{"id":"2723794d-ad9c-4cce-b32e-911eb1a2a267","arxiv_id":"2412.00929","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Log topological restriction homology over O_C is identified, étale locally, with r-Nygaard filtered log prismatic cohomology and the relative log de Rham-Witt complex.","lead":"This paper claims a new bridge between two ways of computing p-adic invariants of logarithmic schemes: topological restriction homology with log poles and the log de Rham-Witt complex. If correct, it would settle Hesselholt's conjecture for p-completely smooth formal schemes over the p-adic complex numbers.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Odd homotopy groups are explicitly ignored in the proof of Theorem 1.2; the étale odd vanishing needed for Theorem 1.1 is only sketched via an unproved K-theory fibration argument, so the motivic graded-piece identifications are not established.","rationale":"The reader's specified weakest assumption is the identification of the algebraic and homotopy-theoretic constructions of completed log prismatic cohomology (Section 4, stated as 'as expected'). That is a genuine external assumption: if false, the replacement of the r-Nygaard filtration by Lη_{ξ^r} A_Omega and the resulting de Rham–Witt comparison fail. However, I find a more load-bearing internal gap: the proof of Theorem 1.2 explicitly says odd homotopy groups are ignored in the log setting, and Theorem 1.1 depends on their étale vanishing via Corollary 4.4, whose proof is only a sketch and relies on an unproved compatibility of motivic filtrations in a fibre sequence. If odd homotopy groups survive, the graded pieces of the motivic filtration are not the stated Nygaard pieces, and the spectral sequence of Theorem 1.1 does not have the asserted form. This concern is independent of the algebraic/homotopy equivalence and would sink the central claim even if that equivalence holds. The reader's rationale did mention the odd-homotopy gap, which is why I mark partial agreement, but their formal weakest-assumption slot was placed on the prismatic comparison. Given that the author themselves flags the result as conditional and the gaps are structural rather than known counterexamples, the existing CONDITIONAL verdict is the right disposition; my stress-test does not move it.","tokens_in":11526,"tokens_out":7374,"duration_ms":72897,"concrete_test":"Compute the log quasisyntomic sheafification of the odd homotopy group π_{2n+1}TR^r for a non-perfectoid log quasisyntomic ring, e.g., the standard log point (O_C, N → O_C, 1 ↦ 0) or the log affine line, by choosing a log quasiregular-semiperfectoid cover and running the Cech-to-derived descent spectral sequence. If a nonzero odd class survives, Corollary 4.4 is false; if it vanishes for all n and r, the gap is fillable. Alternatively, give a complete proof of Corollary 4.4 by writing out the comparison of motivic filtrations across the fibre sequence of Construction 4.3 and showing the boundary map is strictly filtered with vanishing odd contributions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1(1) requires gr^n_M TR^r to be identified with the r-Nygaard filtered log prismatic cohomology. This graded piece is computed from the double-speed Postnikov filtration of TR^r after log quasisyntomic sheafification, so it includes both even and odd homotopy groups. In the proof of Theorem 1.2 (Section 3) the author states: 'the odd homotopy groups are junk terms... We do not know how to do this in the log setting, therefore, we simply ignore the terms.' Consequently, Theorem 1.2 has at best established the even part of the motivic filtration; it has not established the full filtrations and graded-piece identifications that Theorem 1.1 uses. Corollary 4.4 is supposed to supply the missing odd vanishing étale locally, but its proof is a sketch: it asserts that since the motivic filtration of algebraic K-theory is identified with the double-speed Postnikov filtration in the first two columns of the Hesselholt–Madsen fibre sequence, 'the same happens for the third column.' This requires compatibility of filtrations with the fibre sequence, strictness, and control of boundary maps, none of which is shown. The appeal to [And24a, Thm. 1.5] does not transfer because the non-log odd vanishing of [BS22] is exactly what is unavailable in the log setting. Thus the central comparison and the claimed E2-degenerating spectral sequence rest on an unproven vanishing theorem, independently of the algebraic/homotopy equivalence assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that, étale locally for X = spf S a p-completely smooth affine p-adic formal scheme over spf O_C, the log topological restriction homology TR^r((X,M_X); Z_p) carries a complete, exhaustive, descending, multiplicative motivic filtration whose graded pieces identify with r-Nygaard filtered log prismatic cohomology, and that the cohomology of these graded pieces computes Matsuue's p-complete relative log de Rham–Witt complex. It also claims a spectral sequence converging to TR^r that degenerates on E2. The work builds on the author's preceding non-logarithmic framework [And24a] and on the logarithmic motivic filtration of Binda–et al. [Bin+23], and it is explicitly conditional on the as-yet-unproved identification of algebraic and homotopy-theoretic approaches to completed log prismatic cohomology over perfectoid bases.","tokens_in":11872,"tokens_out":5633,"duration_ms":52775,"significance":"If established, the main theorem would prove a form of Hesselholt's conjectures over algebraically closed p-adic fields, extending the non-logarithmic results of Bhatt–Morrow–Scholze and the author's previous work to the logarithmic setting. The paper is honest about its conditional assumptions and clearly identifies the parts of the argument that are deferred to other preprints. The potential significance is high, and the geometric setup is natural. However, the central claims rest on two load-bearing gaps: the odd homotopy groups of the motivic filtration are explicitly not handled in the proof of Theorem 1.2, and the étale odd vanishing of Corollary 4.4 is only asserted rather than proved. The paper is therefore best viewed as a research announcement whose full theorems are not yet established.","major_comments":[{"comment":"Theorem 1.2 is stated for the full motivic filtration of TR^r((S,Q);Z_p) and of its S^1-homotopy fixed points, with associated graded identified with the two-term complexes τ^{[2n-1,2n]}. The proof, however, ends with: 'the odd homotopy groups are junk terms... We do not know how to do this in the log setting, therefore, we simply ignore the terms.' The odd homotopy groups are part of the two-term complexes and of the claimed complete filtration, so Theorem 1.2 is not proven as stated. Since Theorem 1.1(1) invokes Theorem 1.2 for the graded pieces of log TR^r, this is a load-bearing gap; the odd vanishing must be supplied or the theorem statements weakened accordingly.","section":"Section 3, proof of Theorem 1.2"},{"comment":"The entire proof of étale local odd vanishing is the sentence: 'Therefore, in the first two columns in the above diagram above, this filtration on algebraic K-theory is étale locally identified with that coming from TC, under the trace map. Hence, the same happens for the third column, as well.' This requires compatibility of the motivic filtration with the Hesselholt–Madsen fibre sequence, strictness of the trace map, and control of the boundary map's effect on odd homotopy groups; none of these is shown. The appeal to [And24a, Thm. 1.5] does not transfer because the non-log odd vanishing of [BS22] is exactly what is unavailable in the log setting. Corollary 4.4 is essential for Theorem 1.1(1), so this gap is load-bearing.","section":"Section 4.2, proof of Corollary 4.4"},{"comment":"The paper states in the Introduction that it works 'under the additional assumption that the algebraic approach [ČK19; Kos20; KY23] and the homotopy theoretic approach [Bin+23] to completed log prismatic cohomology over a perfectoid base coincide (as expected)', and Section 4 says this identification is used 'freely'. Construction 4.1 relies on this identification to replace N_r^n Δhat with the décalage filtration Lη_{ξ^r}^{≥n} A_Ω, and this replacement underpins both the graded-piece identifications and the de Rham–Witt comparison in Theorem 1.1. Since the equivalence is not proved, Theorem 1.1 is conditional on a nontrivial conjecture; the condition should be stated as part of the theorem or eliminated.","section":"Introduction and Section 4"},{"comment":"Proposition 3.4 is a key technical step used in the proof of Theorem 1.2 to pass between TR^r and its S^1-homotopy fixed points, but it is stated without proof or reference. The displayed formula involving 'v_r' is also notationally unclear (the quotient by v_r is not defined in the text). This proposition is load-bearing for the identification of the even homotopy groups of TR^r with the r-Nygaard filtered pieces, so a proof or a precise citation is needed.","section":"Proposition 3.4"},{"comment":"Proposition 4.2 establishes the relative log de Rham–Witt comparison for semistable p-adic formal schemes spf S, as in [ČK19]. Theorem 1.1, however, is stated for arbitrary p-completely smooth affine p-adic formal schemes over spf O_C. The paper does not explain how the semistable case is extended to the general smooth case, e.g., by log smooth descent, deformation, or a reduction argument. Without such an argument, Theorem 1.1(2) is not justified.","section":"Proposition 4.2 and Theorem 1.1(2)"}],"minor_comments":[{"comment":"There are numerous typographical errors, including 'conj ectures', 'comple x', 'homotopy theoretic approach t o', and inconsistent capitalization of 'r-nygaard'. A thorough proofreading pass is needed.","section":"Throughout"},{"comment":"The statement of Proposition 3.4 is garbled; the chain of equivalences is not typeset correctly and the notation 'v_r' is not defined. Please clarify what is being asserted.","section":"Proposition 3.4 statement"},{"comment":"The Frobenius equalizer sequence in Theorem 1.1(2) is displayed only partially as 'τ≤n i_* j^* μ^{⊗n}_{p^v} i_* N^n_∞ Δhat / p^v i_* N^n_∞ Δhat / p^v 1−F'; this is incomplete and should be written out fully.","section":"Theorem 1.1(2) display"},{"comment":"Reference [Hes05] is listed as 'The absolute de Rham-Witt complex' with no venue or preprint number; for a published item, full bibliographic data should be provided. Several other references are to preprints with only arXiv numbers, which is acceptable but should be checked for consistency.","section":"References"},{"comment":"The paper repeatedly calls its arguments 'proof sketches'. Given that the main theorems are conditional on unproved assumptions and on vanishing results that are not established, the paper should clearly label the results as conditional or as sketches in the theorems themselves, not only in the surrounding text.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Faidon's note states a substantive conjecture-adjacent comparison: log TR^r over O_C should carry a motivic filtration whose graded pieces are r-Nygaard filtered log prismatic cohomology, identified with the log de Rham-Witt complex. If true, this is a real step toward Hesselholt's conjectures. The construction of the r-Nygaard filtration via iterated pullbacks and the motivic filtration framework for log TR^r are genuinely useful. The paper is also honest about its conditional status: it explicitly drops odd homotopy groups, and it flags the identification of two log prismatic approaches as 'as expected'.\n\nThe problem is exactly where the stress-test lands. In the proof of Theorem 1.2, the odd homotopy groups are ignored, so the motivic filtration is only established on even parts. Theorem 1.1 needs odd vanishing étale locally. Corollary 4.4 is supposed to supply this, but the proof is a sketch: the motivic filtration of K-theory matches the double-speed Postnikov filtration in the first two columns of the Hesselholt–Madsen fibre sequence, so 'the same happens' for the third column. Compatibility of filtrations along the fibre sequence, strictness, and control of boundary maps are not shown. The appeal to the author's non-log result does not transfer, because the non-log odd vanishing is exactly the missing input. Separately, the dependence on the algebraic/homotopy-theoretic identification of log prismatic cohomology is load-bearing; if that identification fails, the identification of the Nygaard filtration with décalage on log A_Omega collapses. These are not minor gaps; they are the proof of Theorem 1.1.\n\nThat said, the paper is not circular and not sloppy. It is a research announcement written by someone who knows the machinery. The claims are plausible, and the author has correctly identified what needs to be checked. The audience is log THH and p-adic K-theory people who want a clear target and a map of open points. It deserves a serious referee, but the referee should focus on Corollary 4.4 and the identification of log prismatic approaches. As it stands, I would not cite the theorem as proven, and I would ask for a revision that either proves the odd vanishing or restricts the main theorems to the even part.","headline":"Plausible and honestly conditional, but the main theorem is not proved: the odd homotopy groups are explicitly dropped, and the étale vanishing meant to replace them is asserted without a real proof.","tokens_in":826,"tokens_out":1513,"would_cite":false,"duration_ms":29960,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F30","14F42","19D55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For p-completely smooth log formal schemes over the p-adic complex numbers, log $\\mathrm{TR}^r$ is filtered by Nygaard-graded log prismatic cohomology, whose cohomology is the log de Rham–Witt complex.","keywords":["log topological restriction homology","log de Rham–Witt complex","log prismatic cohomology","r-Nygaard filtration","motivic filtration","p-adic formal schemes","perfectoid rings","topological Hochschild homology with log poles"],"falsifier":"A concrete test would be to compute the $n$-th graded piece of the motivic filtration on log $\\mathrm{TR}^r$ for a specific log smooth base over $\\mathcal{O}_C$, such as the canonical log structure on the $p$-adic affine line, once through the algebraic log $A_\\Omega$/log prismatic construction and once through the homotopy-theoretic log prismatic construction; if the two $r$-Nygaard filtered complexes differ, or if the spectral sequence of Theorem 1.1 fails to degenerate on $E_2$, the central claim is false.","tokens_in":11310,"feed_emoji":"🧮","tokens_out":14411,"duration_ms":116809,"temperature":0.7,"pith_summary":"This paper aims to prove, for p-completely smooth affine p-adic formal schemes over the field of p-adic complex numbers, a conjecture relating topological restriction homology with logarithmic poles ($\\mathrm{TR}^r$) to the absolute log de Rham–Witt complex. The central claim is that, étale locally, $\\mathrm{TR}^r$ carries a motivic filtration whose graded pieces are the graded pieces of the $r$-Nygaard filtration on log prismatic cohomology, identified with $\\tau_{\\leq n}A_\\Omega/\\xi^r$, and that the cohomology of these pieces is the relative log de Rham–Witt cohomology $W_r\\Omega^{n,\\mathrm{cont}}_{(-,-)/\\mathcal{O}_C}$. If this holds, the associated spectral sequence degenerates on the second page and computes log $\\mathrm{TR}^r$ completely, supplying the predicted de Rham–Witt description over $\\mathcal{O}_C$. The proof is conditional on an explicit assumption, stated as 'as expected', that the algebraic and homotopy-theoretic constructions of completed log prismatic cohomology over a perfectoid base agree.","feed_headline":"Logarithmic TR homology matches de Rham–Witt forms","feed_subtitle":"Graded pieces are Nygaard-filtered log prismatic cohomology; their cohomology is the log de Rham–Witt complex.","key_machinery":"The load-bearing object is the log $r$-Nygaard filtration $N^{\\geq n}_r\\widehat{\\Delta}_{(S,Q)/\\mathbb{A}_{\\mathrm{inf}}}\\{n\\}$, an $r$-fold iterated pullback of the ordinary Nygaard filtration on completed log prismatic cohomology. In the smooth case this filtration is identified with the décalage filtration $L\\eta^{\\geq n}_{\\xi^r} A_\\Omega$, and it is what the even parts of the motivic filtration on log $\\mathrm{TR}^r$ and on its $S^1$-fixed points are shown to be. The motivic filtration itself is produced by log quasisyntomic sheafification of the double-speed Postnikov filtration from the log quasiregular-semiperfectoid case, which transfers homotopy-theoretic information about $\\mathrm{THH}$ and its Frobenius into an algebraic filtration. The de Rham–Witt comparison then runs through the conjugate filtration on log $r$-Hodge–Tate cohomology and general properties of the décalage functor.","core_discovery":"The paper's central result is that for a p-completely smooth affine p-adic formal scheme $X=\\operatorname{spf} S$ over $\\operatorname{spf} \\mathcal{O}_C$, equipped with the canonical log structure from its generic fibre, the spectrum $\\mathrm{TR}^r((X,\\mathcal{M}_X);\\mathbb{Z}_p)$ admits, étale locally, a complete, exhaustive, descending, multiplicative motivic filtration. The $n$-th graded piece is identified with the $n$-th graded piece of the $r$-Nygaard filtration on log prismatic cohomology, equivalently with $\\tau_{\\leq n} A_\\Omega/\\xi^r$, and the resulting spectral sequence degenerates on the second page. Taking cohomology of the graded pieces gives the comparison $H^n(N^n_r\\widehat{\\Delta}_{(-,-)/\\mathbb{A}_{\\mathrm{inf}}}) \\simeq W_r\\Omega^{n,\\mathrm{cont}}_{(-,-)/\\mathcal{O}_C}$ with the relative log de Rham–Witt complex, the relation predicted by the conjectures. The argument builds a motivic filtration on log $\\mathrm{TR}^r$ and its $S^1$-fixed points by log quasisyntomic sheafification of double-speed Postnikov filtrations, identifies the even pieces with an $r$-fold iterated pullback version of the Nygaard filtration, and then uses the log $A_\\Omega$ and log de Rham–Witt comparisons.","pith_inferences":["Beyond the paper: since the key identifications are étale local and the motivic filtration is built by log quasisyntomic descent, the affine restriction in Theorem 1.1 should extend to arbitrary p-completely smooth formal schemes over $\\mathcal{O}_C$ by gluing.","Beyond the paper: the assumed agreement between algebraic and homotopy-theoretic log prismatic cohomology could be tested directly on log quasiregular-semiperfectoid rings, where both sides are computable; a mismatch there would refute the conditional theorem.","Beyond the paper: if the comparison holds, the localization fibre sequences relating algebraic $K$-theory to log $\\mathrm{TR}$ would make the log de Rham–Witt complex a computational input for the $K$-theory of semistable $p$-adic schemes, a consequence the paper gestures toward but does not develop.","Beyond the paper: the paper leaves open whether the odd parts vanish in the full log quasisyntomic topology rather than only étale locally; checking this on non-smooth log quasisyntomic bases would clarify how much of Theorem 1.2 survives without the smoothness assumption."],"forward_implications":["The motivic filtration spectral sequence for log $\\mathrm{TR}^r$ degenerates on $E_2$, so the homotopy groups of log $\\mathrm{TR}^r$ are determined by the cohomology of the $r$-Nygaard filtered log prismatic complex.","Étale locally, the graded pieces of log $\\mathrm{TR}^r$ coincide with truncated log $A_\\Omega$-cohomology $\\tau_{\\leq n}A_\\Omega/\\xi^r$, connecting topological restriction homology to the log $A_\\Omega$ formalism.","The comparison $H^n(N^n_r\\widehat{\\Delta}_{(-,-)/\\mathbb{A}_{\\mathrm{inf}}}) \\simeq W_r\\Omega^{n,\\mathrm{cont}}_{(-,-)/\\mathcal{O}_C}$ yields the conjectured relation between log $\\mathrm{TR}^r$ and the relative log de Rham–Witt complex.","The odd parts of the motivic filtration vanish étale locally for smooth $X$, so no odd homotopy-group terms obstruct the log comparison in the étale topology.","Passing to the inverse limit over restriction maps extends these descriptions to $\\mathrm{TR}$ and its $S^1$-fixed points through the $r=\\infty$ Nygaard filtrations."],"supporting_citations":[{"why":"It states the conjectures connecting TR^r with logarithmic poles to the absolute log de Rham–Witt complex, which the paper aims to prove over O_C.","marker":"[Hes05]"},{"why":"It supplies the log THH cyclotomic machinery, log quasisyntomic descent, and the motivic filtration for log prismatic cohomology that the constructions build on.","marker":"[Bin+23]"},{"why":"It provides the identification of log prismatic cohomology with log A_Ω-cohomology in the semistable case, used to express the Nygaard filtration via décalage.","marker":"[ČK19]"},{"why":"It gives the algebraic theory of logarithmic prismatic cohomology whose agreement with the homotopy-theoretic version is assumed in Section 4.","marker":"[Kos20]"},{"why":"It completes the algebraic theory of logarithmic prismatic cohomology and is part of the assumed agreement over perfectoid bases.","marker":"[KY23]"},{"why":"It constructs the p-complete relative log de Rham–Witt complex that is the target of the cohomological comparison.","marker":"[Mat17]"},{"why":"It establishes the non-logarithmic r-Nygaard and motivic filtration results for TR that the logarithmic proof adapts.","marker":"[And24a]"},{"why":"It provides the perfectoid TR^r computation and the motivic filtration spectral sequence framework used over O_C.","marker":"[BMS19]"},{"why":"It supplies the p-adic Cartier comparison between A_inf-cohomology and de Rham–Witt forms for semistable formal schemes, supporting Proposition 4.2.","marker":"[Aok23]"}],"fun_headline_variants":["Motivic filtration on log TR yields de Rham–Witt","Log TR graded pieces are Nygaard filtered prismatic","Spectral sequence collapses: log TR to de Rham–Witt","Log TR homology: prismatic to de Rham–Witt bridge","Graded log TR is Nygaard filtered prismatic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole comparison rests on the assumption, stated in the introduction as 'as expected', that the algebraic and homotopy-theoretic constructions of completed log prismatic cohomology over a perfectoid base coincide; this identification is needed to replace the $r$-Nygaard filtration by the décalage filtration $L\\eta^{\\geq n}_{\\xi^r} A_\\Omega$, and if it fails the main theorem's identifications do not go through.","fun_headline_variants_meta":{"raw":{"variants":["Motivic filtration on log TR yields de Rham–Witt","Log TR graded pieces are Nygaard filtered prismatic","Spectral sequence collapses: log TR to de Rham–Witt","Log TR homology: prismatic to de Rham–Witt bridge","Graded log TR is Nygaard filtered prismatic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000998,"raw_usage":{"total_tokens":4268,"prompt_tokens":1030,"completion_tokens":3238,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":646,"completion_tokens_details":{"reasoning_tokens":3150}},"tokens_in":646,"tokens_out":3238,"duration_ms":22024,"temperature":1.0,"reasoning_tokens":3150,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:51:55.510789+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test would be to compute the $n$-th graded piece of the motivic filtration on log $\\mathrm{TR}^r$ for a specific log smooth base over $\\mathcal{O}_C$, such as the canonical log structure on the $p$-adic affine line, once through the algebraic log $A_\\Omega$/log prismatic construction and once through the homotopy-theoretic log prismatic construction; if the two $r$-Nygaard filtered complexes differ, or if the spectral sequence of Theorem 1.1 fails to degenerate on $E_2$, the central claim is false.","supporting_citations":[],"review_version":1}