{"id":"f12775b2-78ce-42fd-9f8f-54cf6d5e73bc","arxiv_id":"2608.04297","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Homological Real trace methods are developed, yielding spectral sequences that compute the continuous mod two Bredon homology of Real topological negative cyclic homology for Real bordism and related spectra.","lead":"This paper develops new spectral sequence tools for computing the mod two homology of Real topological negative cyclic homology, building on earlier work of Bruner and Rognes in the classical setting. The tools are used to compute the continuous mod two Bredon homology of Real bordism and truncated Real Brown-Peterson spectra, which are stepping stones toward computations in Real algebraic K-theory.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7.12 for BP_R<n> rests on Theorem 6.1(b), whose proof depends on the unproved extended-power homology computation [Wil19, Prop. 2.4.1]; if that input is wrong, the stated E∞-page and Corollary 7.13 fail.","rationale":"The reader's weakest assumption identifies exactly the step I would stress. Theorems 7.4 and 7.6 use only Theorem 6.1(a), which depends on published results, whereas Theorem 7.12 uses Theorem 6.1(b) for the τ-classes, and the proof of that case depends on [Wil19, Prop. 2.4.1], which the paper itself notes is unproved and only sketched in the preprint [CGP25, Prop. 5.6]. This is load-bearing because the entire differential analysis in the universal example determines the permanent cycles that are pushed forward to the BP_R<n> spectral sequence. If the extended-power homology computation is incorrect or incomplete, the E∞-algebra in Theorem 7.12 and Corollary 7.13 would not be justified. The paper is transparent about this dependency, and the main construction of the spectral sequence as well as the MU_R/BP_R computations appear sound; but the BP_R<n> computation should be regarded as conditional on the missing proof. I would therefore keep the reader's CONDITIONAL verdict rather than upgrade or downgrade it. The proposed concrete test directly settles whether the unproved input lands by recomputing the first nontrivial extended-power homology group from first principles.","tokens_in":31091,"tokens_out":15319,"duration_ms":148863,"concrete_test":"Independently compute H_⋆(Dbar_2(S(C^1)_+ ∧ S^{ρ+1})) for the first nontrivial instance of case (b), k=1 and r=1, by building the C2-equivariant cell filtration of the extended power and running the C2-equivariant Künneth spectral sequence, without invoking [Wil19, Prop. 2.4.1]. Compare the resulting span and relations with the asserted generators xδx, Q_{ρ-1}(x), Q_{ρ}(x), Q_{2ρ}(δx), Q_{2ρ+σ}(δx), and their A^{C2}_⋆-module structure. If the output differs, Theorem 6.1(b) fails; if it matches, repeat for r=2. A complementary check is to audit [CGP25, Prop. 5.6] to confirm it is a complete proof rather than a sketch, and re-derive its E2-page and differentials.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the proof of Theorem 6.1(b), which is the only mechanism producing permanent cycles from the τ-classes in the BP_R<n> computation. For |x| = kρ+1, the proof takes the F2-homology of X = Dbar_2(S(C^r)_+ ∧ S^V) to be generated by xδx, Q_{iρ-ε}(x), and Q_{jρ+εσ}(δx) with the stated index ranges, citing [Wil19, Proposition 2.4.1]. The text explicitly says Wilson states this computation without proof and defers a sketch to [CGP25, Proposition 5.6]. This is not a cosmetic gap: the differential computation in the universal example, and hence the permanent-cycle list in Theorem 6.1(b), depends on that additive basis. The MU_R and BP_R computations (Theorems 7.4 and 7.6) use only case (a), which rests on the published [Wil17, Thm. 2.15]. But Theorem 7.12 and Corollary 7.13 for BP_R<n> use case (b) for the τ_{n+i} classes, e.g. d_2(τ_{n+i}) = ybar·xbar_n^{2^{i-1}} and the claimed permanent cycles Q_{2^{n+i-1}ρ+σ}(τ_{n+i}) and Q_{2^{n+i}ρ}(τ_{n+i}) + τ_{n+i}·σbarτ_{n+i}. If [Wil19, Prop. 2.4.1] has a hidden sign, a missing generator, or a wrong index range, the stated E∞-algebra in Theorem 7.12 would not follow. The paper flags the dependence honestly, but does not supply the proof, so the strongest computational claim is conditional on an external unproved statement.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a genuine C2-equivariant analogue of Bruner and Rognes' homological trace methods. For a C2-spectrum with twisted T-action X, the authors construct a conditionally convergent spectral sequence of A^{C2}_*-comodules E2 = H_*(X)⟨ȳ⟩ ⇒ H^c_*(X^{hC2T}; F2), and show that for E∞-algebras it is multiplicative with the relation ȳa = μ_*(a)ȳ. They identify the d2-differential via a Connes-type operator, prove a general permanence theorem (Theorem 6.1) for Dyer–Lashof-derived classes under a nontrivial d_{2r}-differential, and apply it to compute E∞-terms for TCR−(MU_R), TCR−(BP_R), and, under a standing C2-E∞ hypothesis and for −1 ≤ n ≤ 2, TCR−(BP_R⟨n⟩), together with geometric fixed-point variants. The spectral sequence construction and the MU_R/BP_R computations are written in detail, but the BP_R⟨n⟩ results depend on an unproved extended-power homology computation and on an unspecified torsion summand in the stated E∞-terms.","tokens_in":31520,"tokens_out":11139,"duration_ms":107948,"significance":"The construction of the homological parametrized homotopy fixed point spectral sequence is a new and potentially useful tool, and the comparison map to the geometric fixed point spectral sequence is a valuable feature. If the identified gaps are filled, the MU_R and BP_R computations would be solid and would provide the first explicit computations of continuous Bredon homology of Real negative cyclic homology for these examples. The paper is honest about its dependencies: it explicitly flags that [Wil19, Prop. 2.4.1] is stated without proof, that a sketch is deferred to [CGP25, Prop. 5.6], and that BP_R⟨n⟩ is only known to be C2-E∞ for −1 ≤ n ≤ 2. The main risk is not circularity but incompleteness of an external computational input.","major_comments":[{"comment":"The proof of case (b) is entirely dependent on the additive description H_*(X) ≅ H_*{xδx, Q_{iρ−ϵ}(x), Q_{jρ+ϵσ}(δx) | i ≥ k+1, j ≥ k+r, ϵ ∈ {0,1}} for |x| = kρ+1, quoted from [Wil19, Proposition 2.4.1]. The text immediately notes that Wilson states this computation without proof and refers to [CGP25, Proposition 5.6] for a sketch. This description is load-bearing for Theorem 7.12 and Corollary 7.13, because the permanent cycles for the τ_{n+i} classes are read off from these generators after applying the differential formula. A missing generator, an index-range error, or a sign error would change the E∞-page. Please supply a complete proof, or cite a published proof, or alternatively restrict the BP_R⟨n⟩ claims to the part that uses only case (a), which rests on the published [Wil17, Thm. 2.15].","section":"Section 6, proof of Theorem 6.1(b)"},{"comment":"Each of these statements ends with 'plus a simple ¯y-torsion module in filtration zero' (respectively 'simple z-torsion module'). The torsion summand is never described, and the term 'simple' is not defined. Since the theorems purport to determine the E∞-term, an unspecified summand means the computation is incomplete. Please identify this module, or prove that it is zero, or state explicitly what 'simple' means and how the module is determined by the convergence and the rest of the page.","section":"Theorems 7.4, 7.6, 7.12 and Corollaries 7.7, 7.13"},{"comment":"The proof of Lemma 5.7 is a sketch: the reduction to π_0^{C2} of (S^σ × S^σ)_+ is given, but the verification on geometric fixed points is relegated to Figure 2 and the phrase 'readily verified.' This lemma is used in Proposition 5.8 to establish the multiplicative relation ȳa = μ_*(a)ȳ, which is part of Theorem A. A complete proof is needed.","section":"Lemma 5.7 and Proposition 5.8"}],"minor_comments":[{"comment":"The sentence 'the colimit of the tower in Theorem 3.1 is contractible' is imprecise: the contractible object is the colimit of the spaces E C2T(2n), not the tower of function spectra; please rephrase.","section":"Section 3.1, after Theorem 3.1"},{"comment":"The formula d_{2r}(Q_{kρ−ϵ}(x)) = Q_{kρ−ϵ}(d_{2r}(x)) requires an extension of the equivariant Dyer–Lashof operations to the E_{2r}-page in the target bidegree; please state this extension or provide the universal-example verification directly.","section":"Proposition 5.10"},{"comment":"The assertion 'for degree reasons d2(¯ξ_{n+2}) = 0' is not justified; please indicate which target group vanishes.","section":"Theorem 7.12, proof"},{"comment":"The notation μ2 · {h_1, ..., h_{n+1}} and μ2 · b_n is introduced in the statement but not defined before use; please add a sentence explaining the μ2-orbit notation.","section":"Proposition 7.10"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about [Wil19, Prop. 2.4.1] lands: the proof of Theorem 6.1(b) and the BP_R⟨n⟩ computation depend on it, and the paper itself flags that it is unproved. This is not a circularity issue; it is an external dependency that must be resolved before acceptance. The unspecified torsion module in the E∞-statements is a second, independent incompleteness. The authors are transparent about both issues, which makes major revision appropriate rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good to get your take. I've read the paper. The core new thing is real: they construct a homological parametrized homotopy fixed point spectral sequence for C2-spectra with twisted T-action, generalizing Bruner–Rognes to the Real setting. The construction via the O(2)-equivariant filtration on E C2 T is written out in detail, including the multiplicative structure and the Weyl-twisted relation. That part looks solid to me. The computations for MU_R and BP_R (Theorems 7.4 and 7.6) are new and convincing, and the geometric fixed point corollary is a nice payoff.\n\nThe soft spots are real, and they sit exactly where you say. The E∞-terms are stated 'plus a simple y-bar torsion module in filtration zero'—that unspecified summand means the answer is not fully determined, even if the permanent-cycle algebra is. More importantly, Theorem 6.1(b), which is the engine for the BP_R<n> computation, depends on Wilson's extended power homology computation [Wil19, Prop. 2.4.1], which is stated without proof and only deferred to a sketch in [CGP25]. The paper flags this honestly, but the flag doesn't make the computation established. If that additive basis is missing a generator or has a wrong index range, the permanent cycles in Theorem 7.12 and Corollary 7.13 do not follow. That's not a cosmetic concern; it's the load-bearing input for the strongest claims.\n\nI also note the proof of Lemma 5.7 is given as a sketch—plausible, but not fully detailed. The authors are transparent about it, but it adds to the sense that the paper is a bit ahead of its proof.\n\nOverall: the spectral sequence construction is a real contribution and the MU_R/BP_R computations are likely right. The BP_R<n> result is conditional on an unproved external statement, so the paper reads as a strong research announcement plus detailed methods, rather than a finished computation. I'd send it to a serious referee. The referee should demand either a proof of the Wilson input, or a clear statement that the BP_R<n> theorem is conditional on [CGP25]. I'd cite it for the spectral sequence and the MU_R/BP_R computations, and I'd be cautious citing Theorem 7.12 until the extended-power computation is available.","headline":"A genuinely new spectral sequence and computations for Real trace invariants, with the main BP_R<n> result conditional on an unproved extended-power computation.","tokens_in":32029,"tokens_out":2137,"would_cite":true,"duration_ms":22083,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-08T20:19:24.968185+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}