{"id":"608be9f6-962c-425c-ad08-cafa6a13841b","arxiv_id":"1908.08378","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Betti realization identifies the p-complete C2-equivariant stable homotopy category as a localization of the p-complete cellular real motivic stable homotopy category, yielding computable RO(C2)-graded homotopy groups from motivic homotopy groups.","lead":"This paper proves that, after completing at a prime, C2-equivariant stable homotopy theory is a localization of real motivic stable homotopy theory, and gives a recipe for computing C2-equivariant homotopy groups from motivic ones. The result turns hard equivariant computations into simpler motivic computations, with worked examples for Eilenberg-MacLane spectra and Real K-theory.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the localization theorem is internally coherent and the flagged assumptions are checkable rather than flawed.","rationale":"The reader's weakest_assumption points to Prop. 8.3, and I agree that this is the most delicate input to the main theorem. My reading of the proof, however, finds the concern checkable rather than fatal. The recollement argument for Theorem 8.22 is formal once the Borel and geometric pieces are fully faithful. The Borel piece is obtained by lifting complex Betti localization up the rho-completion tower via Prop. 3.10; this requires C(rho) to carry an E-infinity structure and the comparison to Spec(C)_+ to be compatible with the relevant module categories. The paper's construction of that structure through the lax monoidal cellularization functor is non-canonical, but existence of an object-level equivalence to Cell(Spec(C)_+) is enough to transfer an E-infinity structure, and the subsequent tower identifications can be checked rather than assumed. The odd-primary convergence issue in Prop. 8.3 is the only place I found a genuinely terse step: the text says Spec(C)_+ is eta-complete by Prop. 6.7, although that proposition is stated for cellular spectra. The missing verification can be supplied from the facts that rho acts as zero on Spec(C)_+ and that in the post-2-inverted eigenspace decomposition eta acts as zero, which forces the required eta-completeness. Similarly, the Dugger-Isaksen range used in the construction of tau-self maps is compatible with the asymptotics of gamma(i-1), so the computational half is not endangered by an index-range error. None of these observations undermines the stated p-complete localization theorem or the computational method; they only identify places where a careful referee would ask for a line of justification. I therefore see no reason to move the verdict from ACCEPT, and I recommend the paper be accepted unchanged, with the suggestion that the authors add a one-paragraph clarification of the odd-primary eta-completeness and the E-infinity transfer in a final version.","tokens_in":28746,"tokens_out":46107,"duration_ms":497306,"concrete_test":"Independently verify that the E-infinity structure on C(rho)_p^wedge transferred from Cell(Spec(C)_+)_p^wedge along Prop. 8.3 is compatible with the rho-completion tower: check that the unit map of this E-infinity structure has iterated cofibers equivalent to C(rho^n)_p^wedge for all n, so that the module-category tower used in Cor. 8.21 and Prop. 3.10 is the same completion tower as the one defined by the maps S^{-n,-n} -> S^{0,0}. If this compatibility fails, the Borel full faithfulness step would need a different E-infinity structure; if it holds, the flagged non-canonical choice in Prop. 8.3 is harmless.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing concern that would change the verdict. The central claim Theorem 8.22 is assembled from the recollement Lemma 5.1, full faithfulness on the Borel part (Cor. 8.21, via Prop. 3.10, Thm. 8.18, Cor. 8.6), full faithfulness on the geometric part (Bachmann Thm. 8.10), and the vanishing/equivalence conditions Lem. 8.19 and Lem. 8.20. Each formal step is coherent, and the use of Lemma 8.19 is justified because homotopy complete spectra are detected by their underlying spectra. I probed the two places the reader flags. First, Prop. 8.3: the non-canonical comparison C(rho) -> Spec(C)_+ is supported by an HF_p homology computation; at odd primes the MASs convergence point requires C(rho) and Spec(C)_+ to be eta-complete. The appeal to Prop. 6.7 is terse because Spec(C)_+ is not cellular, but the needed eta-completeness is checkable from rho acting as zero on Spec(C)_+, so this is a presentation gap rather than a mathematical flaw. Second, Theorem 7.10's use of the Dugger-Isaksen range i >= 3j - 5 is legitimate: the relevant bidegrees have top degree about i and weight about i - 2*gamma(i-1), with gamma(i-1) approximately i/2, so the inequality holds for all large i. I did not find an internal inconsistency.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves that p-complete C2-equivariant stable homotopy is a localization of p-complete cellular real motivic stable homotopy via C2-Betti realization. The main theorem (Theorem 8.22) states that the right adjoint Cell Sing^{C2}: (Sp^{C2})_p^wedge -> SH^{cell}(R)_p^wedge is fully faithful, and Theorem 8.26 gives a computational formula expressing the homotopy groups of the homotopy completion of the C2-Betti realization in terms of motivic homotopy groups of the tau-inverted rho-completion. The proof combines formal recollement lemmas (Section 5), a monoidal Barr-Beck theorem of Mathew-Naumann-Noel, cellularization techniques (Section 4), Bachmann's real etale localization theorem, and explicit tau-self maps on C(rho^i) constructed in Section 7 using the Dugger-Isaksen isomorphism theorem and Stahn's odd-primary results. The paper also works out examples HF_2, HZ_2, and kgl_2^wedge.","tokens_in":29009,"tokens_out":11426,"duration_ms":106540,"significance":"If correct, this establishes a new and powerful bridge between real motivic and C2-equivariant stable homotopy, with direct computational consequences. The formal framework is reusable, and the examples clearly demonstrate the method. The paper is honest about its reliance on external deep results (Bachmann, Heller-Ormsby, Stahn, Dugger-Isaksen, Lin), and the proof chain appears internally coherent. I specifically checked the two fragile points flagged in review: the Dugger-Isaksen range i >= 3j-5 used in Theorem 7.10 is satisfied in the relevant bidegrees, and the eta-completeness assertion in Proposition 8.3 is true, though the written justification for Spec(C)_+ needs a small addition. The authors also give explicit credit to independent work of Isaksen-Kong-Wang-Xu, though the citation is incomplete.","major_comments":[],"minor_comments":[{"comment":"The sentence \"both C(rho) and Spec C+ are eta-complete by Prop. 6.7\" is not literally correct: Prop. 6.7 is stated only for X in SHcell(K)[1/2], and Spec C+ is not cellular (Remark 8.4). Please add a direct argument for the eta-completeness of Spec C+ (for instance, via the vanishing of rho on Spec C+ and the cellular approximation C(rho)), since this is used to justify convergence of the motivic Adams spectral sequence at odd primes.","section":"Section 8, Proposition 8.3"},{"comment":"The sentence \"which has also been independently obtained by Isaksen-Kong-Wang-Xu\" provides no reference; please add a precise citation or state this as a personal communication, so that readers can locate the independent work.","section":"Introduction, Theorem 1.7"},{"comment":"The phrase \"Ricka proves proves this\" contains a duplicated \"proves\", and \"metioned\" should be \"mentioned\".","section":"Remark 8.17"},{"comment":"The notation pi^{C2}_* is used both for the Z-graded and the RO(C2)-graded equivariant homotopy groups; please disambiguate, especially in the Mayer-Vietoris arguments that mix bigraded and single-graded inputs.","section":"Section 9"}],"recommendation":"minor_revision","confidential_remarks":"The paper is in good shape and the main results appear sound. The only substantive request is the direct justification of eta-completeness of Spec(C)_+ in Proposition 8.3; this is a local fix and should not require major changes. I also recommend clarifying the reference to the independent work of Isaksen-Kong-Wang-Xu."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is the real thing. Behrens and Shah prove that Betti realization exhibits (Sp^{C2})_p^wedge as a localization of SH^{cell}(R)_p^wedge, and they turn the structural statement into a recipe: for p-complete cellular X over R, pi^{C2}_{*,*}(BeC2_p(X)^h) is read off from pi^R_{*,*}(X^wedge_rho[tau^{-1}]), with Bachmann controlling the geometric part. The tau-self maps on C(rho^i) are new and are what makes the calculation work. The examples, especially HF2 and kgl -> kR, actually exhibit the negative cone arising from the motivic input.\n\nWhat is new: Heller-Ormsby had the colocalization direction via the constant functor; Bachmann had the rho-localization. The localization direction here, plus the tau-self map construction, is not in the prior literature. The paper is also careful about what it does not claim: it notes the non-p-complete statement is not settled (Rem. 8.17) and makes no uniqueness claims for the self-maps.\n\nThe proof looks sound to me. The formal sections on localizations, cellularization, and recollements are clear, and Theorem 8.22 is assembled honestly from full faithfulness on the Borel and geometric pieces plus the recollement lemma. I checked the two spots likely to fail. Proposition 8.3 uses a non-canonical comparison map C(rho) -> Spec C_+ depending on a choice of null homotopy; only existence is needed, and the HF_p homology computation is the real content. At odd primes the eta-completeness point is terse because Spec C_+ is not cellular, but it is checkable. The Dugger-Isaksen range in Theorem 7.10 is legitimate in the relevant bidegrees. I found no post hoc data selection and no circularity: the u-self maps are proven first, then lifted.\n\nSoft spots: the external dependency stack is tall—Bachmann, Stahn, Dugger-Isaksen, MNN, Lin/Segal. That lowers my confidence from high to moderate, not because I saw a contradiction but because no single reader can quickly verify every layer. Some infinity-category and spectral sequence details are abbreviated. None of this undermines the p-complete claim.\n\nWho should read it: homotopy theorists at the motivic/equivariant interface. It is a substantial paper and it deserves a serious referee. I would send it out.","headline":"A genuine structural bridge: p-complete C2-equivariant stable homotopy is a localization of cellular real motivic stable homotopy, with a usable computational recipe; the proof is dense and externally dependent but sound.","tokens_in":29588,"tokens_out":3297,"would_cite":true,"duration_ms":33256,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P91","14F42","55P42"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that p-complete C2-equivariant stable homotopy is a localization of p-complete cellular real motivic stable homotopy.","keywords":["C2-equivariant stable homotopy","real motivic stable homotopy","Betti realization","localization","cellular spectra","tau-self maps","isotropy separation","RO(C2)-graded homotopy groups"],"falsifier":"For $p=2$, compute the effect of the comparison map $C(\\rho)^\\wedge_p \\to \\Sigma_+^\\infty \\mathrm{Spec}\\,\\mathbb{C}^\\wedge_p$ on mod $2$ motivic homology: Proposition 8.3 predicts exactly the quotient $\\mathbb{F}_2[\\tau,\\rho] \\to \\mathbb{F}_2[\\tau]$, so any class in the kernel beyond $\\rho$ would disprove the localization theorem.","tokens_in":28502,"feed_emoji":"🔄","tokens_out":10354,"duration_ms":87296,"temperature":0.7,"pith_summary":"This paper proves that Betti realization, from real motivic spectra over $\\mathbb{R}$ to $C_2$-equivariant spectra, becomes a localization after passing to $p$-complete cellular objects: the right adjoint $\\mathrm{Cell}\\,\\mathrm{Sing}^{C_2}$ embeds the $p$-complete $C_2$-equivariant stable homotopy category fully faithfully into the $p$-complete cellular real motivic stable homotopy category. The point is computational as well as categorical. For a $p$-complete cellular real motivic spectrum $X$, the geometric part of its $C_2$-Betti realization is obtained by inverting the Euler class $\\rho$, the homotopy-complete part is obtained by inverting $\\tau$ on the $\\rho$-completion tower, and the Tate part by inverting both. Because the $C_2$-equivariant homotopy type is recovered from an isotropy separation square built from these three pieces, the $RO(C_2)$-graded equivariant homotopy groups of examples such as $\\mathrm{H}\\mathbb{F}_2$, $\\mathrm{H}\\mathbb{Z}_2$, and connective real $K$-theory can be computed from their simpler bigraded motivic homotopy groups.","feed_headline":"p-complete C2-spectra are a localization of real motivic spectra","feed_subtitle":"The fully faithful embedding computes RO(C2)-graded equivariant homotopy groups from simpler motivic homotopy groups.","key_machinery":"The load-bearing comparison map is the non-canonical map $C(\\rho) \\to \\Sigma^\\infty_+ \\mathrm{Spec}\\,\\mathbb{C}$ from the cofiber of the real motivic Euler class $\\rho$ to the complex variety point; Proposition 8.3 proves it becomes an equivalence after $p$-completion and cellularization, even though $\\Sigma^\\infty_+ \\mathrm{Spec}\\,\\mathbb{C}$ itself is not cellular. This turns the complex motivic category into modules over $C(\\rho)$ inside the real cellular category, so the complex $\\tau$-inversion localization can be pulled back to the real side. The remaining input is a supply of $\\tau$-self maps on the spectra $C(\\rho_i)$: at the prime $2$ these are lifted from $u$-self maps on the equivariant spectra $C(a_i)$, using James periodicity and an isomorphism range between real motivic and $C_2$-equivariant spheres; at odd primes a $\\tau^2$-self map exists on every $\\rho$-complete real motivic spectrum. Finally, the isotropy separation square for $C_2$-spectra, equivalently the $\\rho$-arithmetic square, assembles the geometric, homotopy-complete, and Tate localizations into the full equivariant homotopy type.","core_discovery":"The central claim is that the adjunction $\\widehat{\\mathrm{Be}}^{C_2}_p : \\mathrm{SH}^{\\mathrm{cell}}(\\mathbb{R})_p^\\wedge \\rightleftarrows (\\mathrm{Sp}^{C_2})_p^\\wedge : \\mathrm{Cell}\\,\\mathrm{Sing}^{C_2}$ is a localization: $\\mathrm{Cell}\\,\\mathrm{Sing}^{C_2}$ is fully faithful. Equivalently, every $p$-complete $C_2$-equivariant spectrum is, up to equivalence, the $p$-complete cellular Betti realization of a $p$-complete cellular real motivic spectrum. On homotopy groups, Theorem 8.26 gives isomorphisms $\\pi^{\\mathbb{R}}_{*,*}(X^\\wedge_\\rho[\\tau^{-1}]) \\cong \\pi^{C_2}_{*,*}(\\widehat{\\mathrm{Be}}^{C_2}_p(X)^h)$ and, via Theorem 8.10, $\\pi^{\\mathbb{R}}_{*,*}(X[\\rho^{-1}]) \\cong \\pi^{C_2}_{*,*}(\\widehat{\\mathrm{Be}}^{C_2}_p(X)^\\Phi)$. Combining the geometric and homotopy-complete parts with the Tate part through the isotropy separation square recovers the full $RO(C_2)$-graded equivariant homotopy groups.","pith_inferences":["The paper leaves open whether the localization theorem survives without $p$-completion; if it did, the relation between real motivic and $C_2$-equivariant stable homotopy would be a genuine recollement rather than one on completed cellular objects.","Because the proof of the $\\tau$-self maps at $p=2$ uses only a specific isomorphism range between real motivic and equivariant stems, extending that range would automatically produce $\\tau$-self maps in a larger region, making the computational method applicable to more spectra.","The negative cone that appears in $\\pi^{C_2}_{*,*}\\mathrm{H}\\mathbb{F}_2$ is presented in the paper as a consequence of local duality; a concrete test would be to see whether the same duality mechanism predicts the negative cones for $\\mathrm{H}\\mathbb{Z}_2$ and $\\mathrm{kR}$ without separate input."],"forward_implications":["For every $p$-complete cellular real motivic spectrum $X$, the $RO(C_2)$-graded homotopy groups of $\\widehat{\\mathrm{Be}}^{C_2}_p(X)$ are completely determined by the bigraded motivic homotopy groups of $X$ together with the maps in the isotropy separation square.","At odd primes the $C_2$-equivariant Tate spectrum of a Betti realization is contractible, so the equivariant homotopy groups split as a direct sum of the $\\rho$-inverted and the $\\tau^2$-inverted motivic groups.","The full faithfulness of $\\mathrm{Cell}\\,\\mathrm{Sing}^{C_2}$ means that constructions and objects in $p$-complete $C_2$-equivariant stable homotopy can be transferred to the real motivic category; the paper uses this to compute the equivariant homotopy of $\\mathrm{H}\\mathbb{F}_2$, $\\mathrm{H}\\mathbb{Z}_2$, and $\\mathrm{kgl}^\\wedge_2$.","The $\\tau$-self maps give real motivic periodicity: $C(\\rho_i)^\\wedge_p$ admits a $\\tau_j$-self map with $j = 2\\gamma(i-1)$ at $p=2$, and every $\\rho$-complete real motivic spectrum admits a $\\tau^2$-self map at odd primes."],"supporting_citations":[{"why":"Identifies real Betti realization with localization at $\\rho$, supplying the geometric-fixed-point half of the equivariant computation.","marker":"[Bac18]"},{"why":"Provides the isomorphism between R-motivic and C2-equivariant stable stems in the range used to lift equivariant u-self maps to real motivic $\\tau$-self maps.","marker":"[DI17a]"},{"why":"Establishes the complex case: p-complete Betti realization is computed by inverting $\\tau$ on cellular complex motivic spectra at $p=2$.","marker":"[DI10]"},{"why":"Extends the $\\tau$-inversion description of complex Betti realization to odd primes and supplies the odd-primary motivic Adams-Novikov spectral sequence input.","marker":"[Sta16]"},{"why":"Supplies the monoidal descent theorems used to identify stable homotopy categories as module categories.","marker":"[MNN17]"},{"why":"Gives the isotropy separation square that splits a C2-spectrum into homotopy-complete, geometric, and Tate parts.","marker":"[GM95]"},{"why":"Supplies the James-periodicity periodicity of a-torsion in C2-equivariant stems used to construct u-self maps on the $C(a_i)$.","marker":"[Lan69]"}],"fun_headline_variants":["Localizing motivic spectra recovers C2-equivariant homotopy","Motivic localization yields full C2-equivariant homotopy groups","From motivic to C2-spectra: a localization theorem","C2-equivariant homotopy from motivic localization","Localization computes RO(C2)-graded groups from motivic data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole bridge rests on the claim that the real motivic cofiber of the Euler class $\\rho$ can replace the non-cellular spectrum $\\Sigma_+^\\infty \\mathrm{Spec}\\,\\mathbb{C}$ after $p$-completion and cellularization; if that comparison map fails to be an equivalence, the identification of the complex motivic category with modules over $C(\\rho)$ breaks.","fun_headline_variants_meta":{"raw":{"variants":["Localizing motivic spectra recovers C2-equivariant homotopy","Motivic localization yields full C2-equivariant homotopy groups","From motivic to C2-spectra: a localization theorem","C2-equivariant homotopy from motivic localization","Localization computes RO(C2)-graded groups from motivic data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000651,"raw_usage":{"total_tokens":2962,"prompt_tokens":895,"completion_tokens":2067,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":511,"completion_tokens_details":{"reasoning_tokens":1976}},"tokens_in":511,"tokens_out":2067,"duration_ms":14317,"temperature":1.0,"reasoning_tokens":1976,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:40:41.828336+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $p=2$, compute the effect of the comparison map $C(\\rho)^\\wedge_p \\to \\Sigma_+^\\infty \\mathrm{Spec}\\,\\mathbb{C}^\\wedge_p$ on mod $2$ motivic homology: Proposition 8.3 predicts exactly the quotient $\\mathbb{F}_2[\\tau,\\rho] \\to \\mathbb{F}_2[\\tau]$, so any class in the kernel beyond $\\rho$ would disprove the localization theorem.","supporting_citations":[],"review_version":1}