma 14.3, and the proof of Corollary 14.4 assembles the strata into Δ_G. The only unguarded spot is the unrestricted statement of Corollary 14.4, which writes Mst(X,A)=Mst(X) without repeating the no-pseudo-reflection hypothesis; under that hypothesis A=0, as the theorem requires, so this is a presentation issue rather than a threat to Theorem 1.3. Therefore the correct response is to keep the reader's ACCEPT verdict: the concern is real but external, explicitly disclosed, and does not by itself indicate an error in this manuscript.","tokens_in":74192,"tokens_out":22139,"duration_ms":233798,"concrete_test":"Extract the exact well-definedness statement from [TY19] used in §14 and re-prove it for the first wild non-cyclic case, G=(Z/p)^2 acting by its regular representation: use the Artin-Schreier description in §5.3 to stratify Δ_G by the v-function, verify each v^{-1}(s) is a finite-type DM stack, and check that the sum ∑_s {v^{-1}(s)}L^{d-s} converges in the completed Grothendieck ring M'_k,r. If any v-fiber fails to be finite type or the sum diverges, the right-hand side of Theorem 1.3 is not well-defined; if the re-derivation succeeds, the cited precondition is satisfied.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central equality in Theorem 1.3 (Corollaries 14.4 and 16.3) is Mst(A^d_k/G)=∫_{Δ_G}L^{d-v}. The right-hand side is not independently defined in this paper: Section 14 defines ∫_{Δ_H}L^{d-v} as ∑_s {v^{-1}(s)}L^{d-s}, where each v^{-1}(s) is asserted to be represented by a finite-type DM stack C_s→Δ_H and the convergence of the sum is asserted to be proved in [TY19]. Lemma 14.3 only establishes that v is locally constructible; the passage from local constructibility to a canonical finite-type stratification with well-defined classes in the completed Grothendieck ring is delegated to the companion paper. Thus, if [TY19]'s well-definedness theorem were incorrect or inapplicable, Theorem 1.3 would not have a meaningful right-hand side. The paper flags this explicitly in §1.5 ('the well-definedness of this integral was proved in [TY19]') and immediately before Corollary 14.4, so the dependency is not hidden. I found no internal inconsistency in the reduction to [TY19]; the concern is a precondition on the object in the formula rather than a demonstrated flaw in the equality.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of motivic integration for formal Deligne-Mumford stacks over a power series ring in arbitrary characteristic. It introduces untwisting stacks, constructs a space of twisted arcs, proves a change-of-variables formula for twisted arcs, and defines stringy motives for stacky log pairs and their invariance under crepant pseudo-modifications. The main application, Theorem 1.3 (Corollaries 14.4 and 16.3), establishes the wild motivic McKay correspondence for linear actions of arbitrary finite groups, generalizing the previously known cyclic-prime-order case, and yields a motivic version of Bhargava's mass formula as a special case.","tokens_in":74485,"tokens_out":9959,"duration_ms":112467,"significance":"If correct, this is a substantial contribution: it provides the first general wild motivic McKay correspondence for arbitrary finite groups and a motivic Bhargava mass formula, while also unifying and extending earlier tame and formal-scheme theories. The paper is detailed and proof-based, with explicit statements, an appendix on general stack-theoretic results, and careful treatment of the completed Grothendieck ring and its realizations. It is transparent about its reliance on the companion works [TY17] and [TY19]; in particular, the well-definedness of the integral over Δ_G in Theorem 1.3 is cited to [TY19] rather than proved here. I found no internal inconsistency in how that dependency is used, and the reliance is standard mathematical practice when the cited results are available.","major_comments":[],"minor_comments":[{"comment":"The right-hand side of Theorem 1.3 is an integral over Δ_G whose well-definedness as an element of the completed Grothendieck ring is cited to [TY19]. Since this is a necessary precondition for the statement, please state explicitly which theorem of [TY19] is being invoked and include its statement, or at least its theorem number, so that the reader can verify the hypotheses without consulting the companion paper.","section":"Section 1.5 and the definition before Corollary 14.4"},{"comment":"The proof of Lemma 14.3 argues that after removing a constructible subset one obtains an open dense substack on which the boundary is vertical, and then concludes that v is constant on each connected component of Γ_G. The passage from the open dense substack to the full local constructibility statement should mention a noetherian induction or an explicit refinement of the stratification, because the argument as written only establishes the open dense substep.","section":"Lemma 14.3"},{"comment":"When passing from formal DM stacks over Df to DM stacks over k, the notation J∞X and Mst(X,A) is defined by base change to X×Df; it would help the reader if the text explicitly noted that the motivic measure and the shift function are the ones constructed in Section 11 after this base change, since the notation is reused for the non-formal setting.","section":"Section 16"},{"comment":"The proof of Corollary 16.4 is only a sketch and uses the quantity dim ∫_{Δ_G\\{o}} L^{d-v} defined by a supremum over a countably infinite stratification. The authors state that this is independent of the stratification; a brief justification would be useful, especially because the integral may diverge and the supremum may be infinite.","section":"Corollary 16.4"}],"recommendation":"minor_revision","confidential_remarks":"The paper depends heavily on the author's companion works [TY17] and [TY19], and the well-definedness of the central integral in Theorem 1.3 is proved in [TY19], not in this paper. The author is explicit about this dependency, which is appropriate. The editor may wish to confirm that [TY19] is publicly available and that the specific theorem invoked is stated there; this does not affect my recommendation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know about this paper is that it is a genuine advance, not a repackaging. Yasuda constructs motivic integration on formal DM stacks with wild stabilizers, proves a change-of-variables formula with shift functions, and gets the wild McKay correspondence for arbitrary finite groups. That was open beyond cyclic p-groups. He also derives a motivic Bhargava mass formula and crepant invariance of stringy motives. This is the natural culmination of a long line of his own work, and the machinery—untwisting via Hom stacks, the universal twisted formal disk, the shift function—is new and looks substantial.\n\nWhat is good: the paper is proof-heavy and explicit. The change-of-variables theorem is proved in the text after a serious stack-theoretic setup, with appendices covering the needed technical lemmas. The dependency on earlier papers, especially [TY17] and [TY19], is standard practice and is disclosed repeatedly. In particular, the main formula Corollary 14.4 reduces the RHS to an integral over Δ_G, and the paper says in §1.5 that well-definedness was proved in [TY19]. That is not a hidden circularity; it is a precondition.\n\nThe soft spot is exactly that precondition. The right-hand side of Theorem 1.3, ∫_{Δ_G} L^{d−v}, is not independently defined in this paper. Lemma 14.3 proves v is locally constructible, but the passage from local constructibility to a canonical finite-type stratification of Δ_G with classes in the completed Grothendieck ring, and the convergence of the sum, are delegated to [TY19]. The stress-test note is accurate on this point. If [TY19] has a gap, the main equality lacks a meaningful RHS. That does not make the theorem false, but it makes the paper less self-contained than its headline suggests. A referee should be asked to verify that [TY19]'s well-definedness theorem applies, especially because the stratification of Δ_G and the convergence are load-bearing.\n\nMinor points: the text is dense, and some formal-stack technicalities are relegated to appendices, but that is a style choice, not a flaw. The self-citation pattern is appropriate for a body of work that builds on itself.\n\nWho gets value: arithmetic geometers and people working on motivic integration, McKay correspondence, or quotient singularities. The paper deserves a serious referee—it is important enough and the proof is detailed enough that desk rejection would be wrong. My recommendation: send to peer review, and make sure the referee checks the [TY19] dependency explicitly.","headline":"Real advance: motivic integration over wild DM stacks plus the wild McKay correspondence for arbitrary finite groups, but the central formula's right-hand side is defined in a companion paper [TY19], so the theorem is conditional on that well-definedness result.","tokens_in":74982,"tokens_out":1483,"would_cite":true,"duration_ms":20968,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E18","14D23","14E16","14B05","11G25","11S15"],"pacs":[],"model":"deepseek-v4-flash","headline":"A motivic integration theory for wild Deligne-Mumford stacks proves the wild McKay correspondence for arbitrary finite groups.","keywords":["motivic integration","Deligne-Mumford stacks","wild McKay correspondence","stringy motives","formal stacks","log pairs","Grothendieck ring of varieties","ramification"],"falsifier":"Take G=Z/$p^{2}$ acting diagonally on $A^{2}$_k over a perfect field of characteristic p, compute the right side ∫$L^{{2-v}}$ by stratifying Δ_G according to ramification data, compute the left side from a log resolution or from the change-of-variables formula, and compare the two classes in the completed Grothendieck ring; any mismatch would refute the wild McKay correspondence in this case.","tokens_in":73988,"feed_emoji":"🧮","tokens_out":5265,"duration_ms":55908,"temperature":0.7,"pith_summary":"This paper develops motivic integration over Deligne-Mumford stacks whose stabilizers may have order divisible by the characteristic, a case previously out of reach, and uses it to prove the wild McKay correspondence for linear actions of arbitrary finite groups. The central formula says that for a finite group G acting linearly on affine space A^d_k without pseudo-reflections, the stringy motive of the quotient A^d_k/G is the integral over the moduli space Δ_G of G-torsors over the punctured formal disk of $L^{{d-v}}$, where v is a constructible function tied to ramification. Before this paper, the wild case was known only for cyclic groups of prime order. If correct, the result gives a motivic analogue of Bhargava's mass formula and a way to compute discrepancies of quotient singularities from arithmetic data.","feed_headline":"Wild McKay correspondence proven for all finite groups","feed_subtitle":"A motivic integral over G-torsors on the punctured disk now computes quotient stringy motives in any characteristic.","key_machinery":"The load-bearing construction is the untwisting stack. For a formal DM stack X over the power series ring, one fixes the moduli stack Θ of Galoisian group schemes and the universal integral model E_Θ of a G-torsor over the punctured formal disk, whose quotient is the universal twisted formal disk. The untwisting stack is Utg_Γ(X)=Hom^rep(E_Γ,X_Γ), the Hom stack of representable morphisms, modified by passing to the rig-pure part along a flattening stratification Γ→Θ. The stack of twisted arcs is then the arc stack J_∞ Utg_Γ(X)^pur. The paper's change-of-variables formula for proper birational morphisms compares motivic integrals on J_∞Y and J_∞X, with the new shift function s_X measuring the difference between the Jacobian orders of the coarse-modulus projection and the untwisting morphism; in the linear case s_X(γ)=-v(E), where v is the length of a module of equivariant maps, essentially the Artin conductor.","core_discovery":"At the paper's core is the claim that twisted arcs on a wild stack can be untwisted: every representable morphism from a twisted formal disk to a stack is controlled by an ordinary arc on an 'untwisting stack', so that motivic integration on wild stacks reduces to integration on schemes. The paper constructs the untwisting stack as a Hom stack of representable morphisms from the universal integral model of a G-cover, stratifies it to make it flat, and proves a change-of-variables formula carrying two correction terms: the usual Jacobian order and a new shift function s_X. For the quotient stack [A^d_k/G], the shift function equals -v on the moduli space of G-torsors, yielding the wild McKay correspondence M_st(A^d_k/G)=∫_{Δ_G}$L^{{d-v}}$. The same machinery yields invariance of stringy motives of stacky log pairs under crepant morphisms and, by specializing to the symmetric group, the motivic version of Bhargava's mass formula.","pith_inferences":["If the correspondence holds, stringy invariants of wild quotients become arithmetic objects: each stratum of G-torsors contributes a class determined by ramification data, which a cohomological realization would turn into congruences between counts of local field extensions and stringy Hodge numbers.","The invariance under crepant morphisms suggests the stringy motive defines a K-equivalence invariant for singular DM stacks in arbitrary characteristic, a property not established before for wild stabilizers.","A concrete testable extension is the case G=Z/p^2 in dimension two: writing the integral over Δ_G explicitly from Artin-Schreier data and comparing it with a resolution-theoretic computation would verify the whole mechanism in the first genuinely new wild case.","The untwisting technique appears to extend beyond linear actions to arbitrary finite group actions on formal schemes, since the local structure theorem reduces to linear actions only after passing to tangent representations; checking non-linear wild actions would widen the correspondence's scope."],"forward_implications":["For every finite group G with a linear action on A^d_k having no pseudo-reflection, the quotient's stringy motive equals the motivic integral over Δ_G of L^{d-v}, generalizing the known cyclic-prime-order case.","The motivic version of Bhargava's mass formula holds: ∫_{Δ_{S_n}}L^{-a}=Σ_{j=0}^{n-1}P(n,n-j)L^{-j}, with a the Artin conductor.","Crepant birational morphisms of stacky log pairs preserve stringy motives, including in characteristic dividing the stabilizer orders and for singular DM stacks.","Discrepancies of quotient singularities are computable from the integral: discrep(centers⊂X_sing;X)=d-1-max{dim X_sing, dim∫_{Δ_G\\{o}}L^{d-v}}, and convergence of the integral is equivalent to log terminality when a log resolution exists.","Specializing to characteristic zero recovers the classical McKay sum over conjugacy classes Σ L^{d-age(g)}, and specializing to tame stacks recovers the earlier tame motivic McKay correspondence."],"supporting_citations":[{"why":"Proves the well-definedness of the motivic integral over Δ_G that appears as the right-hand side of the wild McKay correspondence.","marker":"[TY19]"},{"why":"Introduced the shift function for wild linear actions and outlined the grand design of the general theory.","marker":"[Yas17a]"},{"why":"Proved the point-counting version of the wild McKay correspondence and supplied the argument translated here into the motivic Bhargava mass formula.","marker":"[Yas17b]"},{"why":"Identified the function v with Artin and Swan conductors, connecting the integral to classical ramification invariants.","marker":"[WY15]"},{"why":"Provides the moduli-theoretic foundations for Galoisian group schemes and formal torsors on which the untwisting construction rests.","marker":"[TY17]"},{"why":"Supplies the original change-of-variables framework for motivic integration that the paper generalizes to wild stacks.","marker":"[DL99]"},{"why":"States the mass formula whose motivic version is derived as a corollary of the wild McKay correspondence.","marker":"[Bha07]"}],"fun_headline_variants":["Wild McKay correspondence generalizes to all finite groups","Motivic McKay for all finite groups, any characteristic","Untwisting wild arcs proves McKay correspondence broadly","Wild quotient stringy motives via untwisted arcs","From cyclic to all finite groups: wild McKay resolved"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The formula's right-hand side presupposes that the integral over Δ_G is well defined in the completed Grothendieck ring; the paper invokes a companion result for that well-definedness rather than proving it here.","fun_headline_variants_meta":{"raw":{"variants":["Wild McKay correspondence generalizes to all finite groups","Motivic McKay for all finite groups, any characteristic","Untwisting wild arcs proves McKay correspondence broadly","Wild quotient stringy motives via untwisted arcs","From cyclic to all finite groups: wild McKay resolved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000541,"raw_usage":{"total_tokens":2549,"prompt_tokens":857,"completion_tokens":1692,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":1617}},"tokens_in":473,"tokens_out":1692,"duration_ms":12973,"temperature":1.0,"reasoning_tokens":1617,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:29:28.290611+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take G=Z/$p^{2}$ acting diagonally on $A^{2}$_k over a perfect field of characteristic p, compute the right side ∫$L^{{2-v}}$ by stratifying Δ_G according to ramification data, compute the left side from a log resolution or from the change-of-variables formula, and compare the two classes in the completed Grothendieck ring; any mismatch would refute the wild McKay correspondence in this case.","supporting_citations":[],"review_version":1}