{"id":"6740b125-f8c2-42a1-a065-d9ea2536a771","arxiv_id":"2507.01412","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For uniformly free, ergodic actions with spectral gap, the motivic coarse assembly map for the cone quotient O∞(X)//G fails to be an equivalence.","lead":"This paper proves that for certain group actions on spaces carrying an ergodic probability measure, the universal coarse motive of the associated cone quotient does not satisfy the coarse assembly property. It shows that a known failure in analytic K-homology is actually visible in a universal homotopy-theoretic framework.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 10.6 applies [MV23, Thm. 6.20] to a non-ample module; if that theorem requires ampleness, the class p is not constructed and Theorem 1.3 has no witness.","rationale":"The central claim of Theorem 1.3 is a non-membership statement in CMcass, witnessed by non-surjectivity of the coarse assembly map for KX on O∞(X)//G. The proof constructs an explicit class u in the cokernel using the Drutu-Nowak projection P̂. Every step after Section 10 inherits from Lemma 10.6: without P̂ ∈ C(Sq(X)//G,Ĥ,μ̂), the class p in (10.4) is undefined, and the trace computation 0 ≠ [(±1)^n] = τ((±1)[p′]) has no object to which it can be applied. The reader's concern about Assumption 1.3.7 concerns the existence of the measure, but even granting the measure, the module (Ĥ, μ̂) and the equality (10.2) are not fully justified: the paper notes the module is not ample, and it cites [MV23] for an equality that may require ampleness. In addition, the formula for μ̂ as a 'projection-valued measure' is written as a scalar measure, which is at best a typo and at worst a gap in the definition of the controlled Hilbert space. I therefore regard Lemma 10.6 as the weakest point in the argument. Other concerns, such as the reliance on the unpublished preprint [Bun25] or the unfinished 'bis hier' artifact, affect confidence and presentation but do not single out a step as specifically vulnerable. A conditional acceptance is appropriate: the result is plausible and the framework coherent, but Lemma 10.6 must be either verified against [MV23] or replaced by a proof that works directly with the stabilized ample module.","tokens_in":28408,"tokens_out":18292,"duration_ms":211601,"concrete_test":"Inspect the statement and proof of [MV23, Thm. 6.20] and verify whether it applies to the non-ample module (Ĥ, μ̂) from Construction 10.5, or whether it requires the module to be ample. Independently, spell out the definition of the module structure: if μ̂ is meant as a projection-valued measure on L²(Z×X, δ×ν), replace the unclear formula by an explicit decomposition of Ĥ over the partition (B_i) and check whether P̂ is a finite-propagation operator with respect to the resulting controlled structure; if the projection-valued measure is instead defined on L²(μ̂), then Ĥ must be replaced by that Hilbert space and all subsequent L²-index computations revisited.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.3 depends on the class p in π₀KX(Sq(X)//G) built in Section 10. Its existence requires Lemma 10.6, which asserts P̂ ∈ C(Sq(X)//G, Ĥ, μ̂). The proof of Lemma 10.6 invokes equation (10.2), an equality C = C^{fp∩lc} imported from [MV23, Thm. 6.20], for the module (Ĥ, μ̂), which the paper explicitly notes is not ample. If [MV23, Thm. 6.20] assumes ampleness or additional bounded-geometry hypotheses that (Ĥ, μ̂) does not satisfy, then the equality (10.2) is unavailable and P̂ need not lie in the Roe algebra. Moreover, μ̂ is defined as a 'finitely additive projection-valued measure' via ∑ ν(B_i)δ_{b_i}; since δ_{b_i} is a scalar Dirac measure, not a projection on Ĥ = L²(Z×X, δ×ν), the module structure itself is under-specified. Without a well-defined P̂ in the Roe algebra, the class p of (10.4) cannot be constructed, and the trace/ghost contradiction in Section 9 has no witness, so Theorem 1.3 is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper adapts the recent counterexample construction of Kitsios–Schick–Vigolo for warped cones to the motivic coarse geometry framework of Bunke–Engel. Its main result, Theorem 1.3, states that for a uniform bornological coarse G-space X satisfying boundedness, a good uniform scale, finite Assouad–Nagata dimension, uniform freeness, finite asymptotic dimension of the group, and the existence of an ergodic invariant non-atomic probability measure with a spectral gap, the coarse motive of the cone quotient O∞(X)//G does not belong to the localizing subcategory CMcass where the motivic coarse assembly map is an equivalence. The proof constructs a nontrivial class p in π0KX(Sq(X)//G) from a Drutu–Nowak projection, proves that the corresponding relative class is a ghost, and combines an L2-index theorem for sequence spaces (Theorem 8.1) with transfer maps along branched coarse G-coverings to get a contradiction with surjectivity of the assembly map in degree one.","tokens_in":1976,"tokens_out":2364,"duration_ms":182023,"significance":"If correct, the paper gives substantial evidence for the usefulness of the motivic framework: it provides a version of the L2-index theorem (Theorem 8.1) adapted to the coarse assembly map in the Bunke–Engel formalism, clarifies the role of transfers in this context, and produces motivic counterexamples to the coarse Baum–Connes surjectivity that are not formally deducible from the classical analytic results of [KSV25]. The paper is carefully structured and many of its technical steps, particularly the motivic cone calculation in Proposition 4.3 and the controlled Hilbert space construction, are argued in detail. However, the construction of the witness class p in Section 10 relies on the module-theoretic Lemma 10.6, and as written that lemma is not sufficiently supported: the module (Ĥ, μ̂) is explicitly non-ample and the definition of μ̂ appears incoherent. Since p is the load-bearing input for the contradiction in Section 9, these issues must be repaired before the main theorem can be regarded as established.","major_comments":[{"comment":"The proof of Lemma 10.6 imports the equality C(X//G, Ĥ, μ̂) = C^{fp∩lc}(X//G, Ĥ, μ̂) from [MV23, Thm. 6.20] and applies it to the module (Ĥ, μ̂), which the paper itself states is not ample. If the cited theorem requires ampleness or further bounded-geometry hypotheses that fail for (Ĥ, μ̂), then (10.2) is unavailable and one cannot conclude that P̂ belongs to the Roe algebra. This is not a cosmetic gap: the class p in (10.4) exists only if P̂ lies in C(Sq(X)//G, Ĥ, μ̂), and without p the contradiction argument in Section 9 has no witness. Please state the precise hypotheses of [MV23, Thm. 6.20] and verify them for the stabilized module (Ĥ s, μ̂ s), or provide a direct proof that P̂ s is a norm limit of controlled locally compact operators on an ample module.","section":"Section 10, Lemma 10.6 and equation (10.2)"},{"comment":"The displayed formula μ̂(Y) := Σ_i ν̂(B_i)δ_{b_i} is not a well-defined projection-valued measure on Ĥ = L²(Z×X, δ×ν). The symbols δ_{b_i} are Dirac measures at the base points, hence scalars, not projections on Ĥ; if they were intended as rank-one projections attached to vectors δ_{b_i}, those vectors do not belong to L²(Z×X, δ×ν) because ν is non-atomic. Moreover, a sum weighted by the positive numbers ν̂(B_i) cannot be projection-valued unless all weights are 0 or 1. Consequently the Roe algebra C(Sq(X)//G, Ĥ, μ̂) is not defined by the given data. The relation between the module (Ĥ, μ̂) and the measure-module (Ĥ, ν̂) used in Remark 10.4 is also not explained. The module structure must be specified precisely (for example, through a disintegration of L² into finite-dimensional fibers over a discrete set of base points) before Lemma 10.6 can be assessed.","section":"Section 10, definition of μ̂ before Lemma 10.6"},{"comment":"Remark 10.4 asserts that P̂ ∈ C^{fp∩lc}(Sq(X_disc)//G, Ĥ, ν̂), where ν̂ is the product measure δ×ν, while Lemma 10.6 works with the different module data (Ĥ, μ̂). If ν̂ and μ̂ determine different module structures, then the controlledness of the operators ρ̂(g) and the local compactness of P̂ must be checked separately for the structure used in Lemma 10.6; the current text does not identify the two modules or prove that the relevant properties transfer from one to the other. This ambiguity affects the definition of the class p and therefore the proof of Theorem 1.3.","section":"Section 10, Remark 10.4 and Lemma 10.6"}],"minor_comments":[{"comment":"The statement q(X//G) ≃ colim_BG q(X) involves a colimit in the homotopy category BCh; please specify whether this is a homotopy colimit and, if so, in which model structure or ∞-categorical sense.","section":"Section 2, Lemma 2.5"},{"comment":"There is a duplicated phrase: “satisfies satisfies f(WG) = W”; please remove the repetition.","section":"Section 5, after Definition 5.8"},{"comment":"In the sentence “In the follwoing we discuss its essential properties,” the word “following” is misspelled.","section":"Section 10, last paragraph"},{"comment":"The text contains “its is natural to ask”; the correct wording is “it is natural to ask.”","section":"Section 1, paragraph after Theorem 1.3"},{"comment":"The notation “bX-controlled” is used without definition; since the Roe algebra in this paper is formed over Sq(X)//G, please clarify whether bX is the underlying bornological coarse space and what “controlled” means in this context.","section":"Section 10, Remark 10.4"},{"comment":"The reference [Saw] is given as a bare URL with no year; please add a year, a preprint number, or a published version if available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper relies substantially on the author's own unpublished or very recent preprints ([Bun25], [BE25]) and on [MV23], which is also a preprint. For a journal publication, the editors may want to confirm that [Bun25] and [MV23] are publicly available and, if possible, that the specific results cited (especially [MV23, Thm. 6.20]) apply at the level of generality claimed. The central mathematical idea is attractive, but the module-theoretic construction in Section 10 needs a serious revision before the main theorem can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a real result, not a repackaging. Bunke proves that the coarse motive of the warped-cone quotient O∞(X)//G is not in CMcass under a spectral gap assumption, which is strictly stronger than the classical analytic K-homology failure in KSV25. The translation is nontrivial: the classical assembly map is not a natural transformation between coarse homology theories, so you cannot just cite KSV25. The paper builds the machinery to do it in the motivic world.\n\nWhat is good: the proof is detailed and carefully structured. Sections 8–10 contain a new L²-index theorem for sequence spaces adapted to the motivic assembly map, and the use of transfers along branched coarse G-coverings is substantial. The paper is honest about what it takes from [Bun25] and what it proves here.\n\nThe soft spots: (1) Heavy dependence on [Bun25], the author's own unpublished preprint. That is normal but makes the proof hard to check. (2) Lemma 10.6 invokes [MV23, Thm. 6.20] for a module that is explicitly not ample. The paper claims the proof applies, but no details are given. If that theorem really needs ampleness, the construction of the class p collapses. A referee must check this. (3) The text has an unfinished-editing tell: the \"bis hier\" in Remark 6.7, which should not be in a final version. (4) The definition of the projection-valued measure μ̂ in Section 10 is under-specified as written; the δ_{b_i} notation looks like a typo for something like the rank-one projection onto the characteristic function of B_i. This is fixable but should be clarified.\n\nNone of these are, on their own, proof of a fatal flaw. The central structure of the argument in Section 9 is sound assuming the technical lemmas. The paper is for people working in coarse geometry, K-theory, and assembly maps; it deserves a serious referee.\n\nRecommendation: send it to refereeing, with the request to check [Bun25] and the application of [MV23] carefully, and to ask the author to clean up the editing and the module definition.","headline":"A serious motivic translation of the warped-cone counterexamples, with a plausible but check-dependent proof and one technical point that needs a careful referee.","tokens_in":29206,"tokens_out":3374,"would_cite":true,"duration_ms":38330,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19K56","46L80","51F30"],"pacs":[],"model":"deepseek-v4-flash","headline":"For uniform bornological coarse spaces with a sufficiently ergodic probability measure, the paper proves that the coarse assembly map on the cone quotient $\\mathcal{O}^{\\infty}(X)//G$ is not an equivalence.","keywords":["coarse geometry","bornological coarse spaces","coarse motives","coarse assembly map","coarse K-homology","warped cones","spectral gap","ergodic group actions"],"falsifier":"If for any $X$ satisfying the seven assumptions one could construct $w\\in\\pi_2 KX(\\mathcal{O}^{\\infty,\\mathrm{strg}}P(\\mathcal{O}^{\\infty}(X)//G))$ with $\\mu(w)=u$ for the class $u$ of Section 9, the theorem's conclusion would be wrong. Equivalently, computing the Drutu-Nowak projection for the $SU(2)$ example and showing its transfer $f^*[\\hat P]$ is non-zero, or showing that the tail traces in (8.2) are unequal, would break the contradiction on which the proof rests.","tokens_in":28144,"feed_emoji":"🌀","tokens_out":9227,"duration_ms":207965,"temperature":0.7,"pith_summary":"This paper proves that the coarse cone quotient $\\mathcal{O}^{\\infty}(X)//G$ of a uniform bornological coarse space $X$ with a $G$-action can have a coarse motive that the motivic coarse assembly map does not control. Under seven hypotheses, including uniform freeness of the action, finite Assouad-Nagata dimension of suitable scales, finite asymptotic dimension of $G$, and the existence of an invariant non-atomic probability measure on which $G$ acts ergodically with a spectral gap, the paper shows that $Yo(\\mathcal{O}^{\\infty}(X)//G)$ is not in $\\mathcal{CM}^{\\mathrm{cass}}$, the localizing subcategory of coarse motives on which the assembly map is an equivalence. This matters because these quotient cones are the bornological-coarse analogue of warped cones: the large-scale geometry of the quotient remembers the representation-theoretic complexity of the action. A concrete class in coarse $K$-homology, built from the Drutu-Nowak projection, is exhibited outside the image of assembly.","feed_headline":"Ergodic group actions defeat the coarse assembly map","feed_subtitle":"With a spectral-gap measure, the motivic assembly map for the quotient cone is not an equivalence.","key_machinery":"The load-bearing construction is the Drutu-Nowak projection $\\hat P$, the orthogonal projection onto functions on $\\mathbb{Z}\\times X$ that are constant on each component $\\{n\\}\\times X$. Ergodicity plus spectral gap makes $\\hat P$ a spectral projection of the averaging operator $\\hat M_S$, so it lives in the Roe algebra of the quotient cone (Lemma 10.6); stabilisation turns it into a coarse $K$-homology class $p$ whose components all have trace $1$. Around this class the paper assembles a machinery of branched coarse $G$-coverings and transfers: the map $f:(G\\ltimes Sq(X)_V)//G\\to Sq(X)_V//G$ is a branched coarse covering, and Theorem 8.1, an $L^2$-index theorem for sequence spaces, says that classes in the image of assembly have equal tail traces before and after transfer. Lemma 10.8 says $\\hat P^s$ is a ghost, so its transfer vanishes; the clash between the non-zero trace and the zero transfer is what proves non-surjectivity.","core_discovery":"The central claim is Theorem 1.3: if $X$ is bornologically bounded with maximal coarse structure, every uniform entourage admits a finite dense subset, $X$ has a uniform scale dominated by Lipschitz scales of finite Assouad-Nagata dimension, the $G$-action is uniformly free, $G$ is finitely generated of finite asymptotic dimension, and $X$ carries a $G$-invariant non-atomic Borel probability measure of full support that is ergodic with spectral gap, then $Yo(\\mathcal{O}^{\\infty}(X)//G)\\notin\\mathcal{CM}^{\\mathrm{cass}}$. The proof constructs a class $u\\in\\pi_1 KX(\\mathcal{O}^{\\infty}(X)//G)$ that, if it were in the image of the assembly map, would contradict the $L^2$-index trace identity of Theorem 8.1: the trace of the transferred class would have to be both the non-zero sequence $(\\pm1)^n$ and zero, because the Drutu-Nowak projection representing the class is a ghost operator. Consequently the coarse $K$-homology assembly map is not surjective, and the motive is not one on which assembly is an equivalence. The argument also shows that $\\mathcal{O}^{\\infty}(X)//G$ and the squeezing quotient $Sq(X)//G$ do not have weakly finite asymptotic dimension.","pith_inferences":["Remark 10.7 indicates that the spectral-gap hypothesis can likely be weakened to strong ergodicity; if so, the same ghost-class argument would apply to actions that are strongly ergodic but may lack a spectral gap, broadening the counterexamples.","The bounded-geometry pullback question in Remark 7.3 becomes testable on these examples: if $\\mathcal{CM}^{\\mathrm{disc}}\\cap\\mathcal{CM}^{\\mathrm{bgeom}}$ were known to equal $\\mathcal{CM}^{\\mathrm{cass}}\\cap\\mathcal{CM}^{\\mathrm{bgeom}}$, then the constructed non-cass motive would have to be genuinely non-discrete in the strong category.","Replacing finite asymptotic dimension by the operator-norm localisation property in Theorem 8.1, as Remarks 8.4 and 8.5 suggest, would allow property-A groups and likely reproduce the original warped-cone counterexamples in the motivic category without finite asymptotic dimension.","The ghost-projection construction may be reusable on any uniformly free action with a strongly ergodic invariant measure, and the same class $p$ could distinguish coarse motives of different actions, a question the paper leaves open."],"forward_implications":["For $G$ a non-abelian free subgroup of $SU(2,\\overline{\\mathbb{Q}})$ acting on $X=SU(2)$, all hypotheses hold, so $Yo(\\mathcal{O}^{\\infty}(X)//G)\\notin\\mathcal{CM}^{\\mathrm{cass}}$ and the coarse $K$-homology assembly map is not surjective.","Because $Yo(\\mathcal{O}^{\\infty}_\\phi(X)//G)\\simeq Yo(\\mathcal{O}^{\\infty}(X)//G)$ for fixed decay rates, the usual Euclidean warped-cone quotients also fail to lie in $\\mathcal{CM}^{\\mathrm{cass}}$.","The quotient spaces $\\mathcal{O}^{\\infty}(X)//G$ and $Sq(X)//G$ admit no cofinal set of coarse entourages whose associated coarse structures have finite asymptotic dimension; they do not have weakly finite asymptotic dimension.","The motivic coarse assembly map for $\\mathcal{O}^{\\infty}(X)//G$ has a non-surjective homotopy group map in degree one, giving a negative result in the motivic setting that cannot be obtained by directly quoting the analytic warped-cone counterexamples."],"supporting_citations":[{"why":"Defines the motivic coarse assembly map and the cone-at-infinity formalism in which the paper's assembly statement is made.","marker":"[BE20a]"},{"why":"Provides the category of bornological coarse spaces, the coarse motive functors $Yo$ and $Yo^{\\mathrm{strg}}$, and the coarse $K$-homology theory $KX$.","marker":"[BE20b]"},{"why":"Supplies the original warped-cone counterexample ideas and the Drutu-Nowak class construction that this paper adapts to the motivic setting.","marker":"[KSV25]"},{"why":"Provides branched coarse $G$-coverings, transfer maps, ghost transfer vanishing, and the $L^2$-index theorem used in the contradiction.","marker":"[Bun25]"},{"why":"Its Lemma 6.5 is the analytic ancestor of Theorem 8.1, the tail-trace identity for classes in the image of the assembly map.","marker":"[WY12]"},{"why":"Introduces the Drutu-Nowak projection and the spectral-gap and ergodicity machinery used to construct the class $p$.","marker":"[DN17]"},{"why":"Gives the spectral gap for finitely generated non-abelian free subgroups of $SU(2)$, making Example 1.4 satisfy the assumptions.","marker":"[BG08]"},{"why":"Shows the topological coarse $K$-homology functor extends to a coarse homology with transfers, so transfers can act on the relevant classes.","marker":"[BE23]"}],"fun_headline_variants":["Spectral-gap measures break coarse assembly","Coarse assembly fails on ergodic cone quotients","Ergodic groups thwart motivic assembly map","No assembly equivalence for coarse cone quotients"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on one assumption: $X$ has a $G$-invariant probability measure with no atoms, full support, on which $G$ acts ergodically and with a spectral gap; without such a measure the constructed class $p$ is not known to exist, and the contradiction that proves the theorem cannot be run.","fun_headline_variants_meta":{"raw":{"variants":["Spectral-gap measures break coarse assembly","Coarse assembly fails on ergodic cone quotients","Ergodic groups thwart motivic assembly map","No assembly equivalence for coarse cone quotients"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1277,"prompt_tokens":920,"completion_tokens":357,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":299}},"tokens_in":536,"tokens_out":357,"duration_ms":4003,"temperature":1.0,"reasoning_tokens":299,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:51:58.548193+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"If for any $X$ satisfying the seven assumptions one could construct $w\\in\\pi_2 KX(\\mathcal{O}^{\\infty,\\mathrm{strg}}P(\\mathcal{O}^{\\infty}(X)//G))$ with $\\mu(w)=u$ for the class $u$ of Section 9, the theorem's conclusion would be wrong. Equivalently, computing the Drutu-Nowak projection for the $SU(2)$ example and showing its transfer $f^*[\\hat P]$ is non-zero, or showing that the tail traces in (8.2) are unequal, would break the contradiction on which the proof rests.","supporting_citations":[],"review_version":1}