{"id":"918fe841-26c0-4e1b-baa7-e432e282d321","arxiv_id":"2506.08492","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs logarithmic topological cyclic homology for E_k-rings with prelogarithmic structure and proves repletion-residue localization sequences for log THH and log TC.","lead":"This paper builds a framework for logarithmic topological Hochschild and cyclic homology of structured ring spectra, and derives localization sequences for these theories. It also supplies a candidate model for the fraction field of topological K-theory and computes the log THH and log TC of even-periodic sphere spectra.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the TC exactness step flagged by the reader is valid because TC(-)_p is an equalizer of exact functors on cyclotomic spectra, so the passage from Theorem 9.3 to Corollary 9.5 is sound.","rationale":"I read Theorem 9.3 and Corollary 9.5 carefully. The reader's concern is about applying TC to a cofiber sequence. This is a natural thing to worry about, since TC is a limit construction. However, in the Nikolaus–Scholze formalism used in the paper, TC(-)_p is a finite limit of exact functors, hence exact. I verified that each ingredient (homotopy fixed points, homotopy orbits, Tate construction, p-completion) preserves cofiber sequences in stable ∞-categories. The paper does not spell this out, and a one-line lemma would improve exposition, but this does not affect the validity of the central claim. The THH sequence is constructed by base change along a cofiber sequence in graded twisted cyclotomic modules, and the TC sequence follows by exactness. The rest of the paper's main results, including the examples, are consistent. I therefore find no load-bearing concern and recommend no change to the reader's verdict.","tokens_in":65052,"tokens_out":25186,"duration_ms":311771,"concrete_test":"Verify Corollary 9.5 by an independent diagram chase: apply the explicit formula TC(X)_p = fib(can − φ^{hT}_p) to the three-term cofiber sequence of Theorem 9.3 and check that the map of equalizer diagrams is a bicartesian square via the 3x3 lemma. If the resulting sequence TC(A//αbar)_p → TC(A)_p → TC(A,ξ,αbar)_p → ΣTC(A//αbar)_p fails to be a cofiber sequence, the exactness premise is false; otherwise the reader's concern is settled.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption is that Corollary 9.5 requires an unproved exactness property of TC(-)_p. I checked this against the paper's Definition 6.1. For a p-cyclotomic spectrum X, TC(X)_p is the equalizer of G∘can and φ^{hT}_p, equivalently the fiber of the difference X^{hT}_p → (X^{tC_p})^{hT}_p. Each functor in this expression preserves cofiber sequences in the stable ∞-category of cyclotomic spectra: homotopy fixed points are right adjoints, homotopy orbits are left adjoints, the C_p-Tate construction is the cofiber of the norm map between these exact functors, the residual T-homotopy fixed points are again a right adjoint, and p-completion is a localization. The fiber of a natural transformation between exact functors is exact by the 3x3 lemma. Hence TC(-)_p preserves the cofiber sequence of cyclotomic THH(A)-modules in Theorem 9.3 and yields the stated TC(A)_p-module cofiber sequence without a hidden premise. I found no other load-bearing gap in the central repletion–residue construction.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces R-based prelog E_k-rings using Picard-graded Thom spectrum functors, defines logarithmic topological Hochschild homology and logarithmic topological cyclic homology in this framework, and proves repletion–residue cofiber sequences relating THH(A) (respectively TC(A)_p) to the corresponding theories of the residue A//αbar. It establishes a multi-generator cube version, proves logification invariance, compares the new constructions with the authors' earlier point-set models, and computes the log THH and log TC of non-negative even-periodic sphere spectra. The main advertised applications are prelog structures on ku, ℓ, and BP⟨n⟩, and a model for the fraction field of topological K-theory.","tokens_in":65313,"tokens_out":12641,"duration_ms":166808,"significance":"If correct, the paper fills the missing localization terms for log TC, giving the first cyclotomic localization sequences in this context and extending the earlier non-cyclotomic results of [RSS15, RSS18]. The Picard-graded Thom spectrum framework is a genuine new construction rather than a repackaging, and the explicit log THH/log TC calculation for S[x] provides concrete, checkable output. The paper is very detailed: Thom spectrum adjunctions, weight-graded cyclotomic structures, and comparisons with point-set models are treated in depth, with the point-set comparisons relegated to appendices. The central TC exactness step is omitted from the text but is in fact valid; the reader's main concern therefore does not invalidate the claims, though it must be addressed in revision.","major_comments":[{"comment":"The TC repletion–residue sequence is obtained from Theorem 9.3 by the phrase 'Passing to TC', with no proof or citation of an exactness theorem. This is not immediate from the definition of TC as an equalizer, and the reader's concern is legitimate. The claim is nevertheless correct in the present ∞-categorical setting: by Definition 6.1, TC(−)_p is the equalizer of the two maps X^{hT}_p → (X^{tC_p})^{hT}_p, hence is the fiber of a natural transformation between exact functors on the stable ∞-category of p-cyclotomic spectra. The functor (−)^{hT} is a right adjoint, (−)^{tC_p} is the cofiber of the norm transformation between exact functors, and p-completion is a Bousfield localization, so the composite functors are exact. Since this exactness is the load-bearing premise for the main new TC statement, the paper should state and prove it, or give a precise reference, before Corollary 9.5 and in Example 10.8.","section":"Corollary 9.5 and Theorem 1.4"}],"minor_comments":[{"comment":"The notation TC(S)[S^1] should be defined explicitly as TC(S) ∧ (S^1)_+ with S^1 carrying the trivial T-action; without this, the reader may confuse it with the circle action underlying TC.","section":"Theorem 11.7"},{"comment":"The proof of logification invariance refers to [RSS15, Thm. 4.24] and only sketches the modifications needed in the present ∞-categorical and Picard-graded setting; a fuller proof or a more precise statement of the transferred argument would improve clarity, although this comparison is not needed for the main localization theorem.","section":"Section 8.15 / Theorem 8.15"},{"comment":"The statement that the lower right-hand square anti-commutes is announced only informally; since the same diagram reappears in Example 10.8, the sign convention should be recorded once in the formal statement as well.","section":"Introduction, diagram (1.2)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is strong and the central mathematical claims appear sound. The only serious issue is the missing justification of TC exactness in Corollary 9.5; the underlying fact is true, so this is a fixable gap rather than a fatal flaw. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: this is the real thing. The paper constructs log TC with cyclotomic structure, proves multi-generator repletion–residue sequences for THH and TC, and gives explicit computations for even-periodic sphere spectra. The one issue the reader flagged—the 'Passing to TC' step—does not hold up as a flaw. TC(-)_p is an equalizer of exact functors on cyclotomic spectra, so it preserves cofiber sequences; the stress-test note is correct. That said, the paper's terse transition from Theorem 9.3 to Corollary 9.5 hides a genuine argument, and the authors would do well to spell it out in a remark, because TC is not exact on arbitrary cofiber sequences of cyclotomic spectra.\n\nWhat is genuinely new: the cyclotomic structure on log THH, the definition of log TC, and the extension from one generator to several E_k prelog structures. The fraction field of topological K-theory is made precise via (ℓ,⟨p,v1⟩), and the even-periodic sphere computations are clean and explicit. The comparison with earlier work is careful, and the appendices are honest about what matches RSS15/RSS18.\n\nSoft spots are real but not fatal. The paper is enormous and will be hard for non-specialists to enter. Several promised connections—Lundemo's comparison with Blumberg–Mandell, the detailed match with AR09/AR12 calculations—are deferred, so parts of the motivation rest on forthcoming work. The one-generator log THH overlaps earlier definitions, so novelty is concentrated in the log TC and multi-generator parts. None of this undermines the central claims.\n\nWho should read it: anyone working on trace methods, localization sequences in THH/TC, or logarithmic structures in stable homotopy. It deserves a serious referee, and I would expect the referee reports to focus on clarity and on checking the exactness argument I mentioned, not on finding a fundamental gap.\n\nRecommendation: send it to peer review. It is a major step for the subject.","headline":"A technically formidable and genuinely new construction of log TC with the missing localization sequences; the flagged TC exactness worry is not a real flaw, though the authors should spell it out.","tokens_in":65853,"tokens_out":2231,"would_cite":true,"duration_ms":29302,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P42","19D55","55P43"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves repletion–residue cofiber sequences for logarithmic THH and p-completed log TC of prelog E2-rings, filling the terms missing from ordinary localization sequences.","keywords":["topological Hochschild homology","topological cyclic homology","logarithmic structures","Thom spectra","cyclotomic spectra","localization sequences","Picard-graded ring spectra","E_k-algebras"],"falsifier":"Take the repletion-residue cofiber sequence of Proposition 9.9 for the sphere base and the non-negative even-periodic sphere spectrum, compute the equalizer defining p-completed TC on it, and check whether the result is the predicted cofiber sequence with last term $\\Sigma\\mathrm{TC}(\\mathbb{S})_p$; any deviation would disprove the exactness premise used to pass from THH to TC.","tokens_in":64858,"feed_emoji":"🧮","tokens_out":9960,"duration_ms":122828,"temperature":0.7,"pith_summary":"This paper introduces new notions of prelog and log $E_k$-ring spectra, built from Picard-graded Thom spectra, and constructs logarithmic topological Hochschild homology and logarithmic topological cyclic homology for them. The central claim is that for a prelog $E_2$-ring generated by a single homotopy class, THH of the ring maps into its logarithmic version with cofiber equal to the suspension of ordinary THH of a collapsed 'residue' quotient, and the same cofiber sequence holds after passing to $p$-completed log TC. Such repletion-residue sequences are exactly the localization sequences that ordinary THH and TC fail to have, so if the claims hold, logarithmic THH and log TC restore the localization behavior that makes algebraic $K$-theory tractable. The paper also proves the multi-generator version, giving cubes of cofiber sequences, and computes log THH and log TC for non-negative even-periodic sphere spectra.","feed_headline":"Log THH and log TC now complete localization sequences","feed_subtitle":"For prelog E2-rings, cofiber sequences match K-theory, with explicit results for even-periodic spheres.","key_machinery":"The machinery is the Picard-graded Thom $R$-algebra functor $\\mathrm{Th}_R$, which builds ring spectra from $E_k$-maps into the space of invertible $R$-modules, together with the replete bar construction: a graded pullback of the cyclic bar construction of the group completion that produces a canonical repletion map from THH of the Thom ring to its logarithmic version. Weight-graded THH with the $L_p$-twisted Tate diagonal, taken from a graded refinement of the cyclotomic-spectrum formalism, supplies the cyclotomic structure, and the 'cyclotomically good' property of a base pair $(R,\\xi_*)$ controls when the repletion fiber is literally ordinary $\\mathrm{THH}(R)$. This combination is what converts a fiber computation into a cofiber sequence of cyclotomic modules and then into a TC statement.","core_discovery":"The main structural theorem states: for a prelog $E_2$-ring $(A,\\xi_{2d},\\bar\\alpha(a))$ of the form built from the Thom spectrum $\\mathrm{Th}_{\\mathbb{S}}(\\xi_{2d})$, there are cofiber sequences $$\\mathrm{THH}(A) \\xrightarrow{\\rho} \\mathrm{THH}(A,\\xi_{2d},\\bar\\$\\alpha$(a)) \\xrightarrow{\\mathrm{res}} \\Sigma\\,\\mathrm{THH}(A/\\!/\\bar\\$\\alpha$(a))$$ of cyclotomic $\\mathrm{THH}(A)$-modules, and the same pattern with $\\mathrm{TC}(A)_p$ in place of $\\mathrm{THH}(A)$. The residue quotient $A/\\!/\\bar\\alpha(a)$ is obtained by collapsing the prelog generator to zero, and the logarithmic theories fill exactly the term between $A$ and its localization that was previously missing. When the monoid has $r$ generators the result is an $r$-dimensional cube of cofiber sequences, and in the even-periodic sphere example the terms are identified explicitly.","pith_inferences":["Should the TC exactness premise hold generally, the cube construction would give localization sequences for every prelog $E_2$-ring over a cyclotomically good base, including examples built over $MU$ and over perfect fields; the paper proves the cyclotomic-good criterion only for a limited list, so this is an extension rather than a stated result.","The identification of log THH as a Thom spectrum over the replete bar construction points toward a logarithmic prismatic cohomology via the even or motivic filtration; the paper names this as a future direction, so treating it as a consequence is an editorial extrapolation.","The multi-generator example $\\ell$ with $\\langle p, v_1\\rangle$ is cast as a model for the fraction field of topological $K$-theory; a natural next test is whether trace maps from algebraic $K$-theory into the log TC of that model reproduce the localization behavior conjectured from earlier calculations, a comparison the paper defers."],"forward_implications":["For the Adams summand $\\ell$ with its $v_1$-prelog structure, the sequence specializes to $\\mathrm{THH}(\\ell) \\to \\mathrm{THH}(\\ell,\\langle v_1\\rangle) \\to \\Sigma\\mathrm{THH}(H\\mathbb{Z}_{(p)})$, and the same after applying $\\mathrm{TC}(-)_p$, matching the localization pattern known for algebraic $K$-theory of the periodic theory.","For connective complex $K$-theory with its Bott element, the same theorem supplies the missing terms in the localization sequence between $ku$ and $KU$.","For truncated Brown-Peterson spectra $BP\\langle n\\rangle$ with the prelog structure generated by $p, v_1, \\dots, v_n$, an $n$-dimensional cube of cofiber sequences is obtained, so log THH and log TC decompose according to the residue quotients such as $k(n)$ and related spectra.","For the non-negative even-periodic sphere spectrum $S[x]$ with its canonical prelog structure, the $p$-completed log TC is explicitly $\\mathrm{TC}(\\mathbb{S})_p[S^1] \\vee \\bigvee_{i>0} \\Sigma((S^{2d})^{\\otimes i})_{hC_i}$, making the invariants computationally accessible."],"supporting_citations":[{"why":"Supplies the cyclotomic spectrum framework and the equalizer definition of TC that the paper uses throughout.","marker":"[NS18]"},{"why":"Provides the graded twisted cyclotomic THH with Tate diagonal that gives log THH its cyclotomic structure.","marker":"[AMMN22]"},{"why":"Earlier single-generator log THH localization sequences that this paper extends and strengthens.","marker":"[RSS15]"},{"why":"Earlier log THH definitions and calculations for topological K-theory spectra that the new construction is compared against.","marker":"[RSS18]"},{"why":"Gives the cellular $E_k$-algebra cell structure used to construct the prelog structure maps $\\bar\\alpha$.","marker":"[GKR W18]"},{"why":"Supplies the relative THH and de Rham-with-log-poles motivation and the finite-field computations used in the cyclotomic-good criterion.","marker":"[HM03]"},{"why":"Proves the Segal-conjecture input that makes the sphere base pair cyclotomically good.","marker":"[Lin80]"},{"why":"Completes the same Segal-conjecture input for the sphere case.","marker":"[Gun81]"},{"why":"Gives the p-completion of the cyclotomic structure map for MU used in proving that MU is cyclotomically good.","marker":"[LNR11]"},{"why":"Provides the classical TC and trace framework used in the even-periodic sphere calculations.","marker":"[BHM93]"}],"fun_headline_variants":["Log THH and TC localization sequences for prelog E2-rings","Prelog rings give log THH and TC localization sequences","Localization sequences proven for log THH and log TC","Even-periodic spheres exemplify log THH and TC sequences","Cube of cofiber sequences for log THH and TC localization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The step that converts the THH-level localization sequence into a TC-level one assumes that p-completed topological cyclic homology preserves the cofiber structure of these specific module sequences; this exactness is invoked without proof, and TC is known not to be exact on arbitrary cofiber sequences.","fun_headline_variants_meta":{"raw":{"variants":["Log THH and TC localization sequences for prelog E2-rings","Prelog rings give log THH and TC localization sequences","Localization sequences proven for log THH and log TC","Even-periodic spheres exemplify log THH and TC sequences","Cube of cofiber sequences for log THH and TC localization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000632,"raw_usage":{"total_tokens":2878,"prompt_tokens":866,"completion_tokens":2012,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":482,"completion_tokens_details":{"reasoning_tokens":1924}},"tokens_in":482,"tokens_out":2012,"duration_ms":16069,"temperature":1.0,"reasoning_tokens":1924,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:11:37.709693+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the repletion-residue cofiber sequence of Proposition 9.9 for the sphere base and the non-negative even-periodic sphere spectrum, compute the equalizer defining p-completed TC on it, and check whether the result is the predicted cofiber sequence with last term $\\Sigma\\mathrm{TC}(\\mathbb{S})_p$; any deviation would disprove the exactness premise used to pass from THH to TC.","supporting_citations":[],"review_version":1}