{"id":"f536aa1e-333f-4a22-a061-88be205ef886","arxiv_id":"2506.14545","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Logarithmic THH with a chosen direction is shown to be the same as the abstract cofiber of the transfer map, resolving a conjecture of Rognes.","lead":"This paper proves a conjecture of John Rognes: logarithmic topological Hochschild homology of a ring or ring spectrum with a chosen element is equivalent to the cofiber appearing in a localization sequence. The result gives a unified framework for topological cyclic homology computations and removes regularity conditions.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader identified evenness as the weakest assumption. This is indeed a scope limitation, but it is explicitly stated and is not a correctness defect in the proven statement: the RSS25 model of log THH is only available for even E2-rings, and the paper covers exactly that setting. The original Rognes conjecture for even-periodic spectra is recovered in Remark 3.31. I found no internal inconsistency in the proof of Theorem 1.9. The most delicate step, Theorem 3.24 and its use in Corollary 3.28, is plausible: the mapping-space inclusions are justified by the split decompositions and Lemma 2.7, and the uniqueness of module lifts follows from Lemma 3.20. The remaining risk is the dependence on two very recent preprints, RSS25 and RSW25, which is a verification risk rather than a flaw in the argument. Since the paper is careful about its limitations and the central claim is well-supported given its hypotheses, I do not see a reason to change the ACCEPT verdict.","tokens_in":28490,"tokens_out":47234,"duration_ms":464699,"concrete_test":"Independently verify the external results used in the proof, especially RSS25 Corollary 4.8 (existence of the E2-map S[t2d] to an even E2-ring), RSS25 Proposition 9.21 (identification of the residue map), and the cyclotomic splitting (2.7)/(2.8) with its weight-zero component Striv ⊕ Striv[1]. If these hold and the Frobenius weight behavior used in Proposition 3.13 is as stated, then the proof of Theorem 1.9 is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the proof of Theorem 1.9 as internally coherent. The graded cyclotomic setup (Lemma 2.7) and the splitting arguments (Proposition 3.13, Proposition 3.22, Theorem 3.24) do not contain an obvious gap, and the evenness hypothesis is used explicitly exactly where the RSS25 model requires it. The main residual risk is external: the term THH(A,<x>) and several key inputs (RSS25 Cor. 4.8, Prop. 9.21, Prop. 11.3) come from the same-week preprint RSS25, and the categorical realization in Theorem 1.3 depends on RSW25. If those external results are sound, the central claim follows as written. The paper also explicitly does not prove the E-infinity refinement (Remark 4.15), so that is a stated limitation rather than a hidden flaw.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a conjecture of Rognes relating the logarithmic topological Hochschild homology THH(A,⟨x⟩) of an even E2-ring A at an element x ∈ π_{2d}(A) to the cofiber THH(A|x) of THH(A/x) → THH(A), as THH(A)-modules in cyclotomic spectra, with the residue maps identified (Theorem 1.9). For discrete commutative rings this gives Theorem 1.1, and via the Ramzi–Sosnilo–Winges construction of localizing motives it yields stable ∞-categories Perf(A,⟨x⟩) realizing the logarithmic term (Theorem 1.3). The paper also proves Theorem 1.7, that the isomorphism of Theorem 1.1 is one of log differential graded rings in the case of complete DVRs, and discusses applications to localization sequences for TR and TC without the usual regularity hypotheses.","tokens_in":28609,"tokens_out":6813,"duration_ms":65393,"significance":"If correct, the main theorem provides a long-sought bridge between two independent constructions of logarithmic THH and gives a clean proof of localization sequences for TC in settings where dévissage fails. The proof is detailed and makes systematic use of graded cyclotomic spectra, including an effective mapping-space lemma (Lemma 2.7) that is used to upgrade equivalences to module-level equivalences. The paper is unusually candid about its limitations: it does not prove an E∞-refinement (Remark 4.15), the (ko,w) case is conditional on constructing a cyclotomic structure (Remark 3.32), and several key inputs are very recent preprints (RSS25, RSW25). These are stated limitations rather than hidden gaps, and they are weighed in the assessment below.","major_comments":[],"minor_comments":[{"comment":"The assertion that the map (4.6) is an equivalence should be justified explicitly: both THH(OK|K) and THH(OK|π) are cofibers of the same transfer map THH(k)→THH(OK), but the compatibility of (4.6) with those cofiber sequences is not spelled out, even though the phrase 'explicit equivalence' in the first paragraph of §4.16 depends on it.","section":"§4.16"},{"comment":"In the proof of Proposition 3.13, the statement that THH(S[t±1])_{<0} contains no cyclotomic summand equivalent to S^{triv} because the p-typical Frobenius multiplies weights by p is plausible but is not justified in detail; a one-sentence argument or a reference would suffice.","section":"§3.12"},{"comment":"The definition of THH(S[t2d],⟨t2d⟩) as the 'weight-connective cover' of THH(S[t±1 2d]) could be made more precise by explicitly stating the splitting of (2.6) and noting that the resulting object is independent of the chosen presentation of the weight grading.","section":"§2.8"},{"comment":"There are a few typographical errors (e.g. 'inolving' near the end of the first page and 'theright-hand' in §4.13) and some sentences in the introduction are overly compressed; a careful proofread is recommended.","section":"Global"}],"recommendation":"minor_revision","confidential_remarks":"The paper is heavily dependent on two very recent preprints (RSS25 and RSW25). If either of these preprints contains errors or is substantially revised, the statements here may need adjustment. The editor may wish to ask the author to confirm that the cited statements from those preprints are in the versions on the arXiv as of the submission date. The paper is a strong fit for the journal and, apart from the local clarifications listed in the minor comments, appears technically sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves Rognes's conjecture: for even E2-rings, logarithmic THH(A,<x>) is equivalent as a THH(A)-module in cyclotomic spectra to the cofiber THH(A|x), with the residue maps identified. That is a real theorem, not a reformulation. It also upgrades the cofiber term to a multiplicative object in a wide range of cases and removes regularity hypotheses for localization sequences in THH, TR, and TC. The main new tool is graded cyclotomic spectra, used to compare module structures and residue maps; that technique is genuinely useful.\n\nWhat is good: the proof is detailed and internally coherent. The author explicitly states what he does not prove: no E-infinity equivalence (Remark 4.15), the ko,w extension is conditional on writing down a cyclotomic structure (Remark 3.32), and Perf(A,<x>) comes from RSW25 rather than being constructed with additional structure. Those are stated limitations, not hidden flaws. The Hesselholt–Madsen comparison at the level of log differential graded rings is also handled cleanly.\n\nSoft spots: the main risk is external. Several key inputs come from RSS25, posted the same week, and the categorical realization depends on RSW25. If those are sound, the theorem follows as written; if not, parts of the framework shift. That is not the author's fault, but it keeps confidence at moderate rather than high. The evenness hypothesis is real: without it the RSS25 model does not define THH(A,<x>), so odd-periodic examples are out. That is a scope restriction, not a flaw. The graded cyclotomic arguments are intricate in places—Proposition 3.19 and Theorem 3.24 in particular—and while I did not find a gap, I would not bet on every infinity-categorical detail without more time. The citation pattern is fine: RSS25 and RSW25 are cited heavily because the construction is built on them, and the central claim is not assumed.\n\nThis paper is for people working in topological cyclic homology, logarithmic THH, and algebraic K-theory of ring spectra. It deserves a serious referee. I would send it to review and ask the referee to check the external dependencies and the graded mapping-space arguments carefully.","headline":"Resolves Rognes's conjecture with a careful, internally coherent proof; the only real risk is the heavy dependence on same-week preprints.","tokens_in":29139,"tokens_out":1512,"would_cite":true,"duration_ms":16232,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19D55","55P42","55P91"],"pacs":[],"model":"deepseek-v4-flash","headline":"Logarithmic THH is the localization cofiber, with matching residue maps.","keywords":["logarithmic topological Hochschild homology","cyclotomic spectra","localization sequences","residue sequences","even E2-rings","topological cyclic homology","log differential graded rings","replete bar construction"],"falsifier":"The central claim would fail if, for some even $\\mathbb{E}_2$-ring $A$ and class $x\\in\\pi_{2d}(A)$, the spectra $\\mathrm{THH}(A,\\langle x\\rangle)$ and $\\mathrm{THH}(A|x)$ had different homotopy groups, or if the two boundary maps disagreed after composition with the equivalence; computing $\\pi_1$ for a concrete even ring outside the paper's examples and comparing the residue map to the predicted log differential quotient would settle the matter.","tokens_in":28277,"feed_emoji":"🔗","tokens_out":7792,"duration_ms":82684,"temperature":0.7,"pith_summary":"Topological Hochschild homology (THH) is an invariant of rings that takes values in cyclotomic spectra, spectra with Frobenius-like maps that feed topological cyclic homology. Because THH is a localizing invariant, localization of rings yields cofiber sequences, but the cofiber term THH(A|x) is usually just a module and is hard to compute. Logarithmic THH THH(A,<x>) is a richer construction with a multiplication and a residue sequence. The paper proves that these two objects are the same: an equivalence of THH(A)-modules in cyclotomic spectra, with the residue boundary map identified with the localization boundary map. This resolves a conjecture and shows that logarithmic THH, TR, and TC extend older localization constructions without needing the regularity hypotheses classically used for devissage.","feed_headline":"Two constructions of logarithmic THH are one object","feed_subtitle":"The residue boundary maps match too, settling a conjecture and removing regularity hypotheses.","key_machinery":"The load-bearing object is the replete bar construction $B^{\\mathrm{rep}}(M)$, the pullback of the cyclic bar construction on a commutative monoid $M$ with the cyclic bar construction on its group completion $M^{\\mathrm{gp}}$; it supplies the logarithmic term $\\mathrm{THH}(A,\\langle x\\rangle)$ by replacing the cyclic bar construction in the definition of $\\mathrm{THH}$. The proof works in graded cyclotomic spectra, where objects carry an integer weight. The central technical control is Lemma 2.7: mapping spectra from objects concentrated in non-negative weights to objects concentrated in non-positive weights forget the module structure, so module-level maps can be read off from weight-zero data. The universal case is the free even $\\mathbb{E}_2$-ring $S[t_{2d}]$ with its class $t_{2d}$; the paper identifies the two constructions there as inclusions of the non-negative weight part, and then base-changes along $S[t_{2d}]\\to A$.","core_discovery":"The central claim is that two a priori different constructions of logarithmic topological Hochschild homology coincide. For an even $\\mathbb{E}_2$-ring $A$ (a homotopy-coherently commutative ring spectrum with homotopy groups concentrated in even degrees) and a class $x\\in\\pi_{2d}(A)$, the paper proves an equivalence $$\\varphi\\colon \\mathrm{THH}(A,\\langle x\\rangle)\\xrightarrow{\\simeq}\\mathrm{THH}(A|x)$$ of $\\mathrm{THH}(A)$-modules in cyclotomic spectra, and it proves that the residue boundary map $\\partial_{\\mathrm{rep}}$ of the logarithmic residue sequence is homotopic to $\\partial\\circ\\varphi$, where $\\partial$ is the boundary map of the cofiber sequence defining $\\mathrm{THH}(A|x)$. For ordinary commutative rings this is Theorem 1.1 with $x$ a non-zero divisor, and the regularity hypotheses classically needed for devissage are not required. The equivalence is first proven on the universal example $\\mathrm{THH}(S[t_{2d}])$ using the weight grading, then base-changed to $A$.","pith_inferences":["If the conjecture on symmetric monoidal realizing categories extends beyond the discrete valuation ring case, logarithmic THH, TR, and TC would become fully fledged localizing invariants with multiplicative structure, potentially yielding new trace maps from logarithmic $K$-theory.","The weight-graded strategy suggests a general template for proving 'residue equals cofiber' identifications: formalize a theory in graded cyclotomic spectra, identify maps from weight-zero information, then base-change; the same pattern may apply to higher logarithmic structures beyond a single cone.","The paper's observation that $K(\\mathrm{Perf}(BP\\langle n\\rangle,\\langle v_n\\rangle))$ differs from $K(E(n))$ suggests that logarithmic THH tracks a nilpotent or ramified variant of $K$-theory rather than ordinary localization, and comparing TC along this difference could yield new chromatic filtrations of TC.","A concrete testable next step is to compute $\\pi_*\\mathrm{THH}(A,\\langle x\\rangle)$ for an even ring beyond the examples treated in the paper and verify that the identified log differential graded ring structure agrees with the cofiber model."],"forward_implications":["For discrete rings, $\\mathrm{THH}(A,\\langle x\\rangle)$ is a multiplicative replacement for the cofiber $\\mathrm{THH}(A|x)$, and the logarithmic residue sequence is literally the localization cofiber sequence.","There exists a stable $\\infty$-category $\\mathrm{Perf}(A,\\langle x\\rangle)$ whose THH is $\\mathrm{THH}(A,\\langle x\\rangle)$, so logarithmic THH, TR, and TC are realized as values of localizing invariants for discrete valuation rings, connective complex $K$-theory, and truncated Brown-Peterson spectra.","For discrete valuation rings, the logarithmic coefficient ring is isomorphic to the earlier Waldhausen-category construction as a log differential graded ring, so known computations of topological cyclic homology transfer to this construction.","The module-level identification of the residue maps is exactly the compatibility needed to compute TC through localization sequences, not just at the level of underlying spectra.","The conjecture that the realizing category can be chosen symmetrically monoidal, with evidence for discrete valuation rings, would make the multiplicative structure of logarithmic THH categorical rather than an extra add-on."],"supporting_citations":[{"why":"Supplies the logarithmic THH construction $\\mathrm{THH}(A,\\langle x\\rangle)$ and the residue cofiber sequence used as the comparison target.","marker":"[RSS25]"},{"why":"Defines logarithmic THH via the replete bar construction, the starting construction of the paper.","marker":"[Rog09]"},{"why":"States the conjecture, in ICM form, that Theorem 1.9 resolves.","marker":"[Rog14]"},{"why":"Provides the framework of cyclotomic spectra and the definition of TC used throughout.","marker":"[NS18]"},{"why":"Supplies the theory of graded cyclotomic spectra and weight-graded THH that controls mapping spectra.","marker":"[AMMN22]"},{"why":"Proves the universal localizing invariant is a Dwyer-Kan localization, enabling the category $\\mathrm{Perf}(A,\\langle x\\rangle)$ realizing the logarithmic term.","marker":"[RSW25]"},{"why":"Gives the Waldhausen-category cofiber term and log differential graded ring structure for discrete valuation rings compared in Theorem 1.7.","marker":"[HM03]"},{"why":"Gives the localization term for ring spectra that the resolved conjecture identifies with the logarithmic term.","marker":"[BM20]"}],"fun_headline_variants":["Log THH equivalence settles Rognes conjecture","Two log THH constructions proven equivalent","Log THH removes regularity hypotheses","Rognes conjecture on log THH confirmed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the ring spectrum $A$ is even, meaning its homotopy groups are concentrated in even degrees, because that is what supplies the map from the universal graded ring $S[t_{2d}]$ to $A$ on which the whole base-change strategy depends.","fun_headline_variants_meta":{"raw":{"variants":["Log THH equivalence settles Rognes conjecture","Two log THH constructions proven equivalent","Log THH removes regularity hypotheses","Rognes conjecture on log THH confirmed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000393,"raw_usage":{"total_tokens":2064,"prompt_tokens":941,"completion_tokens":1123,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":1069}},"tokens_in":557,"tokens_out":1123,"duration_ms":9777,"temperature":1.0,"reasoning_tokens":1069,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:17:13.364750+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The central claim would fail if, for some even $\\mathbb{E}_2$-ring $A$ and class $x\\in\\pi_{2d}(A)$, the spectra $\\mathrm{THH}(A,\\langle x\\rangle)$ and $\\mathrm{THH}(A|x)$ had different homotopy groups, or if the two boundary maps disagreed after composition with the equivalence; computing $\\pi_1$ for a concrete even ring outside the paper's examples and comparing the residue map to the predicted log differential quotient would settle the matter.","supporting_citations":[],"review_version":1}