{"id":"e17117d9-1530-4e5e-880d-a24018577b03","arxiv_id":"2607.05360","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A noncommutative Cartier formula for E1-ring spectra is proven and applied to show that p-curvature of the quantum connection computes quantum Steenrod operations for Calabi-Yau symplectic manifolds.","lead":"The paper proves a formula relating the cap product on topological Hochschild homology to cyclotomic structure maps for any E1-ring spectrum, and applies it to compute p-curvature of quantum connections in terms of quantum Steenrod operations. A smart generalist might read it because it bridges arithmetic geometry (Cartier's formula in characteristic p) with symplectic topology (quantum Steenrod operations), potentially explaining arithmetic properties of mirror symmetry.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The bridge from the spectral Cartier formula to the algebraic Cartier transform (Theorem 8.4) hinges on the commutativity of diagram (8.6), a multi-corner E∞-ring diagram whose verification is asserted but not fully traced through.","rationale":"The reader correctly identified the PVV reproof and the symplectic assumptions as the main risk areas, and the CONDITIONAL verdict is appropriate. My concern is more specific: the single most load-bearing step is the commutativity of diagram (8.6) in the proof of Theorem 8.4, which is the bridge from the spectral to the algebraic world. The reader mentioned this area ('a reproof of a variant of PVV because the comparison is unavailable') but did not pinpoint the specific diagram whose commutativity is asserted rather than verified. The algebraic core (Theorems 1.5, 1.15) appears solid: the proof of Theorem 1.5 is direct, the key diagram (4.5) commutes by construction, and homotopical control via Proposition 2.9 and Lemma 4.10 is sound — the HHR diagonal is an isomorphism on cofibrant objects, and this property extends levelwise to cosimplicial objects. The lifting result (Theorem 9.1) is a standard obstruction-theory argument. The comparison in Lemma 11.4 between the relative Tate diagonal and Kaledin's diagonal is only on homology (not at the chain level, as noted in Remark 11.9), but the applications work at the homology level so this is not a concern. The p-torsion-free condition and the bound p > f(m,n,N,r) are transparently stated. The symplectic Assumptions A–G are clearly conditional and external to the algebraic contribution. I recommend UNCHANGED because the reader's CONDITIONAL verdict already captures the right level of uncertainty: the algebraic core is likely correct but requires expert verification of many diagrams, and the symplectic application is genuinely conditional on unverified assumptions.","tokens_in":67285,"tokens_out":7256,"duration_ms":213848,"concrete_test":"Verify the commutativity of diagram (8.6) on homotopy groups for R = F_p[x] (or F_p[x,x^{-1}]), where all four corners and all maps can be computed explicitly using the HKR theorem (Section 7.3) and Hesselholt's computation of THH(S[x]) (Section 7.2). Specifically, compute the map from the top-left corner HH(R/F_p) through the diagram to the bottom-right corner (Ω^1_{R/F_p}[2])^{tC_p} along both paths, and check they agree. The author computes F'(x^k dx/x) = x^{pk} dx/x in (7.11), but the full diagram (8.6) involves additional objects (involving Z_p and THH(R̃)) whose compatibility with this computation is not explicitly verified. If the two paths through (8.6) disagree on any element, Theorem 8.4 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central algebraic results (Theorems 1.5, 1.15) have clean, direct proofs: the key diagram (4.5) commutes 'essentially by construction,' and homotopical control is handled by Proposition 2.9 and Lemma 4.10, which are sound. The load-bearing step is instead Theorem 8.4, which bridges the spectral noncommutative Cartier formula to the algebraic inverse Cartier transform. Its proof constructs diagram (8.8) and identifies its top row with (1⊗F'⊗1)∘F*(κ')⊗1, then invokes Lemma 8.1 to conclude this is the Cartier transform. This identification depends on the commutativity of diagram (8.6), a large diagram involving E∞-ring maps between HH(R/F_p), HH(R_{Z_p}/Z_p), THH(R̃), their Tate constructions, and Ω^1. The author states this commutativity 'follows from Theorem 7.14 and the diagrams (6.13) and (6.14),' but the verification requires checking compatibility of multiple E∞-ring maps and module structures across a diagram with roughly a dozen objects. This is the single step that cannot be shortcut by citing [93] (because the comparison between Kaledin's algebraic Cartier map and the spectral construction is unavailable), making it the most consequential unverified link in the chain Theorem 1.5 → 1.15 → 6.17 → 8.4 → 1.23 → 1.27. The symplectic Assumptions A–G (especially E) are also genuinely unverified, but these are clearly flagged as conditional by the author and are external to the paper's algebraic contribution.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The paper proves a noncommutative analog of Cartier's formula relating the cap product on topological Hochschild (co)homology to the cyclotomic structure map. The main algebraic results (Theorems 1.5 and 1.15) establish commutative squares in the ∞-category of spectra for arbitrary E1-algebras A, using point-set models in equivariant orthogonal spectra with the convenient model structures of [20]. The relative version (Theorem 1.15) extends this to algebras over a base R. Specializing to R = F_p and using relative Tate diagonals, the author connects the spectral formula to the classical Cartier isomorphism (Theorem 7.14) and computes the p-curvature of the Getzler-Gauss-Manin connection (Theorem 1.23). The paper then applies these results to symplectic geometry (Theorem 1.27), showing that under Assumptions A–G on the Fukaya category, the p-curvature of the quantum connection computes the Quantum Steenrod operations for Calabi-Yau symplectic manifolds. The proofs of Theorems 1.5 and 1.15 proceed via explicit simplicial/cosimplicial constructions using the prismatic subdivision of McClure-Smith, reducing to combinatorial verification. The bridge from the spectral to the algebraic setting (Theorem 8.4) requires an independent reproof of a variant of Petrov-Vaintrob-Vologodsky [93] because no comparison between Kaledin's algebraic Cartier map and the spectral construction is available.","tokens_in":67497,"tokens_out":2032,"duration_ms":177654,"significance":"The paper makes a substantial contribution by providing a uniform spectral-algebraic framework connecting cyclotomic structure on THH with the Cartier isomorphism in characteristic p. The core algebraic theorems (1.5, 1.15) are clean and their proofs are direct: the key diagram (4.5) commutes 'essentially by construction,' and homotopical control is handled by Proposition 2.9 and Lemma 4.10, which are sound. The explicit point-set construction of the cap product via prismatic subdivision (Section 3) and the verification that it agrees with the derived cap product (Lemma 3.23) are valuable. The application to symplectic geometry (Theorem 1.27) is clearly conditional on Assumptions A–G, which the author transparently flags as consequences of the state of Fukaya-categorical foundations. Theorem 1.23 gives a falsifiable, computable prediction for p-curvature. The paper ships a concrete, checkable construction rather than a purely abstract existence result.","major_comments":[{"comment":"Theorem 8.4 is the load-bearing step connecting the spectral noncommutative Cartier formula (Theorem 6.17) to the algebraic inverse Cartier transform. Its proof constructs diagram (8.8) and identifies its top row with (1⊗F'⊗1)∘F*(κ')⊗1, then invokes Lemma 8.1 to conclude this is the Cartier transform. This identification depends on the commutativity of diagram (8.6), a large diagram involving E∞-ring maps between HH(R/F_p), HH(R_{Z_p}/Z_p), THH(R̃), their Tate constructions, and Ω^1. The author states this commutativity 'follows from Theorem 7.14 and the diagrams (6.13) and (6.14),' but the verification requires checking compatibility of multiple E∞-ring maps and module structures across a diagram with roughly a dozen objects. The text does not trace through this verification in sufficient detail for the reader to confirm it. Since this is the single step that cannot be shortcut by a文献引用","section":null},{"comment":"Section 8.2, proof of Theorem 8.4: the identification of the top row of (8.8) with the map (1⊗F'⊗1)∘F*(κ')⊗1 is stated to follow from 'the outer square (consisting of rows 2 and 4 mapping to rows 1 and 5) of the diagram (8.6), together with the factorization claim about F' in Theorem 7.14.' The reader needs to see how the module structures over the various E∞-rings in (8.6) are transported through (8.8), particularly how the T HH(R)-module structure on the domain of ϕ_R interacts with the R_{Z_p}-module structure appearing in the top row of (8.8). A more explicit verification, or at minimum a lemma isolating the compatibility of module structures across (8.6), would strengthen this critical step.","section":null},{"comment":"Theorem 1.23 requires that HH*(C/R) is p-torsion-free, which restricts to p > f(m,n,N,r). The proof of Theorem 1.23 also cannot use Petrov-Vaintrob-Vologodsky [93] directly because 'there is no comparison between Kaledin's noncommutative Cartier map and corresponding spectral constructions available in the literature' (Section 8). This necessitating an independent reproof. While the reproof via Theorem 8.4 is a reasonable strategy, the p-torsion-free hypothesis is essential for the identification in Theorem 8.4 (via Proposition 6.16, which requires dualizability and thus the lift to a standard cyclotomic base). The author should clarify whether the p-torsion-free condition enters only through the applicability of Proposition 6.16, or whether it is also needed for the commutativity of (8.6) itself.","section":null}],"minor_comments":[{"comment":"The paper would benefit from a notation table. The proliferation of symbols (R, R̃, R_{Z_p}, R_p, ˆΩ^1, etc.) across Sections 6–8 makes it difficult to track which ring is the base at each step.","section":null},{"comment":"In diagram (6.7), the statement that 'all rectangles commute except for the right-most rectangle involving THH(F_p) and F_p' is important but easy to miss. Consider highlighting this more prominently, as it is the key reason F_p is not a standard cyclotomic base.","section":null},{"comment":"Section 7.3.2: the claim that HH(k[x]/k) is the trivial square-zero extension k[x]⊕Ω^1_{k[x]/k}[1] as E∞-algebras when the S^1-action is forgotten relies on degree considerations and [76, 7.4.1.18]. A brief indication of why the extension is n-small would help the reader.","section":null},{"comment":"Remark 7.20 discusses completed tensor products and their homotopical meaning. The reference to Appendix I for the condensed mathematics approach is noted, but the main text could briefly state whether the results of Theorem 7.16 for R=k[[x]], k((x)) depend on any properties specific to the condensed framework beyond what is in Lemma 10.19.","section":null},{"comment":"The cyclotomic Deligne conjecture (Appendix A) is stated as a conjecture and not proven in full generality. The paper proves it 'for one particular point in O_∩(p).' This is sufficient for the main theorems but should perhaps be stated more explicitly in the introduction to manage expectations.","section":null},{"comment":"Several references appear to be to unpublished or in-progress work ([97], work in progress). Where these are load-bearing for the symplectic application (Theorem 1.27), the conditional nature is clear, but the reader should be informed of the status.","section":null},{"comment":"In the proof of Theorem 9.1, the obstruction theory computation uses Serre's result that π_i(S)⊗F_p = 0 for 0 < i < 2p−3. The bound N ≥ q/2 + 1 is derived, but the exact relationship between q and the vanishing range could be stated more explicitly.","section":null}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about diagram (8.6) is well-founded. This is the single most consequential unverified link in the chain Theorem 1.5 → 1.15 → 6.17 → 8.4 → 1.23 → 1.27. The author's statement that commutativity 'follows from Theorem 7.14 and the diagrams (6.13) and (6.14)' is plausible but not sufficient for a rigorous verification, given the complexity of the diagram (roughly a dozen objects with multiple E∞-ring maps and module structures). The central algebraic results (Theorems 1.5, 1.15) are sound and well-proven. The symplectic Assumptions A–G are clearly flagged as conditional and external to the paper's algebraic contribution. The recommendation of major revision is driven by the need for a more detailed verification of diagram (8.6) and the module-structure compatibilities in Theorem 8.4, which are load-bearing for Theorem 1.23 and thus for the paper's main advertised application (Theorem 1.27)."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and substantive report. The core algebraic results (Theorems 1.5, 1.15) and the construction of the cap product via prismatic subdivision are acknowledged as sound. The three major comments all concern the proof of Theorem 8.4 in Section 8.2, specifically the verification of commutativity of diagram (8.6) and the role of the p-torsion-free hypothesis. We address these below.","responses":[{"response":"The referee is correct that the verification of diagram (8.6) is insufficiently detailed in the current manuscript. The diagram (8.6) is assembled from three pieces: (i) the middle three rows, which are constructed by Theorem 7.14 and the diagrams (6.13)–(6.14); (ii) the top row, obtained by pushout along Z_p → F_p; and (iii) the bottom row, obtained by pushout along Z_p^{tCp} → F_p^{tCp}. The commutativity of the outer rectangle involving the second and fourth columns of (6.7) induces the vertical arrows connecting rows 2–3 and rows 4–5 of (8.6). What is missing is an explicit lemma that isolates the compatibility of the E∞-ring maps and module structures across this assembly. We will add such a lemma (to be labeled Lemma 8.5 in the revision) that states precisely which compatibilities are needed and verifies them. Specifically, the key points are: (a) the map Z_p → THH(F_p) of E∞-rings in cyclotomic spectra (constructed from TC(F_p) or from the cyclotomic trace) is compatible with the collapse maps to F_p and Z_p respectively, as verified in Lemma 6.6; (b) the commutativity of (6.13) and (6.14) as diagrams of E∞-S^1-rings, which follows from the construction of the cyclotomic structure maps on THH of commutative ring spectra as natural E∞-maps; and (c) the compatibility of the lax monoidal structure of the Tate construction with the pushouts, which is a formal consequence of the symmetric monoidal structure on the Tate construction for cofibrant objects in the model structure of Proposition 2.9. The lemma will make explicit that these three ingredients suffice, and that no additional compatibilities are needed. We believe this addresses the referee's concern, but acknowledge that the current text is inadequate without it.","revision_made":"yes","referee_comment":"Theorem 8.4 is the load-bearing step connecting the spectral noncommutative Cartier formula (Theorem 6.17) to the algebraic inverse Cartier transform. Its proof constructs diagram (8.8) and identifies its top row with (1⊗F'⊗1)∘F*(κ')⊗1, then invokes Lemma 8.1 to conclude this is the Cartier transform. This identification depends on the commutativity of diagram (8.6), a large diagram involving E∞-ring maps between HH(R/F_p), HH(R_{Z_p}/Z_p), THH(R̃), their Tate constructions, and Ω^1. The author states this commutativity 'follows from Theorem 7.14 and the diagrams (6.13) and (6.14),' but the verification requires checking compatibility of multiple E∞-ring maps and module structures across a diagram with roughly a dozen objects. The text does not trace through this verification in sufficient detail for the reader to confirm it. Since this is the single step that cannot be shortcut by a文献引用"},{"response":"This comment is closely related to the first, and we agree that the transport of module structures through (8.8) needs to be made explicit. The key subtlety is that the domain of ϕ_R is THH(A) ∧_{THH(R)} R^{tCp}, where R^{tCp} is an R-module via the Tate-valued Frobenius, while the top row of (8.8) involves HH(A_{Z_p}/R_{Z_p}) ⊗_{R_{Z_p}} R^{tCp}, which is an R_{Z_p}-module. The bridge between these is provided by Proposition 6.12, which identifies THH(A) ∧_{THH(R)} R^{tCp} with HH(A/R) ⊗_R R^{tCp} when R = R̃ ∧ F_p for R̃ a standard cyclotomic base. The identification proceeds through the chain of equivalences in (6.15), which passes through Z_p-coefficients. The module structure compatibility that needs to be checked is that the THH(R)-module structure on the domain of ϕ_R, when transported through the equivalences of (6.15), agrees with the R_{Z_p}-module structure on HH(A_{Z_p}/R_{Z_p}) ⊗_{R_{Z_p}} R^{tCp}. This follows from the fact that all maps in the left column of (8.6) are maps of E∞-S^1-rings, and the module structures are induced by restriction along these maps. We will add an explicit verification of this compatibility as part of the revised Lemma 8.5 mentioned above, and will also expand the discussion surrounding (8.8) to spell out how the module structures are transported at each step.","revision_made":"yes","referee_comment":"Section 8.2, proof of Theorem 8.4: the identification of the top row of (8.8) with the map (1⊗F'⊗1)∘F*(κ')⊗1 is stated to follow from 'the outer square (consisting of rows 2 and 4 mapping to rows 1 and 5) of the diagram (8.6), together with the factorization claim about F' in Theorem 7.14.' The reader needs to see how the module structures over the various E∞-rings in (8.6) are transported through (8.8), particularly how the T HH(R)-module structure on the domain of ϕ_R interacts with the R_{Z_p}-module structure appearing in the top row of (8.8). A more explicit verification, or at minimum a lemma isolating the compatibility of module structures across (8.6), would strengthen this critical step."},{"response":"We can clarify this point. The p-torsion-free condition enters in two distinct places, and the commutativity of (8.6) itself does not require it. Specifically: (1) The commutativity of diagram (8.6) is a statement about E∞-ring maps and module structures involving THH, HH, and Tate constructions of the base rings R, R_{Z_p}, and F_p. This commutativity holds unconditionally — it is a formal consequence of the functoriality of THH as a symmetric monoidal functor to cyclotomic spectra, the properties of the map Z_p → THH(F_p), and the lax monoidal structure of the Tate construction. No p-torsion-free hypothesis is needed here. (2) The p-torsion-free condition enters through Proposition 6.16, which requires that à be dualizable over R̃ (i.e., smooth and proper) so that ϕ_R is an equivalence. The p-torsion-free condition on HH*(C/R) is used to ensure that the lift à over S[1/N] (or a related spherical lift) exists and is smooth and proper, via Theorem 9.1 and Proposition 9.12. The p-torsion-free condition ensures that the relevant obstruction groups vanish for p large enough, allowing the inductive construction of the lift. (3) Additionally, the p-torsion-free condition is used in the comparison of Theorem 7.16, where two connections on a free module with the same flat sections are identified; this requires the module to be p-torsion-free (see Remark 10.13). We will add a remark after Theorem 8.4 making this threefold role of the p-torsion-free condition explicit.","revision_made":"yes","referee_comment":"Theorem 1.23 requires that HH*(C/R) is p-torsion-free, which restricts to p > f(m,n,N,r). The proof of Theorem 1.23 also cannot use Petrov-Vaintrob-Vologodsky [93] directly because 'there is no comparison between Kaledin's noncommutative Cartier map and corresponding spectral constructions available in the literature' (Section 8). This necessitating an independent reproof. While the reproof via Theorem 8.4 is a reasonable strategy, the p-torsion-free hypothesis is essential for the identification in Theorem 8.4 (via Proposition 6.16, which requires dualizability and thus the lift to a standard cyclotomic base). The author should clarify whether the p-torsion-free condition enters only through the applicability of Proposition 6.16, or whether it is also needed for the commutativity of (8.6) itself."}],"tokens_in":67239,"tokens_out":1951,"duration_ms":213503,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The headline result is a noncommutative Cartier formula: for any E1-ring spectrum A, the cap product on THH(A) is compatible with the cyclotomic structure map via the HHR diagonal and p-fold cap product (Theorem 1.5), with a relative version over a base R (Theorem 1.15). This is a new result and the proof strategy is clean and convincing. The author uses explicit point-set models — the prismatic subdivision of McClure-Smith for the cap product, and the HHR norm for the cyclotomic structure — and the key diagram (4.5) commutes essentially by construction. Homotopical control is handled by the convenient model structures of [20] (Proposition 2.9), which is the right tool. The proof of Theorem 1.5 is about as straightforward as one could hope for. The relative version (Theorem 1.15) follows the same pattern, with an additional subtlety about THH(R)-module structures that is handled carefully in Appendix C. Section 7 does a good job connecting the spectral formula back to classical algebra, recovering Cartier's formula for F_p[x] via Hesselholt's computation of THH(S[x]). This part is concrete and checks out. The lifting argument in Section 9 (obstruction theory to lift Z[1/N]-algebras to S[1/N]-algebras) is a nice ingredient that makes the results applicable to Fukaya categories. The soft spot is Theorem 8.4, which bridges the spectral Cartier formula to the algebraic inverse Cartier transform. The proof constructs diagram (8.8) and identifies its top row with the Cartier transform, relying on the commutativity of diagram (8.6) — a large diagram with roughly a dozen objects involving HH(R/F_p), HH(R_Zp/Z_p), THH constructions, and Ω^1. The author states this commutativity follows from Theorem 7.14 and diagrams (6.13) and (6.14), but the verification requires checking compatibility of multiple E∞-ring maps and module structures across the diagram, and this is not fully traced through. This is the one step that cannot be shortcut by citing Petrov-Vaintrob-Vologodsky [93], because the comparison between Kaledin's algebraic Cartier map and the spectral construction is unavailable — hence the independent reproof. The individual ingredients are established earlier with reasonable arguments, so the assembly is plausible, but this is where an expert referee should focus. The symplectic application (Theorem 1.27) is clearly flagged as conditional on Assumptions A-G, particularly Assumption E (equivariant open-closed comparison). The author is transparent that these are consequences of the state of Fukaya-categorical foundations, not geometric hypotheses. This is fine — the algebraic contribution stands independently. The cyclotomic Deligne conjecture (Appendix A) is stated as a conjecture and the paper only verifies one instance; the paper does not depend on the general version. I agree with the reader's conditional verdict. The significance is high if the proofs hold up. The paper deserves a serious referee who can carefully check the diagram chases in Section 8, particularly the commutativity of (8.6).","headline":"Algebraic core (Theorems 1.5, 1.15) is genuinely new and the proof strategy is sound; the bridge to algebra (Theorem 8.4) is the load-bearing soft spot that needs careful checking.","tokens_in":68123,"tokens_out":1768,"would_cite":true,"duration_ms":96410,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Cap product meets cyclotomic structure in noncommutative Cartier formula","keywords":[],"falsifier":"Find a smooth proper dg category C over Z[1/N][[x]] where the p-curvature of the Getzler-Gauss-Manin connection provably disagrees with the equivariant p-fold cap product formula of Theorem 1.23, or exhibit a Calabi-Yau symplectic manifold satisfying Assumptions A-G where the p-curvature of the quantum connection disagrees with the Quantum Steenrod operations.","tokens_in":67369,"feed_emoji":"🔄","tokens_out":830,"duration_ms":83629,"temperature":0.7,"pith_summary":"The paper proves that for every E_1-algebra A, the cap product action of topological Hochschild cohomology (THC) on topological Hochschild homology (THH) is compatible with the cyclotomic structure map on THH. This compatibility is expressed through a commutative square relating the ordinary cap product, the p-fold cap product via the Hill-Hopkins-Ravenel norm, and the cyclotomic structure map. The author extends this to a relative setting over a base ring R, and shows that when specialized to R = F_p, the formula recovers and generalizes the classical Cartier formula describing how the Cartier isomorphism conjugates interior products on differential forms. The paper then applies this algebraic machinery to compute the p-curvature of the Getzler-Gauss-Manin connection on periodic cyclic homology as an equivariant p-fold cap product with the Kodaira-Spencer class. Finally, under standard assumptions on the Fukaya category, the author shows that for Calabi-Yau symplectic manifolds with rational symplectic form, the p-curvature of the quantum connection computes the Quantum Steenrod operations, giving an algebro-geometric interpretation of these equivariant Gromov-Witten invariants.","feed_headline":"Cap product meets cyclotomic structure in noncommutative Cartier formula","feed_subtitle":"Formula linking cap products and cyclotomic maps on THH recovers classical Cartier isomorphism and computes quantum Steenrod operations viap","key_machinery":"The key objects are: (1) the p-fold cap product, constructed using the edgewise subdivision and the Hill-Hopkins-Ravenel norm; (2) the HHR diagonal, which is a point-set homeomorphism on cofibrant objects; (3) the relative Tate diagonal, which serves as a noncommutative analog of the Cartier isomorphism; (4) the Kodaira-Spencer class, whose cap product computes the Getzler-Gauss-Manin connection; and (5) the inverse Cartier transform, whose p-curvature is computed and matched with the p-curvature of the GGM connection via the cyclotomic structure.","core_discovery":"The central mechanism is a commutative square (Theorem 1.5) showing that the cyclotomic structure map on THH intertwines the cap product with the p-fold cap product via the HHR diagonal. This square is proved by constructing the cap product using the prismatic subdivision of McClure-Smith and verifying that it commutes with the cyclotomic structure defined through the norm, reducing to a combinatorial check. The relative version (Theorem 1.15) identifies the domain of the relative cyclotomic structure map with classical Hochschild invariants when the base is F_p, using the key diagram relating Z_p, THH(F_p), and F_p via Tate constructions. This identification, combined with the comparison of","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Cyclotomic structure on THH intertwines with cap product via noncommutative Cartier formul","Cap product and cyclotomic structure commute on THH: a noncommutative Cartier formula","Noncommutative Cartier formula links cap products to cyclotomic maps on THH","Cap product commutes with cyclotomic structure on THH: noncommutative Cartier formula","Commutative square on THH yields noncommutative Cartier formula over F_p"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The symplectic application (Theorem 1.27) depends on Assumptions A-G, particularly that the open-closed map is an isomorphism (nondegeneracy), which is a geometric condition requiring enough Lagrangian submanifolds, and on equivariant open-closed comparison results whose foundations are still being developed.","fun_headline_variants_meta":{"raw":{"variants":["Cyclotomic structure on THH intertwines with cap product via noncommutative Cartier formula","Cap product and cyclotomic structure commute on THH: a noncommutative Cartier formula","Noncommutative Cartier formula links cap products to cyclotomic maps on THH","Cap product commutes with cyclotomic structure on THH: noncommutative Cartier formula","Commutative square on THH yields noncommutative Cartier formula over F_p","Noncommutative Cartier formula computes p-curvature of quantum connection","Cyclotomic THH cap product square recovers classical Cartier isomorphism","Quantum Steenrod operations arise as p-curvature via noncommutative Cartier formula"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1269,"prompt_tokens":642,"completion_tokens":627,"prompt_tokens_details":null},"tokens_in":642,"tokens_out":627,"duration_ms":42921,"temperature":1.0,"reasoning_tokens":431,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-07T14:47:12.427961+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Find a smooth proper dg category C over Z[1/N][[x]] where the p-curvature of the Getzler-Gauss-Manin connection provably disagrees with the equivariant p-fold cap product formula of Theorem 1.23, or exhibit a Calabi-Yau symplectic manifold satisfying Assumptions A-G where the p-curvature of the quantum connection disagrees with the Quantum Steenrod operations.","supporting_citations":[],"review_version":1}