{"id":"dfd3e3b9-efbd-403b-8392-e5d4592a1cd3","arxiv_id":"2411.19929","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The motivic filtration on THH(R;Z_p) is shown to be a p-typical cyclotomic synthetic spectrum, with applications to TC and syntomic cohomology bounds.","lead":"This paper constructs a new category of filtered spectra with circle and Frobenius structure, cyclotomic synthetic spectra, and proves that the motivic filtration on topological Hochschild homology naturally lives in it. A generalist should care because this gives a conceptual home for filtrations used in algebraic K-theory and yields new degree bounds for syntomic cohomology.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof that F^⋆_mot THH(R;Z_p) lifts to CycSyn rests on a connectivity bound for the even filtration that is only attributed to a private communication; without a public proof or a verified coincidence with Pstrągowski's filtration, Theorem A's comparison lemmas are not fully verifiable.","rationale":"The paper's central contribution is the construction of CycSyn and the proof that the motivic filtration on THH is an object of it. The key new ingredient is the synthetic circle Tev, built from the even filtration. The whole construction depends on Tev being connective in the Postnikov t-structure and on the comparison of Tev-fixed points with the classical equivariant even filtrations. Both rely on the lower connectivity bound for gr^i_ev. The paper flags this bound as a private communication, and the only public alternative requires the even filtrations of [18] and [33] to agree, which is asserted rather than proven for the THH objects in question. Because Theorems 3.27 and 3.29 (Theorem A) are deduced through Lemma 3.22, a gap in the bound's availability is a direct gap in the main theorem. The nonconnective statement of Theorem A makes the gap wider, since Lemma 3.22 assumes connectivity. The proposed test—comparing the two even filtrations on the concrete objects that generate the descent—would settle whether the concern lands: if the filtrations agree, the bound is public; if not, the proof of Theorem A requires further hypotheses. The reader's weakest_assumption identified exactly this dependence, and the conditional verdict is appropriate.","tokens_in":60690,"tokens_out":13528,"duration_ms":119782,"concrete_test":"On the base cases THH(F_p;Z_p) and THH(Z_p;Z_p), compute gr^i_ev using the Novikov descent formula of [18, Cor. 2.2.17] and compare with the associated graded of Pstrągowski's even filtration; verify that the two filtrations agree and that gr^i_ev lies in D(Z_p)^{[i,2i]}. If the agreement holds on these base cases and on a quasiregular semiperfectoid algebra, then the private-communication bound is replaceable by [33, Thm. 1.7] for the descent argument of Theorem A; otherwise Lemma 3.22 is unproven for the generality in which Theorem A is stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Remark 2.13 states that for a connective E∞-ring A, gr^i_ev A lies in D(S)^{[i,2i]}, attributing the lower bound to a Burklund–Krause private communication, with [33, Thm. 1.7] available only when the Hahn–Raksit–Wilson even filtration coincides with Pstrągowski's. This bound is used in Lemma 2.37 (identification of F_ev(Z[S^1]) with T_fil), Lemma 2.68 (Postnikov connectivity of Tev), and Lemma 3.22 (comparison of Tev-fixed points with the equivariant even filtration), which in turn feeds Theorems 3.27 and 3.29, i.e., Theorem A. The paper does not prove the coincidence with Pstrągowski's filtration for the relevant THH(R;Z_p), nor does it supply a proof of the private communication. Moreover, Theorem A's second part omits the 'connective' hypothesis that Lemma 3.22 assumes, so the stated domain is broader than the proof supports. If the bound fails or is unavailable for some objects in the stated domain, the comparison F_ev THH(R;Z_p) ≃ F_mot THH(R;Z_p) can fail, and the synthetic cyclotomic structure on the motivic filtration would not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an ∞-category CycSyn of p-typical cyclotomic synthetic spectra, built from synthetic spectra with an action of a synthetic circle Tev defined via the even filtration. The central result (Theorem A) asserts that the motivic filtration on THH(R;Z_p), constructed by Bhatt–Morrow–Scholze for p-quasisyntomic rings and by Hahn–Raksit–Wilson for chromatically p-quasisyntomic ring spectra, naturally refines to an E∞-algebra object in CycSyn. The paper also develops a theory of synthetic orbits, fixed points, and Tate constructions; proves a synthetic Tate orbit lemma; constructs a cyclotomic t-structure whose heart is identified with derived V-complete η-deformed Cartier complexes (Theorem B); embeds formal p-divisible groups into module categories over synthetic THH (Theorem E); and proves a filtered Beilinson fiber square yielding new bounds on syntomic cohomology (Theorem F).","tokens_in":61034,"tokens_out":7175,"duration_ms":63074,"significance":"If the main theorems hold, the paper provides a conceptually satisfying home for the motivic filtration on topological Hochschild homology, explaining the compatibility of the cyclotomic Frobenius with the filtration. The synthetic Tate orbit lemma, the identification of the heart via Cartier modules, and the filtered Beilinson fiber square are likely to be reusable tools. The paper is ambitious and inventive, and the main results would be a substantial contribution to the field. However, several load-bearing inputs are attributed to private communications, and the connective hypotheses in the comparison theorems are not fully matched with the stated domains, so the current version is not yet fully verifiable.","major_comments":[{"comment":"The connectivity bound gr^i_ev A ∈ D(S)^{[i,2i]} for connective E∞-ring spectra is attributed to a private communication from Burklund–Krause, with [33, Thm. 1.7] available only if the even filtration coincides with Pstrągowski's. This bound is load-bearing: it is used to prove Lemma 2.37 (F_ev(Z[S^1]) ≃ T_fil), Lemma 2.68 (Tev is Postnikov connective), Construction 2.15 (Sev is connective), and, through Lemma 2.75, the comparison results in Section 3. Without a public proof or a verified coincidence with Pstrągowski's filtration for the objects used in Theorem A, the construction of the synthetic circle and the main theorem are not fully verified. The authors should provide a proof in the paper or explicitly state the results as conditional on this bound.","section":"Remark 2.13; Lemma 2.37; Lemma 2.68"},{"comment":"Lemma 3.22 assumes that R is a connective E∞-ring with S1-action admitting an S1-equivariant eff cover by an even E∞-ring. However, Theorem A and Theorems 3.27(b) and 3.29(b) are stated for all chromatically p-quasisyntomic E∞-ring spectra as defined in Definition 3.24, where no connective hypothesis appears. Since the comparison F_ev THH(R;Z_p) ≃ F_mot THH(R;Z_p) and the synthetic cyclotomic structure are proved via Lemma 3.22, the stated domain is broader than the proof supports. Please either add the missing connective hypothesis to the statements (and to the definition of chromatically p-quasisyntomic, if intended) or extend Lemma 3.22 and the underlying connectivity bound to the non-connective case.","section":"Lemma 3.22; Theorem 3.27; Theorem 3.29"},{"comment":"Lemma 5.8 relies on 'work of Sanath Devalapurkar and Arpon Raksit (private communications)' for the identification τ≥0 jtCp ≃ THH(Zp). This identification is used in Lemma 5.20 and Corollary 5.21 to establish the filtered Beilinson fiber square and the K(1)-local TC fiber sequence. Like the connectivity bound, this is a load-bearing unpublished input. The authors should either prove this statement, give a precise public reference, or flag the theorem as conditional on forthcoming work.","section":"Lemma 5.8"}],"minor_comments":[{"comment":"In the proof of Theorem E, 'Deiudonné' should be 'Dieudonné'.","section":"Section 4.7"},{"comment":"The phrase 'by counting ranks' in the proof that Z[η,d]/(2η, d^2−ηd) → π_0^P(T_ev) is injective is terse; a short argument using the known rank of the homology of S[S^1] would be clearer.","section":"Section 2.4, Lemma 2.69"},{"comment":"The footnote apologizing for using B for both the E∞-ring and the classifying space is understandable, but the proof would be easier to follow if the ring were renamed (e.g., A or C).","section":"Section 2.3, Lemma 2.49"},{"comment":"The filtrations F^⋆_HRW and F^⋆_BMS are not defined in the manuscript; please cite the precise definitions in [18] and [7] and clarify the p-completion conventions used in the comparisons.","section":"Section 3.3, Theorems 3.27 and 3.29"},{"comment":"Proposition 2.54 depends on Lemma 2.37, and therefore on the privately communicated connectivity bound; this dependency should be noted explicitly at the statement.","section":"Section 2.2, Proposition 2.54"}],"recommendation":"major_revision","confidential_remarks":"The manuscript contains several genuinely new constructions and the main results are plausible. However, the reliance on load-bearing private communications (Remark 2.13 and Lemma 5.8) is substantial for a foundational paper, and the connective hypothesis mismatch in Theorem A is a clear gap. The editor may wish to require the authors to provide full proofs of these inputs or to clearly mark the affected theorems as conditional before sending the paper to press."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: this is the right upgrade path for the motivic filtration. Defining CycSyn and proving that F_mot THH(R;Z_p) lifts to an E_infinity-algebra in it is a genuine structural claim, unifying the BMS and HRW filtrations. The paper is honest about what it builds on, and the comparison theorems use the published filtrations as external benchmarks rather than engineering the conclusion.\n\nWhat is new and good: the synthetic circle Tev, the cyclotomic structure, the identification of the heart with derived V-complete eta-deformed Cartier complexes, and the filtered Beilinson square yielding new TC bounds. Theorem F recovers rather than assumes the earlier bound from [2]. The writing is clear, and the arguments in Sections 2 and 3 are detailed through most of the main constructions.\n\nThe soft spots are real but not fatal. Remark 2.13 attributes the lower bound gr^i_ev A in D(S)^{[i,2i]} to a private communication of Burklund-Krause; that bound is used in Lemma 2.37, Lemma 2.68, and Lemma 3.22, so the Postnikov connectivity of Tev and the comparison of Tev-fixed points with the equivariant even filtration rest on it. The paper notes an alternative via [33, Thm. 1.7] only when the even filtration coincides with Pstragowski's, and it does not prove that coincidence for THH(R;Z_p). That is a gap in verifiability, not a sign of a wrong argument, but it should be made public before the construction is considered fully checked. Similarly, Lemma 5.8 relies on Devalapurkar-Raksit's THH(Zp) identification, which is forthcoming. The second clause of Theorem A also drops the connective hypothesis that Lemma 3.22 assumes; either the proof covers the stated generality or the statement should be trimmed. A few proofs are summarized as left to the reader (for example, parts of Lemma 3.18 and 3.22), but they look like standard completions.\n\nWho this is for: homotopy theorists working on TC, prismatic cohomology, and Cartier modules. This is a serious preprint and deserves a serious referee. My recommendation: send it to peer review. The referee will need to chase the private communication or insist that it be made public, but the core construction and the classification of the heart are important enough to warrant that effort.","headline":"Genuine advance: CycSyn and the cyclotomic lift of the motivic filtration on THH, but the proof leans on a private connectivity bound that should be public.","tokens_in":61552,"tokens_out":2130,"would_cite":true,"duration_ms":22265,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19D55","55P42"],"pacs":[],"model":"deepseek-v4-flash","headline":"Cyclotomic synthetic spectra upgrade the motivic filtration on topological Hochschild homology to a full cyclotomic object.","keywords":["cyclotomic synthetic spectra","motivic filtration","topological Hochschild homology","synthetic spectra","even filtration","topological cyclic homology","syntomic cohomology","Cartier modules"],"falsifier":"Compute $\\mathrm{gr}^i_{\\mathrm{ev}}\\mathrm{THH}(R;\\mathbf{Z}_p)$ for a chromatically $p$-quasisyntomic $R$ and find some $i$ where it is not $i$-connective; such an example would break the Postnikov t-structure and the construction of $T_{\\mathrm{ev}}$, and with them Theorem A.","tokens_in":60509,"feed_emoji":"🌀","tokens_out":6680,"duration_ms":62959,"temperature":0.7,"pith_summary":"This paper defines an infinity-category $\\mathrm{CycSyn}$ of $p$-typical cyclotomic synthetic spectra and proves that the motivic filtration on $\\mathrm{THH}(R;\\mathbf{Z}_p)$, previously known only as a filtered spectrum with circle action, is naturally an $E_\\infty$-algebra in this category. The lift matters because it explains what the motivic filtration actually is and because the synthetic analogue of topological cyclic homology recovers the motivic filtrations on $\\mathrm{TC}$, $\\mathrm{TC}^-$, and $\\mathrm{TP}$. The paper also constructs a t-structure on $\\mathrm{CycSyn}$ whose heart is a category of $\\eta$-deformed Cartier complexes, and uses this to obtain new degree bounds on syntomic cohomology of connective chromatically $p$-quasisyntomic ring spectra.","feed_headline":"Motivic filtration on THH carries full cyclotomic structure","feed_subtitle":"A new category, CycSyn, lifts the filtered circle to synthetic spectra and yields new bounds on syntomic cohomology.","key_machinery":"The load-bearing object is the synthetic circle $T_{\\mathrm{ev}}=F^\\star_{\\mathrm{ev}}(\\mathbf{Z}[S^1])$, a lift of the filtered circle $T_{\\mathrm{fil}}$ of [35] to synthetic spectra, equipped with a bicommutative bialgebra structure. From it the paper builds synthetic analogues of $C_n$-orbits, fixed points, and Tate constructions, and defines $\\mathrm{CycSyn}$ as the lax equalizer of the identity and the $T_{\\mathrm{ev}}$-equivariant $C_{p,\\mathrm{ev}}$-Tate functor. The t-structure analysis passes through synthetic Cartier modules, and the identification of the heart with derived $V$-complete $\\eta$-deformed Cartier complexes transfers the computation to graded objects with operators $d,F,V$.","core_discovery":"The paper's central claim is Theorem A: when $R$ is $p$-quasisyntomic, or chromatically $p$-quasisyntomic, the motivic filtration $F^\\star_\\mathrm{mot}\\mathrm{THH}(R;\\mathbf{Z}_p)$ naturally carries the structure of an $E_\\infty$-algebra in $\\mathrm{CycSyn}$. Here $\\mathrm{CycSyn}$ is the category of pairs $(M,\\phi_p)$ where $M$ is a synthetic spectrum with an action of the synthetic circle $T_{\\mathrm{ev}}$ and $\\phi_p\\colon M\\to M^{tC_{p,\\mathrm{ev}}}$ is a $T_{\\mathrm{ev}}$-equivariant Frobenius map. The paper shows that applying its synthetic $\\mathrm{TC}$ functor recovers the motivic filtration on $\\mathrm{TC}(R;\\mathbf{Z}_p)$, with analogous results for $\\mathrm{TC}^-$ and $\\mathrm{TP}$, and identifies the heart of the Postnikov t-structure on $\\mathrm{CycSyn}$ with derived $V$-complete $\\eta$-deformed Cartier complexes.","pith_inferences":["A testable extension the paper leaves implicit is the non-$p$-typical variant of $\\mathrm{CycSyn}$, using Frobenii for all primes; the paper says the relevant definitions carry over directly.","The synthetic circle $T_{\\mathrm{ev}}$ is the image of $\\mathbf{G}_m$ under the known equivalence between synthetic spectra and cellular motivic spectra over $\\mathbf{C}$, so Theorem A can be read as a motivic statement about normed motivic algebras with a $\\mathbf{G}_m$-action.","The paper's open deformation question for the embedding of formal $p$-divisible groups could be approached through the prismatic $F$-gauge connection it cites, turning the fully faithful embedding of Theorem E into a statement about flat modules.","If the communicated synthetic Segal conjecture holds, the unit of $\\mathrm{CycSyn}$ becomes $p$-adically self-dual, which would simplify the equalizer formula defining synthetic $\\mathrm{TC}$."],"forward_implications":["For every connective chromatically $p$-quasisyntomic $E_\\infty$-ring spectrum $R$, the $i$-th motivic graded piece $\\mathrm{gr}^i_\\mathrm{mot}\\mathrm{TC}(R;\\mathbf{Z}_p)$ lies in $D(\\mathbf{Z}_p)[i-1,2i]$, extending the discrete-ring bound of [2].","The filtered Beilinson fiber square of Section 5 gives pullback squares relating $\\mathrm{TC}$, $\\mathrm{TC}^-$, and $\\mathrm{TP}$ in synthetic spectra, refining the classical Beilinson fiber square.","For a smooth $\\mathbf{F}_p$-algebra $A$, the heart computation yields $\\pi^{\\mathrm{cyc},P}_0(F^\\star_{\\mathrm{ev}}\\mathrm{THH}(A))\\simeq W\\Omega^\\bullet_A$, the de Rham-Witt complex, as objects of the synthetic cyclotomic category.","Formal $p$-divisible groups over smooth algebras over perfect $\\mathbf{F}_p$-algebras embed fully faithfully into cyclotomic synthetic $F^\\star_{\\mathrm{ev}}\\mathrm{THH}$-modules."],"supporting_citations":[{"why":"Supplies the cyclotomic spectrum formalism, the Tate orbit lemma, and the TC equalizer formula on which the synthetic analogues are modeled.","marker":"[32]"},{"why":"Constructs the motivic and even filtrations on THH, TC-, TP, and TC in the chromatically quasisyntomic case that Theorem A lifts.","marker":"[18]"},{"why":"Constructs the motivic filtration on THH for p-quasisyntomic rings that the paper compares against in Theorem 3.29.","marker":"[7]"},{"why":"Introduces the filtered circle and the filtered spectra formalism that the synthetic circle deforms.","marker":"[35]"},{"why":"Provides the even filtration and connectivity bounds used to prove Postnikov connectiveness of synthetic objects.","marker":"[33]"},{"why":"Supplies Cartier modules and the t-structure on cyclotomic spectra that the paper adapts to the synthetic setting.","marker":"[3]"},{"why":"Provides the filtered-spectra model of synthetic spectra and the identification of the synthetic sphere.","marker":"[15]"},{"why":"Gives the Beilinson fiber square whose filtered and synthetic analogue is established in Section 5.","marker":"[2]"}],"fun_headline_variants":["CycSyn: a new home for the THH motivic filtration","Motivic THH filtration gets full cyclotomic synthetic structure","New category CycSyn lifts THH filtration to cyclotomic spectra","Syntomic cohomology bounds via cyclotomic synthetic spectra","THH motivic filtration is a cyclotomic synthetic spectrum"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the even-filtration bound that for a connective $E_\\infty$-ring $A$, the graded piece $\\mathrm{gr}^i_{\\mathrm{ev}}A$ lies in $D(\\mathbf{S})^{[i,2i]}$, a result the paper cites as a private communication in Remark 2.13; if that bound fails for the relevant rings or for $\\mathbf{Z}[S^1]$, the synthetic circle and Theorem A fail with it.","fun_headline_variants_meta":{"raw":{"variants":["CycSyn: a new home for the THH motivic filtration","Motivic THH filtration gets full cyclotomic synthetic structure","New category CycSyn lifts THH filtration to cyclotomic spectra","Syntomic cohomology bounds via cyclotomic synthetic spectra","THH motivic filtration is a cyclotomic synthetic spectrum"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000897,"raw_usage":{"total_tokens":3828,"prompt_tokens":876,"completion_tokens":2952,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":2870}},"tokens_in":492,"tokens_out":2952,"duration_ms":16674,"temperature":1.0,"reasoning_tokens":2870,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:40:22.530200+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\mathrm{gr}^i_{\\mathrm{ev}}\\mathrm{THH}(R;\\mathbf{Z}_p)$ for a chromatically $p$-quasisyntomic $R$ and find some $i$ where it is not $i$-connective; such an example would break the Postnikov t-structure and the construction of $T_{\\mathrm{ev}}$, and with them Theorem A.","supporting_citations":[{"cited_title":"221 (2018), no","cited_arxiv_id":null,"evidence_quote":"Supplies the cyclotomic spectrum formalism, the Tate orbit lemma, and the TC equalizer formula on which the synthetic analogues are modeled."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the motivic filtration on THH for p-quasisyntomic rings that the paper compares against in Theorem 3.29."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Cartier modules and the t-structure on cyclotomic spectra that the paper adapts to the synthetic setting."},{"cited_title":"Isaksen, Achim Krause, and N icolas Ricka, C-motivic modular forms , J","cited_arxiv_id":null,"evidence_quote":"Provides the filtered-spectra model of synthetic spectra and the identification of the synthetic sphere."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Beilinson fiber square whose filtered and synthetic analogue is established in Section 5."}],"review_version":1}