{"id":"569ad7eb-b761-4589-a0b9-b30338bba141","arxiv_id":"2412.00914","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper defines an r-Nygaard filtration on prismatic cohomology and identifies the motivic filtration graded pieces of TR^r and its S1-fixed points with this filtration.","lead":"The paper introduces an r-Nygaard filtration on prismatic cohomology and uses it to describe the motivic filtration of topological restriction homology TR^r and its S1-fixed points. It extends the known r=1 story of Bhatt-Morrow-Scholze to all r, with applications to Witt vectors and de Rham-Witt complexes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of local odd vanishing for TR^r (r≥3) in §4.2 uses a diagonal precomposition that cannot be surjective onto the product; the cited BS22 theorem is misapplied.","rationale":"The reader correctly identified the invocation of [BS22, Sec. 14] as the point where the proof of Theorem 1.5 is least secure. My stress test sharpens this: the precomposition with the diagonal map not only lacks a hypothesis check; it is formally incapable of producing the claimed surjectivity for r≥3 unless the target product collapses. The perfectoid calculations in §3 appear sound, and the QRSPerfd case in §4.1 has a plausible exact-sequence argument, but the passage to all quasisyntomic rings via descent is where the main theorem's validity hinges. Since the theorem may still be true (parts are independently addressed in [DR23], [DM23]), the appropriate verdict is CONDITIONAL: the paper must supply a correct proof of local odd vanishing or a precise citation to a result that covers TR^r for all r. My proposed test is a direct algebraic check that would confirm the flaw or show the argument is repairable.","tokens_in":42354,"tokens_out":20909,"duration_ms":186603,"concrete_test":"Fix r=3 and let α: (N^{≥i}\\hatΔ_S)^3 → (\\hatΔ_S)^2 be the map (a,b,c) ↦ (a-φ_i(b), b-φ_i(c)) obtained from the equalizer presentation. Verify whether α is surjective after quasisyntomic sheafification when the single map x ↦ x-φ_i(x) is surjective. Concretely, work in the algebraic model where M = \\hatΔ_S is a module, N = N^{≥i}\\hatΔ_S is a submodule, and φ: N → M satisfies that n ↦ n-φ(n) is surjective; determine whether (a,b,c) ↦ (a-φ(b), b-φ(c)) is surjective on M^2. Exhibit a counterexample if not. This settles whether the diagonal-precomposition proof in §4.2 can be repaired or must be replaced by a different argument for r≥3.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most load-bearing gap is in the proof of Theorem 1.5(2), §4.2. To show that π_{2i-1}TR^r(S;Z_p)^{hS1} vanishes locally on QSyn, the author considers the exact sequence 0 → N^{≥i}_r\\hatΔ_S{i} → ∏_{1≤k≤r} N^{≥i}\\hatΔ_S{i} →^{α} ∏_{1≤k≤r-1} \\hatΔ_S{i} → π_{2i-1}TR^r(S;Z_p)^{hS1} → 0, and precomposes α with the diagonal diag: N^{≥i}\\hatΔ_S{i} → ∏_{1≤k≤r} N^{≥i}\\hatΔ_S{i}, claiming that BS22 Sec. 14 makes α∘diag surjective on a quasisyntomic cover. But from the iterated pullback presentation of TR^r hS1 (§2.2), the map α is the banded difference (a_1,...,a_r) ↦ (can(a_1)-φ(a_2), ..., can(a_{r-1})-φ(a_r)). Hence α∘diag(x) = (can(x)-φ(x), ..., can(x)-φ(x)), whose image lies in the diagonal of ∏_{1≤k≤r-1}\\hatΔ_S{i}. For r≥3, the diagonal is a proper subset of the product, so α∘diag cannot be surjective; the BS22 vanishing theorem, which concerns the single map can-φ, does not imply surjectivity of this product-valued diagonal map. Thus the local odd vanishing for TR^r is not established by the cited argument. This is load-bearing: without it, the graded pieces of the motivic filtration are not identified with N^{≥i}_r\\hatΔ_S{i}[2i] for general quasisyntomic rings, and Theorem 1.5(2)-(3), Theorem 1.6, and the corollaries in §6 rest on this step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces, for an animated ring S and 1 <= r <= infinity, an 'r-Nygaard filtration' N^{>=i}_r Delta_S, defined as an iterated pullback of the usual Nygaard filtration under the divided prismatic Frobenius, and studies its relation to the motivic filtration of TR^r(S;Z_p) and of its S^1-homotopy fixed points. The main theorem (Theorem 1.5) asserts that, for quasisyntomic S, gr^i_M TR^r(S;Z_p)^{hS^1} is equivalent to N^{>=i}_r \\hatDelta_S{i}[2i] and gr^i_M TR^r(S;Z_p) is equivalent to N^i_r \\hatDelta_S{i}[2i], with analogous statements for TR, TC-related invariants, and the mixed- and positive-characteristic cases. The paper also proposes algebraic constructions on the prismatization stack, including r-Hodge-Tate divisors and a conjugate filtration, and announces a comparison with de Rham-Witt complexes.","tokens_in":42833,"tokens_out":7667,"duration_ms":67317,"significance":"The potential significance is high: the paper offers a natural r-parameter generalization of the Bhatt-Morrow-Scholze picture, giving motivic filtrations of TR^r in terms of prismatic cohomology with an r-fold Nygaard filtration. The perfectoid calculations in Section 3 are detailed and explicit, and the definitions are intrinsic and parameter-free. The paper also makes concrete structural predictions, such as the spectral sequences in Theorem 1.5(3). However, the general quasisyntomic claim is not established by the arguments given: the proof of local odd vanishing in Section 4.2 contains a substantive gap, and the algebraic descriptions in Section 5 are partly asserted rather than proved. At this stage the paper reads as a promising research announcement with a useful framework, but not as a complete proof of the main theorems.","major_comments":[{"comment":"The local odd-vanishing argument is not valid as written. In the exact sequence 0 -> N^{>=i}_r \\hatDelta_S{i} -> ∏_{1≤k≤r} N^{>=i}\\hatDelta_S{i} -> ∏_{1≤k≤r-1} \\hatDelta_S{i} -> π_{2i-1}TR^r(S;Z_p)^{hS^1} -> 0, the map α is induced by the banded difference (a_1,...,a_r) ↦ (can(a_1)-φ(a_2), ..., can(a_{r-1})-φ(a_r)). Precomposing with the diagonal gives α∘diag(x) = (can(x)-φ(x), ..., can(x)-φ(x)), whose image is contained in the diagonal of ∏_{1≤k≤r-1} \\hatDelta_S{i}. For r ≥ 3 the diagonal is a proper submodule of the product, so surjectivity of α∘diag onto the product after a quasisyntomic cover cannot follow from the Bhatt-Scholze vanishing theorem, which concerns the single map can-φ. Consequently the local vanishing of π_{2i-1}TR^r(-;Z_p)^{hS^1} for r ≥ 3 is not established. Since this vanishing is used to identify gr^i_M TR^r and gr^i_M TR^{r,hS^1} with N^{>=i}_r \\hatDelta_S{i}[2i] and N^i_r \\hatDelta_S{i}[2i], and to pass to TR and TC, Theorem 1.5(2)-(3), Theorem 1.6, and Section 6 rest on an unproved step. The author should either prove the required surjectivity directly for the product map or give a different argument for local odd vanishing.","section":"§4.2, proof of Theorem 1.5(2)"},{"comment":"Proposition 5.6 is load-bearing but is not proved; the text says 'one arrives at the following generalization' after Proposition 5.5. The claimed recursive identification (φ^r)^* N^{>=i}_r \\hatDelta_S ≃ φ^*N^{>=i}\\hatDelta_S ⊗_{\\hatDelta_S} ... ⊗_{\\hatDelta_S} (φ^r)^*N^{>=i}\\hatDelta_S is used in Lemma 5.7, Proposition 5.9, Corollary 5.14, and the conjugate-filtration construction in Section 5.2. Without a proof or a precise reference, the algebraic definition of the r-Nygaard filtration is not shown to coincide with the homotopy-theoretic filtration outside the quasiregular-semiperfectoid case.","section":"§5.1, Proposition 5.6"},{"comment":"The conjugate filtration on r-Hodge-Tate cohomology and its identification with the Postnikov filtration are asserted as direct corollaries of replacing I^• by I_r^• in [BL22a]. This is not a formal substitution: the Beilinson t-structure and décalage arguments for the pair (I, φ) do not automatically carry over to the pair (I_r, φ^r), especially since Proposition 5.6, which would supply the needed Nygaard-connectivity estimates, is unproved. Because Theorem 1.3 and the relative de Rham-Witt comparison (6) depend on these identifications, the relative statements currently have the status of announced results rather than proved theorems.","section":"§5.2, Corollary 5.15"},{"comment":"The passage from quasiregular-semiperfectoid rings to all quasisyntomic rings is delegated to 'completely analogous' to [BMS19]. For TR^r this requires checking that the quasisyntomic sheafifications of τ_{[2i-1,2i]}TR^r(-;Z_p)^{hS^1} and τ_{[2i-1,2i]}TR^r(-;Z_p) are the two-term complexes associated to the r-Nygaard filtration, and that the motivic filtrations are the sheafifications of the double-speed Postnikov filtrations. These identifications are not themselves established in the QRSPerfd calculation; they are part of what the theorem must prove. A detailed descent argument is needed, particularly because the local odd-vanishing claim in the previous comment is the only place where the QRSPerfd calculation is shown to survive quasisyntomic sheafification.","section":"§4.2, proof of Theorem 1.5 (general case)"}],"minor_comments":[{"comment":"The sentence 'the identification regarding \\hatDelta_S/ξ_r is a direct corollary of Proposition ??' contains an unresolved cross-reference; the cited proposition should be numbered explicitly.","section":"§4.1"},{"comment":"The stated source and target of the Restriction and Frobenius maps are inconsistent: R,F: TR^{r+1}(R_0;Z_p) -> TR^r(R_0;Z_p)^{hS^1} does not match Theorem 3.7, where the corresponding maps are between TR^{r+1} and TR^r, with homotopy fixed points handled separately.","section":"§3.2, Proposition 3.8"},{"comment":"The spectra TCr are defined using TR^r and TR^{r-1}; for r = 1 the object TR^0 is not defined, and the claimed interpolation at r = 1 should be stated with an explicit convention or a separate base-case formula.","section":"§2.4, Construction 2.18"},{"comment":"There are numerous typographical errors, including 'desription' in Section 1, 'semperfect' in Definition 2.21, and 'cnstructions' in Definition 5.2; a careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The editor may wish to ask the author to clarify the relation between this manuscript and the PhD thesis [And24]; the acknowledgements state that the article stems from the thesis, but the text does not specify which results are already contained in the thesis. The paper also cites [DR23] and [DM23] in Theorem 1.8; the novelty in the positive-characteristic case should be stated more precisely."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The r-Nygaard filtration is a genuine new construction and the perfectoid computations look right, but the proof of Theorem 1.5 has a load-bearing gap: the local odd vanishing argument for r ≥ 3 does not work as written. The stress-test concern lands, and the referee should be pointed at it directly.\n\nWhat is actually new: the filtration N^{≥i}_r Δ_S, built by gluing r copies of the usual Nygaard filtration via the iterated pullback, with its Restriction/Frobenius/Verschiebung symmetries, the r-divided prismatic Frobenius, and the r-Hodge–Tate cohomology. For perfectoid inputs it becomes the ξ_r-adic filtration on A_inf, and Section 3 is the strongest part of the paper: the computation π_*TR^r(R_0)^{hS1} ≅ A_inf[u_r,v_r]/(u_r v_r − ξ_r), the passage to TR^r by killing v_r, and the R/F/V bookkeeping all appear correct. The algebraic definitions in Section 5 and the formal part of Theorem 1.6 are reasonable and worth keeping.\n\nThe soft spot is in §4.2. To show π_{2i-1}TR^r(S)^{hS1} vanishes locally on the quasisyntomic site, the author precomposes the map α: ∏_r N^{≥i}Δ̂_S → ∏_{r-1} Δ̂_S with the diagonal and claims BS22 Sec. 14 makes α∘diag surjective on a cover. But α is the banded difference, so α∘diag(x) = ((ι−φ_i)(x), …, (ι−φ_i)(x)); its image is the diagonal of the product. For r ≥ 3 the diagonal is a proper submodule, and no quasisyntomic cover changes that. The argument proves nothing about α. This step carries Theorem 1.5(2)–(3), Theorem 1.6, and the §6 corollaries. For r = 2 the diagonal argument would work, and r = 1 is BMS19; the gap is r ≥ 3, which is most of the statement.\n\nMinor but real: the same diagonal move appears in the §6.2 proof of Theorem 1.8 with a mismatched product target; the overlap with DR23 and DM23 is mentioned but not delineated — if Theorem 1.8's content is in those papers, the author should say so and lean on it; there is a broken internal reference ('Proposition ??' in §4.1); and several 'completely analogous'/'direct corollary' claims (Proposition 5.6, the conjugate filtration identifications in §5.2) are unproved in the text.\n\nThe audience is researchers in p-adic Hodge theory, especially the THH/TR side and de Rham–Witt comparisons; the filtration and the perfectoid section carry real value for them. The reader's 'conditional' verdict is about right, with the stress-test showing where the condition bites. Recommendation: send to peer review, with a referee brief focused on the local odd vanishing for r ≥ 3 and the precise relationship to DR23/DM23.","headline":"The r-Nygaard filtration and the perfectoid computations are real contributions, but the quasisyntomic descent step in §4.2 rests on a diagonal-precomposition argument that cannot work for r ≥ 3, so Theorem 1.5 is not yet proven as stated.","tokens_in":43344,"tokens_out":14809,"would_cite":true,"duration_ms":124426,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","19D55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes that the motivic filtrations of TR^r and its S^1-fixed points have graded pieces given by the r-Nygaard filtered prismatic cohomology.","keywords":["prismatic cohomology","Nygaard filtration","topological restriction homology","motivic filtration","quasisyntomic descent","topological cyclic homology","de Rham–Witt complex","perfectoid rings"],"falsifier":"For $S=\\mathbb{Z}_p\\langle x\\rangle$, compute the cokernel of the map $\\alpha$ in the exact sequence of Section 4.2 after quasisyntomic sheafification; a nonzero local odd class in $\\pi_{2i-1}\\mathrm{TR}^r(S;\\mathbb{Z}_p)^{hS^1}$ would falsify the identification of the graded pieces in Theorem 1.5.","tokens_in":42135,"feed_emoji":"📐","tokens_out":11794,"duration_ms":95845,"temperature":0.7,"pith_summary":"Topological restriction homology $\\mathrm{TR}^r$ is an invariant that packages the $r$-truncated Witt vectors and, through the Frobenius map $1-F$, builds topological cyclic homology, a close relative of algebraic $K$-theory. The paper sets out to prove that, for quasisyntomic rings, $\\mathrm{TR}^r$ and its $S^1$-homotopy fixed points carry a motivic filtration whose even graded pieces are computed by prismatic cohomology equipped with a new filtration, called the $r$-Nygaard filtration. The odd graded pieces vanish locally in the quasisyntomic topology, so the filtration is completely governed by the even layers. If the main theorem is right, it connects the $\\mathrm{TR}^r$ side of p-adic homotopy theory to prismatic cohomology and gives a concrete target for a prismatic comparison with de Rham–Witt complexes.","feed_headline":"A new r-Nygaard filtration computes TR^r's motivic layers","feed_subtitle":"Even graded pieces of TR^r and its S1-fixed points match prismatic cohomology filtered r times by Nygaard.","key_machinery":"The central object is the $r$-Nygaard filtration on absolute prismatic cohomology, defined for $1\\le r\\le\\infty$ by the iterated pullback $N^{\\ge i}_r\\Delta_S\\{i\\} := N^{\\ge i}\\Delta_S\\{i\\}\\times_{\\Delta_S\\{i\\}}\\cdots\\times_{\\Delta_S\\{i\\}} N^{\\ge i}\\Delta_S\\{i\\}$ along the canonical inclusion and the divided prismatic Frobenius. On the homotopy-theoretic side the matching tool is the iterated pullback presentation $\\mathrm{TR}^r(S;\\mathbb{Z}_p)^{hS^1}\\simeq\\mathrm{TC}^-(S;\\mathbb{Z}_p)\\times_{\\mathrm{TP}(S;\\mathbb{Z}_p)}\\cdots\\times_{\\mathrm{TP}(S;\\mathbb{Z}_p)}\\mathrm{TC}^-(S;\\mathbb{Z}_p)$, which lets the even homotopy groups be read off as the filtration pieces; passing from the $S^1$-fixed points back to $\\mathrm{TR}^r$ is done by killing the periodicity class $v_r$. For perfectoid rings the filtration is simply the $\\xi_r$-adic filtration on $A_{\\mathrm{inf}}$. The $r$-divided prismatic Frobenius $\\phi_{r,i}$ and its graded version to the $r$-Hodge–Tate cohomology connect the algebraic filtration to the higher Frobenius maps on $\\mathrm{TR}^r$.","core_discovery":"Theorem 1.5 states that for a quasisyntomic ring $S$ and $1\\le r\\le\\infty$, the motivic filtrations of $\\mathrm{TR}^r(S;\\mathbb{Z}_p)^{hS^1}\\to\\mathrm{TR}^r(S;\\mathbb{Z}_p)$ are complete, exhaustive, multiplicative and $\\mathbb{Z}$-indexed, and that their graded pieces are identified after quasisyntomic descent with the $r$-Nygaard filtered Nygaard-completed prismatic cohomology: $\\mathrm{gr}^{i,\\mathrm{even}}_M\\mathrm{TR}^r(S;\\mathbb{Z}_p)^{hS^1}\\simeq N^{\\ge i}_r\\hat\\Delta_S\\{i\\}[2i]$ and $\\mathrm{gr}^{i,\\mathrm{even}}_M\\mathrm{TR}^r(S;\\mathbb{Z}_p)\\simeq N^i_r\\hat\\Delta_S\\{i\\}[2i]$ for finite $r$, with $r=\\infty$ obtained by taking the derived limit over Restriction maps. The odd graded pieces vanish locally in the quasisyntomic topology. The same filtered data controls the spectra $\\mathrm{TC}_r(\\mathrm{TR})$ and $\\widetilde{\\mathrm{TC}}_r(\\mathrm{TR})$, which interpolate between ordinary topological cyclic homology and topological cyclic homology of $\\mathrm{TR}$.","pith_inferences":["Editorial extension: the paper's general principle suggests every Nygaard-filtration statement should have an $r$-fold analogue; the natural next test is a prismatic–de Rham–Witt comparison identifying the absolute conjugate filtration on $r$-Hodge–Tate cohomology with the absolute de Rham–Witt forms, which the paper leaves open.","Editorial extension: because the motivic filtration is defined by quasisyntomic sheafification of double-speed Postnikov filtrations, the $r$-Nygaard filtration should be regarded as a canonical invariant of $\\mathrm{TR}^r$ itself; one could exploit this to define $r$-Nygaard filtered analogs of syntomic cohomology and compare them with étale motivic cohomology for $r>1$.","Editorial extension: the perfectoid base case $\\pi_*\\mathrm{TR}^r(R_0;\\mathbb{Z}_p)^{hS^1}\\simeq A_{\\mathrm{inf}}(R_0)[u_r,v_r]/(u_rv_r-\\xi_r)$ suggests a direct computational check of the theorem on $p$-complete polynomial rings, where the filtration should be the connective cover of the $I_r$-adic filtration."],"forward_implications":["For a quasisyntomic ring and finite $r$, $\\mathrm{TR}^r(S;\\mathbb{Z}_p)^{hS^1}$ and $\\mathrm{TR}^r(S;\\mathbb{Z}_p)$ are locally even, so their motivic spectral sequences degenerate locally.","The even motivic layers are exactly $N^{\\ge i}_r\\hat\\Delta_S\\{i\\}[2i]$ for the $S^1$-fixed points and $N^i_r\\hat\\Delta_S\\{i\\}[2i]$ for $\\mathrm{TR}^r$, so the $r$-Nygaard filtration completely controls the even layers.","Taking the limit over Restriction maps identifies the layers of $\\mathrm{TR}$ and $\\mathrm{TR}^{hS^1}$ with $N^{\\ge i}_\\infty\\hat\\Delta_S\\{i\\}[2i]$ and $N^i_\\infty\\hat\\Delta_S\\{i\\}[2i]$, with odd layers coming from $\\mathrm{Rlim}^1$ terms.","The spectra $\\mathrm{TC}_r(\\mathrm{TR})$ and $\\widetilde{\\mathrm{TC}}_r(\\mathrm{TR})$ carry motivic filtrations whose graded pieces are fibers of $R-F$ or $\\mathrm{can}-\\phi^{hS^1}$ on the $r$-Nygaard pieces, interpolating between $\\mathrm{TC}(\\mathrm{TR})$ and ordinary $\\mathrm{TC}$.","In mixed and positive characteristic the filtration recovers $A_\\Omega$-cohomology and, for smooth algebras over perfect fields, the $r$-truncated de Rham–Witt forms, $\\mathrm{gr}^i_M\\mathrm{TR}^r(S;\\mathbb{Z}_p)\\simeq\\tau_{\\le i}W_r\\Omega^\\bullet_{S/k}[2i]$."],"supporting_citations":[{"why":"Supplies the prismatic cohomology foundations and the vanishing theorem used to pass from quasiregular-semiperfectoid rings to all quasisyntomic rings.","marker":"[BS22]"},{"why":"Provides the motivic-filtration method, quasisyntomic descent, and the perfectoid and QRSPerfd computations that the paper adapts from TC^- and THH to TR^r.","marker":"[BMS19]"},{"why":"Provides the A_inf and A_Omega machinery, the Fontaine-style maps, and the mixed-characteristic de Rham–Witt comparisons that the r-Nygaard filtration specializes to.","marker":"[BMS18]"},{"why":"Provides the Cartier-module formalism and the quotient trick (killing v_r) used to pass from TR^r^{hS1} to TR^r.","marker":"[AN21]"},{"why":"Provides the cyclotomic-spectrum foundations, the Tate orbit lemma, and the description of TC as a fiber of 1-F.","marker":"[NS18]"},{"why":"Provides the stacky prismatic framework for the Nygaard filtration, the Beilinson t-structure, and left Kan extension to animated rings.","marker":"[BL22a]"},{"why":"Provides the Hesselholt–Madsen theorem identifying pi_0 TR^r(A) with W_r(pi_0 A), anchoring the Witt-vector layer N^0_r.","marker":"[HM97]"},{"why":"Provides the relative de Rham–Witt comparison used in the perfect-prism case to identify conjugate-filtration graded pieces with Langer–Zink forms.","marker":"[Mol20]"}],"fun_headline_variants":["r-Nygaard filtration ties TR^r to prismatic cohomology","TR^r's motivic filtration from r-Nygaard prismatic cohomology","r-Nygaard glues Nygaard filtrations to compute TR^r","Motivic grading of TR^r via r-Nygaard filtration","r-Nygaard filtration: TR^r's motivic layers matched"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The step that carries the whole extension from computable rings to all quasisyntomic rings is the assertion that a general vanishing theorem applies to the TR^r pullback complex, forcing the odd homotopy groups to vanish locally; the paper invokes this theorem rather than verifying its hypotheses for TR^r.","fun_headline_variants_meta":{"raw":{"variants":["r-Nygaard filtration ties TR^r to prismatic cohomology","TR^r's motivic filtration from r-Nygaard prismatic cohomology","r-Nygaard glues Nygaard filtrations to compute TR^r","Motivic grading of TR^r via r-Nygaard filtration","r-Nygaard filtration: TR^r's motivic layers matched"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001005,"raw_usage":{"total_tokens":4280,"prompt_tokens":1005,"completion_tokens":3275,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":3174}},"tokens_in":621,"tokens_out":3275,"duration_ms":25298,"temperature":1.0,"reasoning_tokens":3174,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:52:48.557740+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $S=\\mathbb{Z}_p\\langle x\\rangle$, compute the cokernel of the map $\\alpha$ in the exact sequence of Section 4.2 after quasisyntomic sheafification; a nonzero local odd class in $\\pi_{2i-1}\\mathrm{TR}^r(S;\\mathbb{Z}_p)^{hS^1}$ would falsify the identification of the graded pieces in Theorem 1.5.","supporting_citations":[],"review_version":1}