{"id":"f75c8f08-7729-49ea-a933-d21136c7d3b8","arxiv_id":"2501.15651","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Real etale motivic homotopy theory over a scheme S is equivalent to sheaves of spaces on the real spectrum of S, and on pointed connected motivic spaces the real etale localization is the rho-telescope.","lead":"This paper proves that real etale motivic homotopy theory over any base scheme is equivalent to sheaves of spaces on the real spectrum of the base. It also shows that for pointed connected motivic spaces, real etale localization is simply the rho-telescope operation.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 6.11 contains an unjustified assertion Jρ = ∗, but the needed conclusion follows directly from ΣJρ = ∗; the gap is localized and does not invalidate the main theorem.","rationale":"Both the reader and I identify Lemma 6.11 as the critical soft spot: it is used in Theorem 6.12 to prove that Σ|Y•| ∧ Jρ is contractible for real-étale covers, which is the descent input for the ρ-periodization description. The proof's positive-characteristic case invokes an unsupported assertion (Jρ = ∗). I verified that the weaker statement ΣJρ = ∗ from Corollary 6.8 is sufficient, because the suspension in Lemma 6.11 is of the smash product |Cech| ∧ Jρ, and in a symmetric monoidal pointed ∞-category Σ(A ∧ B) ≃ A ∧ ΣB. Hence the gap is a one-line fix in exposition rather than a mathematical obstruction. No other independent concern emerged from a line-by-line reading: the main equivalence (Theorem 5.13) rests on cited ∞-topos results and a plausible interval argument; the transfer construction in Lemma 6.9 is intricate but the primitive-element step is standard for finite étale algebras. Because the written proof contains a false intermediate claim, the CONDITIONAL verdict is appropriate; the authors should correct Lemma 6.11's proof. The central claims are very likely correct.","tokens_in":34045,"tokens_out":15230,"duration_ms":133799,"concrete_test":"Check the identity Σ(A ∧ B) ≃ A ∧ ΣB in the pointed symmetric monoidal ∞-category Spc(k) (e.g., via the pushout definition of suspension), with A = |Spec(l)^{ו+1}| and B = Jρ. If it holds, Lemma 6.11's positive characteristic conclusion follows from ΣJρ ≃ ∗ (Corollary 6.8) without the unjustified assertion Jρ ≃ ∗. Also scan §6 to confirm no other use of Jρ ≃ ∗ appears; if none, the central theorem is unaffected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest step in the proof of Theorem 6.12 is Lemma 6.11, where the positive characteristic case is dismissed by asserting 'Jρ = ∗' in the proof, citing Corollary 6.8. But Corollary 6.8 proves only ΣJρ ≃ ∗ for unorderable fields (all positive-characteristic fields). A contractible suspension does not imply contractibility in an ∞-topos, so the assertion as written is unjustified. However, the lemma's conclusion does not require Jρ = ∗. For any pointed motivic space A, Σ(A ∧ Jρ) ≃ A ∧ ΣJρ by associativity and symmetry of the smash product and the definition Σ(−) = S^1 ∧ (−). Taking A = |Spec(l)^{ו+1}|, we get Σ(|Cech| ∧ Jρ) ≃ |Cech| ∧ ΣJρ ≃ |Cech| ∧ ∗ ≃ ∗. Thus Lemma 6.11 holds in positive characteristic directly from Corollary 6.8, and the sentence claiming Jρ = ∗ is an overstatement. The proof should be amended to use ΣJρ = ∗; no later step in Section 6 appears to require Jρ = ∗. This is a localized, easily repairable gap, not a fatal flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an unstable version of real-étale motivic homotopy theory. The main theorem (Theorem 1.1, proved as Theorems 5.13 and 6.12) states that for any base scheme S, the category of A1-invariant real-étale sheaves of spaces on Sm_S is equivalent to Shv(S_rét), the ∞-topos of sheaves on the real spectrum of S. Moreover, for a pointed connected motivic space X, the real-étale localization L_rét X is equivalent to the ρ-periodization X[ρ^{-1}] (equivalently X ∧ Jρ). The proof combines a James-construction analysis of ρ-localization in a symmetric monoidal ∞-category, a detailed study of the real spectrum of the affine line over real closed fields that establishes local contractibility, a transfer argument for real-étale covers in characteristic 0, and an ∞-topos continuity argument to pass to arbitrary bases. Section 7 derives applications to real realizations and to the real-étale localization of motivic Eilenberg–Mac Lane spaces.","tokens_in":34255,"tokens_out":13010,"duration_ms":112481,"significance":"If correct, the main theorem is a substantial destabilization of Bachmann's stable results [Bac18] and provides a clean description of real-étale motivic homotopy theory in terms of sheaves on spaces that carry no motivic structure. The equivalence of real-étale localization with ρ-periodization for connected spaces is a powerful computational tool, and the applications in Section 7 (real realization, Eilenberg–Mac Lane spaces, Bott periodicity) demonstrate its utility. The paper is generally careful and self-contained in its key geometric arguments, and it credits external references for many ∞-categorical and stable facts. The main concerns are a localized but load-bearing gap in Lemma 6.11 and a reliance on a continuity assertion in Theorem 5.13 that should be stated more explicitly. Both appear repairable, but the manuscript as written needs revision.","major_comments":[{"comment":"The proof of the positive-characteristic case asserts \"If k has positive characteristic then Jρ = ∗ by Corollary 6.8\", but Corollary 6.8 proves only ΣJρ = ∗, and a contractible suspension does not imply contractibility of the underlying space in an ∞-topos. This is a load-bearing step: Lemma 6.11 is used in the proof of Theorem 6.12 to establish real-étale descent. The gap is localized and repairable: writing C for the Čech nerve, the object N = (Σ C) ∧ Jρ is equivalent to Σ(C ∧ Jρ), hence is connected by Remark 6.6; the argument in the proof shows that [ΣX,N] ≃ [X ∧ ΣJρ,N] for every pointed X, and when ΣJρ = ∗ this gives [ΣX,N] ≃ π0(N) = ∗, so N = ∗. The proof should be amended to use ΣJρ = ∗ rather than Jρ = ∗.","section":"Lemma 6.11"},{"comment":"The reduction from arbitrary S to finite-type affine Z-schemes invokes a continuity assertion for the functors Shv(S_rét) and Spc_rét(S), said to be an \"∞-topos-theoretic enhancement of [Sch94, Proposition 3.4.1]\" with a pointer to [BH21b, Proof of Theorem 4.2]. Since this continuity is load-bearing for the equivalence over arbitrary base schemes, the paper should either state the precise ∞-categorical continuity result and prove it, or indicate exactly where in the references the required statement appears.","section":"Theorem 5.13, proof"}],"minor_comments":[{"comment":"There are several typographical errors: \"abut\" in the Section 4 title should be \"about\", \"specalize\" on p. 15 should be \"specialize\", \"impplies\" on p. 16 should be \"implies\", \"subscateg ory\" on p. 18 should be \"subcategory\", and \"m ap\" in the abstract should be \"map\".","section":"Throughout"},{"comment":"The symbol \"/BD\" appears in many displayed formulas (e.g., Definition 3.3, Lemma 3.4, Proposition 3.6). If this is a typesetting artifact for the initial object, please use a standard notation so that the text is readable.","section":"Section 3"},{"comment":"The proof invokes the claim \"we use that k is perfect, so that ΩBM ≃ M for connected monoids M\" without a reference; please add a citation or a proof, since the claim is not immediate for motivic spaces.","section":"Lemma 6.7"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper by established authors, and the main theorem is important. The localized gap in Lemma 6.11 is easily repairable, and the continuity issue in Theorem 5.13 is likely a matter of making the reference precise. I expect a straightforward revision to resolve these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper delivers the unstable real-etale motivic homotopy theory equivalences — sheaves on the real spectrum and rho-periodization for connected spaces — and the main theorems are new and likely correct. The proof is not just a destabilization of [Bac18]; it needs new geometric arguments, and the authors supply them.\n\nWhat stands out: Theorem 1.1 and Theorem 6.12. The use of the non-subcanonical nature of the real-etale topology is clever, and the construction of interval-shaped neighborhoods in RA1 is a genuinely new input. The James construction section gives a clean framework for localization, and the paper is honest about which ingredients come from prior work.\n\nSoft spots. Lemma 6.11, in positive characteristic, asserts J_rho = * citing Corollary 6.8, which only proves Sigma J_rho = *. That implication does not hold in an infinity-topos. But the stress-test note is right: the descent proof later only needs the suspension version, and it follows directly from Corollary 6.8. So the gap is localized and repairable; the authors should rewrite that sentence. Also, Theorem 5.13 leans on a continuity statement delegated to [BH21b]/[Sch94]; it is cited, not proved, and a referee should press on that. I did not machine-verify the infinity-categorical steps, but the overall structure is coherent and the dependence on prior published results is legitimate.\n\nThe citation pattern is fine. Self-citations point to published or posted work, including the stable result being destabilized.\n\nVerdict: a serious paper, worth peer review. The gap in Lemma 6.11 is a typo-level overstatement in effect, not a fatal flaw. I'd send it to a good referee and expect it to come back with minor revisions.\n\nWho should read it: anyone working in motivic homotopy theory or real algebraic geometry. I'd bring it to a reading group and cite it.","headline":"Destabilizes Bachmann's stable result to a full unstable equivalence and rho-periodization for connected spaces, with one localized and easily repairable gap in Lemma 6.11.","tokens_in":34812,"tokens_out":2332,"would_cite":true,"duration_ms":21688,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F42","14P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any base scheme $S$, unstable real-étale motivic homotopy theory is equivalent to sheaves of spaces on the real spectrum of $S$; on pointed connected motivic spaces the real-étale localization is the $\\rho$-periodization.","keywords":["motivic homotopy theory","real étale topology","semialgebraic topology","real spectrum","rho-localization","unstable homotopy theory","sheaves of spaces","James construction"],"falsifier":"Evaluate whether Lemma 6.11 needs its full strength: over an unorderable field of positive characteristic such as $\\mathbb{F}_2$, compute whether $J_\\rho$ is contractible or merely whether $\\Sigma J_\\rho$ is contractible. If $\\pi_0(J_\\rho)$ or $\\mathrm{Map}(S^0, J_\\rho)$ is nonzero even though $\\Sigma J_\\rho = *$, the lemma as written is false, and the descent proof should be read as using only the suspension statement, which is all the surrounding argument appears to require.","tokens_in":33829,"feed_emoji":"📐","tokens_out":12574,"duration_ms":93889,"temperature":0.7,"pith_summary":"The paper proves a coincidence of two homotopy theories: over any base scheme $S$, the unstable motivic homotopy theory built with the real-étale topology is equivalent to semialgebraic topology over $S$, meaning sheaves of spaces on the real spectrum of $S$. It shows further that for pointed connected motivic spaces, real-étale localization is implemented by smashing with the telescope of the map $\\rho: S^0 \\to \\mathbb{G}_m$ that sends the non-basepoint to $-1$. If the theorem is right, algebraic invariants that are $\\mathbb{A}^1$-invariant and satisfy real-étale descent are exactly invariants of the real points of a scheme, so computations over ordered fields become topological computations. The paper also computes the real-étale localization of motivic Eilenberg-Mac Lane spaces, showing the resulting homotopy rings are polynomial rings on explicit generators $\\rho$ and $t$.","feed_headline":"Real-étale motivic homotopy equals semialgebraic topology","feed_subtitle":"Algebraic invariants over ordered fields become topological invariants of real points; connected spaces localize by ρ-periodization.","key_machinery":"Two mechanisms carry the argument. The first is the map $\\rho: S^0 \\to \\mathbb{G}_m$, sending the non-basepoint to $-1$, together with its James construction $J_\\rho$ — the free $E_1$-algebra with unit factoring through $\\rho$ — and the $\\rho$-periodization $X[\\rho^{-1}] = X \\wedge J_\\rho$. The authors prove $\\rho$ is $S^1$-central in pointed motivic spaces, so for connected $X$ the object $X \\wedge J_\\rho$ is the $\\Sigma\\rho$-localization of $X$. The second is the real spectrum $RS$ and the class of interval-shaped open subsets of $R(\\mathbb{A}^1_X)$: an open set is interval-shaped when every fiber interval connecting a point to a polynomial section lies inside it. These are real-étale open neighborhoods that are $\\mathbb{A}^1$-homotopy equivalent to their images in $RX$, giving the local contractibility that makes every smooth scheme real-étale locally $\\mathbb{A}^1$-contractible and forces $\\mathbb{A}^1$-invariant real-étale sheaves to be locally constant.","core_discovery":"The paper's main theorem (Theorem 1.1, proved as Theorems 5.13 and 6.12) asserts that for any scheme $S$, the $\\infty$-category $\\mathrm{Spc}_{\\mathrm{r\\'et}}(S)$ of $\\mathbb{A}^1$-invariant real-étale sheaves on smooth $S$-schemes is equivalent to $\\mathrm{Shv}(S_{\\mathrm{r\\'et}})$, the $\\infty$-topos of sheaves of spaces on the small real-étale site of $S$, equivalently on its real spectrum $RS$. Concretely: any real-étale sheaf of spaces on $\\mathrm{Sm}_S$ is locally constant in the real-étale topology, the map $\\rho^*: \\Omega_{\\mathbb{G}_m} X \\to X$ is an equivalence for every such $X$, and locally constant real-étale sheaves are automatically $\\mathbb{A}^1$-invariant. For pointed connected $X \\in \\mathrm{Spc}(S)_*$, the localization $L_{\\mathrm{r\\'et}}X$ is the initial $\\mathbb{A}^1$-invariant real-étale sheaf under $X$, and it is given by the $\\rho$-periodization $X[\\rho^{-1}] = \\mathrm{colim}(X \\to \\mathbb{G}_m \\wedge X \\to \\mathbb{G}_m^{\\wedge 2} \\wedge X \\to \\cdots) \\simeq X \\wedge J_\\rho$, which is also the $\\rho$-localization of $X$. The authors establish this by proving local contractibility of the affine line over real closed fields with respect to the real-étale topology, and by showing that the relevant descent obstruction is killed by a transfer argument for real-étale covers.","pith_inferences":["The theorem suggests a general design principle: a 'designer' Grothendieck topology can be promoted to a homotopy theory whose localization is a single smashing operation whenever the topology is generated by inverting one map; the real-étale topology is the case of inverting $\\rho$.","The $C_2$-equivariant analog in the paper works without connectivity, while the paper shows connectivity is necessary over $\\mathbb{C}$; one could test intermediate hypotheses (nilpotence, finiteness) under which $\\rho$-periodization and real-étale localization still agree for non-connected spaces.","The interval-shaped open method may transfer to other settings with a real line carrying orderings, such as o-minimal structures or spaces of orderings, producing analogous local-contractibility statements for semialgebraic-like topologies.","If Lemma 6.11's $J_\\rho = *$ over positive characteristic is an overstatement, the descent proof still appears to go through using only $\\Sigma J_\\rho = *$; this suggests the theorem is more robust than the written proof, and the gap is repairable."],"forward_implications":["For every base scheme $S$, real-étale motivic spaces are exactly sheaves of spaces on the real spectrum, so real-étale motivic invariants can be computed stalkwise at real closed fields.","Any pointed connected motivic space $X$ has $L_{\\mathrm{r\\'et}}X \\simeq X[\\rho^{-1}]$; in particular real-étale localization is a single smashing operation, not an iteration of separate $\\mathbb{A}^1$ and sheafification steps.","The equivalence $X[\\rho^{-1}] \\simeq L_{\\mathrm{r\\'et}}X$ gives a concrete formula for the real-étale localization of loop spaces, classifying spaces, and other connected spaces of interest in $\\mathbb{A}^1$-homotopy theory.","Over $\\mathbb{R}$, the real realization functor agrees with real-étale localization, so homotopy types of real points of schemes (e.g. real Grassmannians and loop spaces of split reductive groups) are computed by the localized motivic space.","The real-étale localizations of motivic Eilenberg-Mac Lane spaces are constant sheaves with explicit homotopy rings, e.g. $\\pi_* L_{\\mathrm{r\\'et}}K(\\mathbb{Z}(\\star),\\star) \\cong \\mathbb{Z}[\\rho,t]/(2\\rho)$ and $\\pi_* L_{\\mathrm{r\\'et}}K(\\tilde{\\mathbb{Z}}(\\star),\\star) \\cong \\mathbb{Z}[\\rho,t]/(2t\\rho)$, with mod-2 Bockstein behavior fixed."],"supporting_citations":[{"why":"establishes the stable analog that real-étale localization is $\\rho$-inversion; this paper destabilizes that result.","marker":"[Bac18]"},{"why":"supplies the real-étale topology, the comparison of sheaves on $RX$ and $X_{\\mathrm{r\\'et}}$, and the local-homeomorphism property of étale maps on real spectra.","marker":"[Sch94]"},{"why":"promotes the topos comparison to an equivalence of $\\infty$-topoi and gives hypercompleteness, used throughout the category-level arguments.","marker":"[ES21]"},{"why":"provides the foundations of unstable motivic homotopy theory, including the localization theorem used for pointwise contractibility checks.","marker":"[MV99]"},{"why":"supplies computations of homotopy sheaves of smashed motivic spheres and the unstable connectivity theorem used in the nilpotent and connectivity arguments.","marker":"[Mor12]"},{"why":"describes the points and orderings of the real affine line as generalized Dedekind cuts, feeding the interval-shaped open analysis.","marker":"[KS22]"},{"why":"computes the colimit of Milnor-Witt groups that yields $\\pi_1 \\Sigma J_\\rho$ in the proof of Corollary 6.8.","marker":"[Jac17]"},{"why":"provides hyperdescent and cohomological dimension bounds used to control Postnikov convergence in real-étale topoi.","marker":"[CM21]"}],"fun_headline_variants":["Real-étale homotopy equals semialgebraic topology","Real-étale motivic is semialgebraic topology","Real-étale = semialgebraic topology over any scheme","Semialgebraic topology is real-étale homotopy","For every scheme, real-étale = semialgebraic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of descent for real-étale covers uses Lemma 6.11, which asserts that the James construction $J_\\rho$ is contractible in positive characteristic; the cited Corollary 6.8 only proves its suspension $\\Sigma J_\\rho$ is contractible, and a contractible suspension does not generally make the space contractible.","fun_headline_variants_meta":{"raw":{"variants":["Real-étale homotopy equals semialgebraic topology","Real-étale motivic is semialgebraic topology","Real-étale = semialgebraic topology over any scheme","Semialgebraic topology is real-étale homotopy","For every scheme, real-étale = semialgebraic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001621,"raw_usage":{"total_tokens":6478,"prompt_tokens":1003,"completion_tokens":5475,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":619,"completion_tokens_details":{"reasoning_tokens":5385}},"tokens_in":619,"tokens_out":5475,"duration_ms":35258,"temperature":1.0,"reasoning_tokens":5385,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:05:34.821373+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate whether Lemma 6.11 needs its full strength: over an unorderable field of positive characteristic such as $\\mathbb{F}_2$, compute whether $J_\\rho$ is contractible or merely whether $\\Sigma J_\\rho$ is contractible. If $\\pi_0(J_\\rho)$ or $\\mathrm{Map}(S^0, J_\\rho)$ is nonzero even though $\\Sigma J_\\rho = *$, the lemma as written is false, and the descent proof should be read as using only the suspension statement, which is all the surrounding argument appears to require.","supporting_citations":[],"review_version":1}