{"id":"11955741-8758-4b4f-95e0-503e70b825f7","arxiv_id":"2511.22438","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Bounded-geometry metric spaces coarsely embeddable into ℓ^p have the property that geometric ideals in their Roe algebras have the same K-theory as the corresponding ghostly ideals.","lead":"A theorem in coarse index theory: for bounded-geometry metric spaces coarsely embeddable into ℓ^p spaces, the inclusion from any geometric ideal into the corresponding ghostly ideal in the Roe algebra induces an isomorphism on K-theory. This yields relative and maximal coarse Baum-Connes conjectures and an operator-norm localization property for such spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 3.15's asymptotic-closeness-to-K-theory step is asserted, not proved; it is the pivotal new bridge in Theorem 2.6 and should be checked explicitly.","rationale":"The reader's conditional verdict is reasonable. I focus on Corollary 3.15 because it is the non-black-box juncture unique to this paper: the final diagram chase depends on the inclusion A_I(U)(X,E)→A_G(U)(X,E) inducing a K-theory isomorphism. The paper asserts this from asymptotic closeness without providing the mapping-cone/homotopy argument. Lemma 3.2's misprint is real, but the intended statement is a known Mazur-map estimate from [WXYZ24], so I would not rest the main objection on it alone. The concrete check—supplying the missing K-theory argument or finding a counterexample—would settle whether the concern lands. Since this is essentially the same class of gap the reader flagged in the rationale, the conditional verdict should stand unchanged.","tokens_in":24824,"tokens_out":26336,"duration_ms":234147,"concrete_test":"Give a rigorous proof of Corollary 3.15 by explicitly constructing the K-theory isomorphism: either verify that [WY20, Lemma 6.4.11] applies to A_I(U)(X,E) and A_G(U)(X,E) by checking that its equivalence relation respects Definition 3.5(2) and the ε-support condition, and then write down the induced isomorphism on K_0 and K_1; or exhibit a concrete pair of subalgebras satisfying Lemma 3.14's approximation but with different K-theory, showing the cited implication is false. If the proof goes through, also correct Lemma 3.2's formula to Ψ(v)=∥v∥_p ψ(v/∥v∥_p) and supply or locate the Lipschitz estimate needed for Proposition 3.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The pivotal new step is Corollary 3.15: it turns Lemma 3.14's tail-uniform approximation into a K-theory isomorphism between A_I(U)(X,E) and A_G(U)(X,E). The proof is one sentence: K-theory 'only depends on asymptotic behavior as s→∞,' citing [WY20, Lemma 6.4.11], with details left to the reader. Lemma 3.14 alone produces, for each ε>0 and T∈A_G(U), some S and T'∈A_I(U) with sup_{s≥S}∥T_{s,n}−T'_{s,n}∥≤ε. This does not by itself supply a homotopy or show that the quotient A_G(U)/A_I(U) has vanishing K-theory; moreover, both algebras are constrained by the E-propagation condition in Definition 3.5(2), so the hypotheses of the cited lemma must be verified for these ideals. If the implication fails, the diagram chase after (3) does not prove Theorem 2.6. Secondary: Lemma 3.2 is visibly misprinted—ψ is defined on S_p(E) but evaluated at v/∥v∥_2, and the claimed Lipschitz bound L=1+p·2^p is not proved—but this is likely repairable by referencing [WXYZ24].","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that for a bounded-geometry metric space X admitting a coarse embedding into an ℓ^p-space (1≤p<∞), the inclusion i:I(X,U)→G(X,U) from any geometric ideal to the associated ghostly ideal induces an isomorphism on K-theory (Theorem 2.6). The proof follows the ℓ^p Bott–Dirac machinery of [WXYZ24] and [WY20]: it reduces to sparse spaces, constructs a twisted Roe algebra A(X,E), defines index maps with a dimension-dependent scale shift s_n=L(dim E_n)^2, restricts the construction to ideals, and attempts to show that the twisted geometric and ghostly ideals have the same asymptotic behaviour. From this, the paper derives the relative coarse Baum–Connes conjecture, the operator norm localization property ONL_PFin, and a maximal coarse Baum–Connes statement.","tokens_in":25088,"tokens_out":4193,"duration_ms":37842,"significance":"If the main theorem is correct, it is a substantial extension of the Hilbert-space result of [WFZ25] and of earlier ghost-ideal K-theory results, and it provides a uniform route to several corollaries that are currently open in general. The paper is honest about its reliance on prior ℓ^p machinery, and the high-level strategy is coherent. The main weakness is that the pivotal new step (Corollary 3.15) is only sketched, and some supporting technical claims (Lemma 3.2, Proposition 3.13) are either misprinted or insufficiently proved. The result is plausible and likely repairable, but the manuscript as it stands leaves load-bearing details to the reader.","major_comments":[{"comment":"This is the pivotal new step of the paper, but its proof is one sentence: the K-theory of A_I(U)(X,E) and A_G(U)(X,E) 'only depends on their asymptotic behavior as s→∞', citing [WY20, Lemma 6.4.11], and Lemma 3.14 is said to show they have the same asymptotic behaviour. Lemma 3.14 only produces, for each element T and ε>0, a tail approximation by an element T' of A_I(U)[X,E] with sup norm ≤ε. It does not supply a homotopy, a mapping cone argument, or a verification that the hypotheses of the cited lemma are satisfied for these ideals, which are simultaneously constrained by the E-propagation condition in Definition 3.5(2). The diagram chase after (3) depends entirely on this implication. This needs to be proved explicitly.","section":"§3.4, Corollary 3.15"},{"comment":"As printed, the formula for the extended Mazur map is not well-defined: ψ is defined on the ℓ^p unit sphere S_p(E), but the displayed formula evaluates ψ at v/||v||_2, which lies in S_2(E). Also, the asserted Lipschitz bound L=1+p·2^p for p≥2 is not proved in this paper and is only referred to [WXYZ24, Lemma 2.6]. This lemma is load-bearing: it enters Proposition 3.4, Definition 3.6 (via s_n=L(dim E_n)^2), and the spectral estimates. The authors should either give the correct formula and proof or quote the precise statement from the literature.","section":"§3.1, Lemma 3.2"},{"comment":"The proof of Proposition 3.13, which asserts that the evaluation map after the index map is an isomorphism on ideals, is only sketched and contains a serious technical issue in the 'stacking argument'. The path P_N(t) defined near the end of the proof is written as a difference of two diagonal block projection matrices; the individual diagonal entries of this difference are differences of projections, and as written P_N(t) is not a projection-valued path. Consequently [P_N(t)] is not defined in K-theory. Since this proposition is used to identify the vertical maps in diagram (3), it must be repaired or replaced by a complete argument.","section":"§3.3, Proposition 3.13"},{"comment":"The maximal coarse Baum–Connes conjecture is asserted after a paragraph of informal discussion, with statements such as 'one can verify that the proofs in [WY20, Section 12.3 & 12.4] also hold' and 'combining these observations, we obtain'. No detailed proof is supplied. Corollary 1.4 depends on Theorem 4.7. If the maximal result is to be claimed, it needs a real proof; otherwise it should be explicitly marked as conditional or deferred.","section":"§4.2, Theorem 4.7 and Corollary 1.4"}],"minor_comments":[{"comment":"The internal cross-references are inconsistent: Theorem 2.6 is called 'Definition 2.6', Proposition 3.13 is called 'Definition 3.13', and Lemma 3.14 is called 'Definition 3.14'. This makes the paper difficult to read.","section":"Throughout"},{"comment":"The equality A[X,E]∩A_G(U)(X,E)=A_I(U)[X,E] is asserted without proof in the last lines of the lemma. It is plausible but requires justification, since A_I(U)[X,E] was defined by generators with supports in U and the intersection statement is not immediate.","section":"§3.4, Lemma 3.14"},{"comment":"The homogeneous extension of the Mazur map is written with Ψ:(E,||·||_p)→(E,||·||_2), but the display evaluates ψ at v/||v||_2. In addition, the normalization factor ||v||_p appears to be inconsistent with the claimed Lipschitz estimate for ||Ψ(v)-Ψ(u)||_2 with respect to ||v-u||_p. This should be cleaned up.","section":"§3.1, display before Proposition 3.4"},{"comment":"The sentence 'Note that f_n(X_n)⊂E_n is uniformly bounded by M∈N' is unclear: M is used both as a natural number and as a bound, and the conclusion χ_K p_n^(k)=χ_K p_n^(∞) for k≥M depends on this bound. The notation should be clarified.","section":"§3.3, proof of Proposition 3.13"},{"comment":"The argument for non-separable L^p-spaces is informal. The claim that 'the family of all finite-dimensional subspaces of any L^p-space admits a uniformly coarse embedding into ℓ^p' is not proved and is not an immediate consequence of finite representability as stated. This remark is not used in the main proof, but the wording should be softened or the argument expanded.","section":"§3, Remark 3.18"},{"comment":"There are typographical issues in the references and text, e.g. 'P . Nowak' and the use of 'Definition 2.6' for the main theorem. The references [WZ25], [WFZ25], [GWZ25], and [WXYZ24] should be cited with precise theorem/lemma numbers where they are used.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper's core idea is a natural extension of existing ℓ^p machinery, and the main theorem is likely correct. However, the proof of Corollary 3.15 is too thin for the pivotal step, Lemma 3.2 is visibly misprinted, and Proposition 3.13 contains a flawed stacking homotopy. These are fixable in principle, but they are load-bearing, so the revision should be checked carefully. The maximal coarse Baum–Connes assertion is also much more thinly supported than the title/abstract suggests."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The result is real and worth taking seriously. For metric spaces of bounded geometry that coarsely embed into some ℓ^p (or L^p) space, the paper proves that every geometric ideal I(X,U) and its ghostly counterpart G(X,U) have isomorphic K-theory. This genuinely extends the Hilbert-space case of Finn-Sell and of Wang–Fu–Zhang, and the full-Roe-algebra case treated by WXYZ24. It also brings the relative and maximal coarse Baum–Connes statements and ONL_{P_Fin} as corollaries. If the proof is solid, this is a substantial step, especially because K-amenability of the coarse groupoid is still open for ℓ^p-embeddable spaces. The architecture is sensible: reduce to sparse spaces, restrict the ℓ^p Dirac-dual-Dirac construction to the ideals, then compare the twisted geometric and ghostly ideals. The authors give real credit where the machinery comes from and do not overclaim novelty.\n\nThe soft spots are real but mostly local. The biggest is Corollary 3.15. Lemma 3.14 shows that every element of A_G(U) can be approximated, asymptotically in s, by an element of A_I(U). But the jump from that approximation to a K-theory isomorphism is made in a single sentence, citing WY20 Lemma 6.4.11, with details left to the reader. That is the pivotal new step in Theorem 2.6, and it is not obvious: one needs a homotopy or an evaluation-at-infinity argument that actually identifies the K-theory of the two twisted ideals, and the E-propagation conditions in Definition 3.5(2) have to be checked. This should be written out.\n\nThe other issues are minor but should be cleaned up. Lemma 3.2 has a visible typo: the extended Mazur map Ψ is evaluated at v/‖v‖_2 even though the domain is the ℓ^p sphere, and the global Lipschitz bound L = 1 + p·2^p is stated without proof. That is likely repairable by citing the corresponding estimates from WXYZ24, but as printed it is a hole in a lemma the whole Bott-Dirac estimate depends on. Proposition 2.12 also relies on an external sparse-cover lemma without a precise citation, and Theorem 4.7 on the maximal conjecture is more sketch than proof. None of these look fatal; they are gaps in presentation and verification, not obvious contradictions.\n\nWho is this for? Anyone working in coarse index theory, ghost ideals, or the coarse Baum–Connes family of conjectures. The paper deserves a serious referee, not a desk rejection, provided the referee is willing to chase down the WXYZ24 machinery. I would want the Corollary 3.15 argument supplied, Lemma 3.2 repaired, and the maximal-version proof expanded before I would call it settled.","headline":"A genuinely new K-theory isomorphism for ℓ^p-embeddable spaces, built on the ℓ^p Bott–Dirac machine; the main idea is right, but the pivotal Corollary 3.15 is asserted rather than proved and should be fixed before this is accepted as written.","tokens_in":25679,"tokens_out":964,"would_cite":true,"duration_ms":10674,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19K56","47L20"],"pacs":[],"model":"deepseek-v4-flash","headline":"In ℓ^p-embeddable spaces, geometric and ghostly ideals always have the same K-theory.","keywords":["coarse geometry","Roe algebra","ghost ideal","K-theory","ℓ^p-spaces","coarse Baum-Connes conjecture","operator norm localization","Mazur map"],"falsifier":"Compute the Lipschitz constant of the extended Mazur map on finite-dimensional ℓ^p_n for p>2 and large n: if the ratio ||Ψ(v)-Ψ(u)||_2 / ||v-u||_p ever exceeds the bound asserted in Lemma 3.2 (or grows with n), the dimension-uniform estimates in Proposition 3.4 break and the index map into the twisted Roe algebra is not defined. Alternatively, find a bounded-geometry ℓ^p-embeddable space (p>2) not embeddable into Hilbert space and check whether K_*(G(X,U)/I(X,U)) vanishes for some invariant open U; any nonzero group would contradict the theorem.","tokens_in":24638,"feed_emoji":"👻","tokens_out":10605,"duration_ms":78725,"temperature":0.7,"pith_summary":"This paper proves that for any metric space with bounded geometry that admits a coarse embedding into some ℓ^p-space (1 ≤ p < ∞), the ghostly ideal in the Roe algebra carries no extra K-theory beyond its geometric sub-ideal: the inclusion induces an isomorphism in K-theory, equivalently the quotient has zero K-theory. Previously this was known only for spaces coarsely embeddable into Hilbert space, via K-amenability of the coarse groupoid; here that tool is replaced by an ℓ^p Bott-Dirac construction, which is restricted to the geometric and ghostly ideals inside a twisted Roe algebra. A sympathetic reader should care because the theorem, if correct, settles a natural family of higher-index conjectures for all ℓ^p-embeddable bounded-geometry spaces: the relative coarse Baum-Connes conjecture, the maximal coarse Baum-Connes conjecture, and the operator norm localization property for finite-rank projections. The argument reduces the problem to sparse subspaces and then shows that in the twisted setting the ghostly and geometric ideals have the same asymptotic behavior, forcing the K-theory isomorphisms.","feed_headline":"K-theory sees no ghosts in ℓ^p-embeddable spaces","feed_subtitle":"It proves the relative coarse Baum-Connes conjecture and a finite-rank localization property for these spaces.","key_machinery":"The machinery is the ℓ^p Dirac-dual-Dirac construction, transplanted from Hilbert-space coarse index theory to a twisted Roe algebra. On each finite-dimensional ℓ^p approximation E_n, one uses the extended Mazur map Ψ between the ℓ^p and Euclidean unit spheres to define a Bott generator C_v and then a Bott-Dirac operator B_{s,v} = s^{-1}D + C_v on L^2(E_n) with Clifford coefficients. Its bounded transform Φ(B_{s,v}) is packaged into a twisted Roe algebra A(X,E) of functions of the zoom parameter s; the crucial input is a set of dimension-uniform estimates on the propagation and Lipschitz continuity of Φ(B_{s,v}) in the ℓ^p direction, which let the index map descend to ideals and eventually e","core_discovery":"The central claim is Theorem 2.6: if a bounded-geometry metric space X coarsely embeds into an ℓ^p-space (p ∈ [1,∞)), then for every invariant open U ⊆ βX the inclusion I(X,U) → G(X,U) induces a K-theory isomorphism; equivalently, G(X,U)/I(X,U) has trivial K-theory. The proof generalizes the ℓ^p Dirac-dual-Dirac construction: after reducing to sparse subspaces, it builds a twisted Roe algebra and shows the geometric and ghostly ideals coincide there asymptotically. Consequences drawn in the paper: the relative coarse Baum-Connes conjecture for all subspaces, the finite-rank operator norm localization property, and maximal-versus-reduced Roe algebra K-theory isomorphisms.","pith_inferences":["We infer that the twisted Bott-Dirac framework may give a direct route to K-amenability of the coarse groupoid for ℓ^p-embeddable spaces; the paper leaves this as an open question, but its maximal-versus-reduced K-theory isomorphisms are the usual signature of K-amenability.","Going beyond the paper, the equivalence it proves for sparse subspaces suggests ONL_PFin might itself be characterized by vanishing of K_*(G(X)/I(X)) for the full ghostly ideal, giving a testable higher-index criterion for spaces of unknown embeddability.","Because ℓ^p- and Hilbert-space embeddability are equivalent for p ≤ 2, the theorem is only genuinely new for p > 2; it therefore predicts the desired K-theory vanishing in the one regime where no separating examples are yet known."],"forward_implications":["For any ℓ^p-embeddable bounded-geometry space, the relative coarse Baum-Connes conjecture holds for every subspace Y ⊆ X, including the boundary coarse Baum-Connes conjecture.","Such spaces have the operator norm localization property for equi-approximable finite-rank projections (ONL_PFin), meaning ghost projections on sparse subspaces are forced to be compact.","The maximal coarse Baum-Connes conjecture holds for these spaces, and the canonical quotient maps from maximal to reduced Roe algebras (and their relative versions) induce K-theory isomorphisms.","For every invariant open U, the K-theory of the ghostly quotient G(X,U)/I(X,U) vanishes, so ghosts are invisible to K-theory in all directions.","The main result also extends from ℓ^p to general L^p-spaces, including non-separable ones (Theorem 3.17)."],"fun_headline_variants":["ℓ^p spaces: ghost ideals K-theoretically trivial","Coarse ℓ^p embedding erases ghost K-theory","K-theory: no ghostly gap for ℓ^p-embeddable spaces","ℓ^p coarsely embeddable: ghosts match in K-theory","Ghostly K-theory collapses for ℓ^p-embeddable spaces"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on a dimension-uniform bound quantifying how much the ℓ^p-to-Euclidean sphere map stretches distances; the paper asserts this bound but the key formula is not proved here, and without it the main construction does not go through.","fun_headline_variants_meta":{"raw":{"variants":["ℓ^p spaces: ghost ideals K-theoretically trivial","Coarse ℓ^p embedding erases ghost K-theory","K-theory: no ghostly gap for ℓ^p-embeddable spaces","ℓ^p coarsely embeddable: ghosts match in K-theory","Ghostly K-theory collapses for ℓ^p-embeddable spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000159,"raw_usage":{"total_tokens":1018,"prompt_tokens":646,"completion_tokens":372,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":390,"completion_tokens_details":{"reasoning_tokens":272}},"tokens_in":390,"tokens_out":372,"duration_ms":4075,"temperature":1.0,"reasoning_tokens":272,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T19:47:37.327677+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Lipschitz constant of the extended Mazur map on finite-dimensional ℓ^p_n for p>2 and large n: if the ratio ||Ψ(v)-Ψ(u)||_2 / ||v-u||_p ever exceeds the bound asserted in Lemma 3.2 (or grows with n), the dimension-uniform estimates in Proposition 3.4 break and the index map into the twisted Roe algebra is not defined. Alternatively, find a bounded-geometry ℓ^p-embeddable space (p>2) not embeddable into Hilbert space and check whether K_*(G(X,U)/I(X,U)) vanishes for some invariant open U; any nonzero group would contradict the theorem.","supporting_citations":[],"review_version":1}