{"id":"551410a2-f8d3-4738-b9df-5cdcbb17aea6","arxiv_id":"1908.09164","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the Thom spectra y(n) between S and HF2, the paper proves that K(m)_*(TP(y(n))) vanishes for 1≤m≤n, so TP raises chromatic height by one.","lead":"This paper computes the chromatic complexity of topological periodic cyclic homology for a family of Thom spectra y(n), showing their complexity increases by one height. It gives evidence for the Ausoni-Rognes red-shift conjecture and introduces a homological method that applies to associative ring spectra without full coefficient ring knowledge.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Missing HF_p-nilpotent completeness check for {TP(y(n))[i]} leaves the passage from continuous Morava K-theory vanishing to K(m)_*(TP(y(n)))=0 unproven.","rationale":"The paper's main contribution is a homological method for computing chromatic complexity of TP and TC^- for the Thom spectra y(n). The internal calculations in Sections 3–5 are detailed and provide strong evidence for the continuous Morava K-theory vanishing. However, the headline result Theorem 1.2 is about ordinary Morava K-theory, and the only bridge from continuous to ordinary is Theorem 6.1 of the companion paper [5]. The reader correctly identifies that one of the hypotheses, HF_p-nilpotent completeness, is not verified in Section 6.1. This is the single most load-bearing concern because if it fails, the main theorem reduces to a statement about a different, continuous variant of Morava K-theory. The gap may be easily fillable—for instance, bounded below spectra with finite-type Z_2-homotopy groups are often HF_2-nilpotently complete by standard convergence theorems—but as written the proof is incomplete. We therefore recommend keeping the verdict CONDITIONAL, with the concrete test above to determine whether the gap closes.","tokens_in":36058,"tokens_out":6144,"duration_ms":56054,"concrete_test":"Verify that TP(y(n))[i] is HF_2-nilpotently complete. Since Lemma 6.2 shows π_*(TP(y(n))[i]) is bounded below and a finite-type Z_2-module, try to invoke a standard convergence theorem for the HF_2-based Adams spectral sequence (e.g., Bousfield or Miller) to conclude that the HF_2-nilpotent completion map is an equivalence. Alternatively, inspect [5] for a lemma automatically supplying nilpotent completeness for bounded-below finite-type spectra and check its hypotheses. If neither works, explicitly construct the HF_2-Adams tower for a fixed i (e.g., i=0) and show the homotopy limit is equivalent to TP(y(n))[i], or find a counterexample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 6.5 invokes [5, Thm 3.5] to conclude K(m)_*(TP(y(n)))=0 from the continuous vanishing in Theorem 4.15. One of the explicit hypotheses of [5, Thm 3.5] is that each spectrum Y[i] in the tower be HF_p-nilpotently complete. Section 6.1 verifies finite type (Lemma 6.2), finite generation (Cor 6.3), primitive degree bounds (Lemma 6.4), and vanishing of the limit of Margolis homology (Thm 4.15), but never states or proves nilpotent completeness. Without this condition, [5, Thm 3.5] cannot be applied, and the step from K(m)^c_*(TP(y(n)))=0 to K(m)_*(TP(y(n)))=0 does not follow. Thus the main theorem 1.2 is conditionally dependent on a hypothesis that is not established in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the chromatic complexity—defined via the vanishing range of Morava K-theory—of the trace invariants TP, TC^-, TC, and algebraic K-theory of the Mahowald Thom spectra y(n), which interpolate between the sphere spectrum and HF_2 and satisfy K(m)_*(y(n)) = 0 for 0 ≤ m < n while K(n)_*(y(n)) ≠ 0. The main theorem (Theorem 1.2) asserts that K(m)_*(TP(y(n))) = 0 for 1 ≤ m ≤ n, so topological periodic cyclic homology raises the Morava K-theory vanishing threshold by one for every member of the y(n) family, giving evidence for a TP-version of the Ausoni–Rognes red-shift conjecture at all chromatic heights. The paper also proves that the relative algebraic K-theory K(y(n), HF_2) is K(m)-acyclic for 0 ≤ m ≤ n−1 (Theorem 6.8).","tokens_in":36266,"tokens_out":27718,"duration_ms":247550,"significance":"If the main theorem holds, the paper is a significant step: it gives the first family of ring spectra indexed by arbitrary chromatic height for which TP is shown to increase the Morava K-theory vanishing range by one, supporting the red-shift program at all heights. The methodology—using the homological Tate spectral sequence and Margolis homology in place of unavailable homotopy-group computations of THH(y(n))—is new and likely to have further applications, and the identification of H_*^c(TP(y(n))) with P(t^{±1}) ⊗ H_*(z(n)/v_n) as E_*-comodules (Proposition 4.9) is a clean structural result. The paper is careful in its claims: it explicitly corrects its earlier TC^- statement (Remark 5.9), separates proven statements from conjectures (Conjecture 4.11, footnote 3), and provides a fully worked example showing K(1)^c_*(TC^-(y(1))) ≠ 0. These features indicate careful scholarship. The main deficit, detailed below, is that the bridge from continuous to ordinary Morava K-theory via [5, Thm 3.5] rests on an unverified nilpotent-completeness hypothesis and on the companion preprint [5]; closing that gap is required before the main theorem can be considered established.","major_comments":[{"comment":"The proof of Theorem 6.5 applies Theorem 6.1 of [5] to conclude K(m)_*(TP(y(n))) = 0 from the continuous vanishing computed in Theorem 4.15. The quoted Theorem 6.1 explicitly lists as a hypothesis that each spectrum Y[i] in the sequence be HF_p-nilpotently complete. The verification in the proof of Theorem 6.5 refers only to Lemma 6.2 (bounded below finite type), Corollary 6.3 (finite generation), Lemma 6.4 (primitive degree bounds), and Theorem 4.15 (vanishing of the limit of Margolis homologies); none of these statements states or proves that TP(y(n))[i] is HF_2-nilpotently complete for each i. Since this condition is precisely what justifies passing from lim_i K(m)_*(TP(y(n))[i]) = 0 to K(m)_*(holim_i TP(y(n))[i]) = 0, the isomorphism (11), and with it Theorem 1.2, is conditional on an unverified hypothesis. The authors should either prove nilpotent completeness directly or supply a precise citation of a Bousfield- or Miller-type convergence theorem that applies to the bounded-below finite-type spectra TP(y(n))[i], and they should state the direction of the limit in the Greenlees tower so that it matches the sequence indexed by increasing i in Theorem 6.1.","section":"§6.1, Theorem 6.5 (alleged application of [5, Thm 3.5])"},{"comment":"Lemma 6.4 supplies the uniform bound M = 2^{n+1} on the degree of comodule primitives that is required by Theorem 6.1, but the proof is not rigorous as written. First, the displayed equality '2^{n+1}−1 = |∏_{j=1}^n σξ_j|' is arithmetically incorrect: |σξ_j| = 2^j, so |∏_{j=1}^n σξ_j| = ∑_{j=1}^n 2^j = 2^{n+1}−2; the claimed strict bound M = 2^{n+1} still holds, but the displayed equality should be corrected. Second, the final paragraph of the proof ('we need to choose a final sequence such that elements that are not divisible by t always appear, but this is possible') is not an argument, and the claim that coactions on t^a x for a > 0 cannot be canceled by adding higher-filtration terms needs a complete justification. Since the primitive-degree bound is an explicit hypothesis of [5, Thm 3.5], the proof should be expanded or replaced by a precise reference.","section":"§6.1, Lemma 6.4"},{"comment":"The passage from continuous to ordinary Morava K-theory—the only route to the main theorem—goes through Theorem 3.5 of the companion preprint [5] (Angelini-Knoll–Salch, arXiv:2003.03510), which is quoted verbatim but not proved in the paper. Because Theorem 1.2 rests on this external result, the paper cannot be fully evaluated unless [5] is stable, publicly available, and its hypotheses match the statement quoted here. The authors should either include the proof of the quoted theorem in an appendix or confirm the publication status of [5] so that the referee can verify the argument.","section":"§6, Theorems 6.1 and 6.5 (dependency on companion preprint)"}],"minor_comments":[{"comment":"There is a typo in the quoted hypothesis list: 'HFp-nilpotently compete' should read 'HF_p-nilpotently complete'.","section":"§6.1, Theorem 6.1 (statement)"},{"comment":"In the second paragraph of the proof, 'H(V(i);Q_n)' should be 'H(V(i);Q_m)', since m is the index of the Morava K-theory in the statement.","section":"§4.4, proof of Theorem 4.15"},{"comment":"The abstract states that TP(y(n)) 'has chromatic complexity n+1', but the results prove only the vanishing statements K(m)_*(TP(y(n))) = 0 for 1 ≤ m ≤ n; nonvanishing of K(n+1)_*(TP(y(n))) is not established, so the phrasing should be 'chromatic complexity at least n+1' or 'raises the vanishing threshold by one'.","section":"§1, Abstract and Question 1.1"},{"comment":"The direction of the limit 'lim_i' in the definition of continuous homology and in Theorems 4.15 and 5.8 is not pinned down explicitly; since [5, Thm 3.5] is stated for a sequence Y[2]→Y[1]→Y[0], the authors should specify the indexing convention for the Greenlees tower so that the match between the two towers is unambiguous.","section":"§4.2, Definition 4.2 and Theorem 4.15"}],"recommendation":"major_revision","confidential_remarks":"The main theorem depends on Theorem 3.5 of the first author's companion preprint [5], which is posted contemporaneously with the present manuscript; the editor may wish to confirm that [5] is accepted for publication (or at least stable) before final acceptance of this paper, since the referee cannot fully audit the correctness of the quoted theorem from the present manuscript alone. Footnote 3 of the paper shows the authors are aware of the hypotheses of [5] in one direction, which makes the absence of a nilpotent-completeness check in Section 6.1 more conspicuous. The overlap with Land–Meier–Tamme [25] is acknowledged in Section 1, so there is no novelty disclosure concern. The paper is a good fit for the journal's topology and homotopy theory scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This one is worth reading. The main new result is Theorem 1.2: TP(y(n)) has vanishing K(m)-homology for 1≤m≤n, which for m=n gives a height shift at every chromatic height in the y(n) family. That is the first family of this kind for TP, and it is solid evidence for the red-shift question. The homological Tate spectral sequence computations for THH, TP, and TC^- of y(n) are the real workhorse, and they look careful. I checked Remark 5.9, where they correct their own earlier claim about TC^-; that kind of self-correction is a good sign. Theorem 1.3 on relative K-theory is not new — Land-Meier-Tamme got it independently, and the paper says so — but it is not the main point.\n\nThe soft spot is the passage from continuous to ordinary Morava K-theory. Theorem 6.5 applies [5, Thm 3.5] to the tower {TP(y(n))[i]}, and one of the explicit hypotheses of that theorem is that each spectrum be HF_p-nilpotently complete. Section 6.1 verifies finite type, finite generation, primitive degree bounds, and vanishing of the Margolis homology limit, but there is no proof and no reference for nilpotent completeness. It might be automatic from the other hypotheses, or it might be standard in the Greenlees filtration, but the paper does not say. If the condition fails, the equality K(m)^c_* = K(m)_* does not follow, and Theorem 1.2 is not proven as written. This is a real gap, though I suspect it is fillable. I would not call it fatal; I would call it the main thing a referee should demand.\n\nThe rest of the argument is honest. The computations are benchmarked against HF2 and the sphere, not fitted to the target, so the circularity burden is low. The reliance on [5] is a self-citation, but [5] is a general theorem, not cooked up for this paper.\n\nBottom line: this deserves a serious referee. I would recommend peer review with the request that the authors either prove HF_p-nilpotent completeness for the tower or point to a theorem that does it. Once that is settled, the paper is likely acceptable. I'd bring it to reading group; the spectral sequence arguments are worth seeing even if the main theorem is temporarily conditional.","headline":"A genuinely new TP red-shift family at all heights, but the main theorem's bridge from continuous to ordinary Morava K-theory rests on an HF_p-nilpotent completeness hypothesis the paper never verifies.","tokens_in":36792,"tokens_out":4998,"would_cite":true,"duration_ms":49587,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55P42","55P43","55T99","19D55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that topological periodic cyclic homology raises the chromatic height of every Thom spectrum y(n) by one.","keywords":["chromatic red-shift conjecture","topological periodic cyclic homology","Morava K-theory","Thom spectra y(n)","homological Tate spectral sequence","Margolis homology","topological cyclic homology","algebraic K-theory"],"falsifier":"Test the missing hypothesis directly: for each n and i, check whether TP(y(n))[i] is nilpotently complete with respect to HF_2 (for instance by computing the relevant completion map or $lim^{1}$ term). A single failure would remove the bridge from continuous to ordinary Morava K-theory, leaving Theorem 1.2 without its final step; alternatively, compute K(m)_*(TP(y(n))) for m=n in a small case such as n=1 with an independent method and look for a nonvanishing class.","tokens_in":35850,"feed_emoji":"🔺","tokens_out":11384,"duration_ms":104557,"temperature":0.7,"pith_summary":"This paper studies the chromatic complexity of algebraic K-theory through its approximations, topological periodic cyclic homology and topological negative cyclic homology. For the family of Thom spectra y(n) that interpolate between the sphere spectrum and the mod-2 Eilenberg–MacLane spectrum, it proves that the m-th Morava K-theory of TP(y(n)) vanishes for every 1≤m≤n, so TP raises the chromatic height by at least one at every level of the family. This is intended as uniform evidence for the chromatic red-shift conjecture, stated in terms of Morava K-theory vanishing rather than finite-presentation height. The method is homological: a Tate spectral sequence computes the continuous homology of TP(y(n)), Margolis homology detects the vanishing, and a comparison step passes from continuous to ordinary Morava K-theory.","feed_headline":"Periodic cyclic homology lifts chromatic height by one","feed_subtitle":"For each Thom spectrum y(n), Morava K-theory vanishes through height n, a uniform red-shift signal.","key_machinery":"The load-bearing machinery is the homological Tate spectral sequence, which computes the continuous homology H^c_*(TP(R)) = lim_i H_*(TP(R)[i]) from the filtration defining it. Its E2-page is P($t^{{±1}}$) ⊗ H_*(THH(R)), and its differentials are governed by the circle action; for R=y(n), those differentials are evaluated using the spectral sequence computing H_*(THH(y(n))) and a precise description of the map THH(y(n))→THH(HF2). The key identification is H^c_*(TP(y(n))) ≅ P($t^{{±1}}$) ⊗ H_*(z(n)/v_n) as comodules over the subalgebra generated by the Milnor primitives, which transfers the problem to a spectrum whose Margolis homology is computed by elementary chain complexes. The localized Adams spectral sequence then converts Margolis homology vanishing into continuous Morava K-theory vanishing, and a theorem on commuting unbounded homotopy limits with Morava K-theory converts that into ordinary vanishing.","core_discovery":"The central discovery is a computation of the continuous homology of TP(y(n)): it is isomorphic, as a comodule over the subalgebra generated by Milnor primitives, to P($t^{{±1}}$) ⊗ H_*(z(n)/v_n), where z(n) is an integral analog of y(n) and z(n)/v_n is the cofiber of a v_n self-map. Because the Margolis homology of z(n)/v_n vanishes for Q_1,...,Q_n, the localized Adams spectral sequence gives K(m)^c_*(TP(y(n))) = 0 for 1≤m≤n. A technical comparison then upgrades this to ordinary K(m)_*(TP(y(n))) = 0, proving the main theorem. The same machinery yields vanishing for relative topological cyclic homology and for relative algebraic K-theory K(y(n), HF2) in the range 0≤m≤n−1, and it reveals a nonvanishing obstruction in the top continuous Morava K-theory of TC^-(y(n)) that makes TP look like the better red-shift detector.","pith_inferences":["If the missing nilpotent-completeness condition for TP(y(n))[i] is verified, the same continuous-to-ordinary comparison would likely apply to any E1 ring spectrum whose topological Hochschild homology is understood as a comodule algebra.","Conjecture 4.11, that TP(y(n)) is equivalent to the Tate construction on z(n)/v_n after completion, would turn TP computations into Tate constructions of finite-type spectra; proving it would give a general red-shift machine for inputs without finite presentation.","The paper's TC^- obstruction at the top continuous Morava K-theory suggests an explicit test: compute the analogous T0 contribution for higher n and see whether it always obstructs K(n)-vanishing of TC^-(y(n)).","The z(n) spectra and their v_n-cofibers carry exactly the information needed for these TP calculations; they may be useful as integral building blocks for other chromatic-shift questions."],"forward_implications":["If correct, the main theorem makes TP(y(n)) K(m)-acyclic for every 1≤m≤n, so its chromatic complexity is at least n+1 while y(n) itself has type n: an explicit height shift at every level of the family.","Relative algebraic K-theory K(y(n), HF2) is K(m)-acyclic for 0≤m≤n−1, so algebraic K-theory of this family at least preserves chromatic complexity.","The same method proves vanishing for relative topological cyclic homology TC(y(n), HF2) in the range 0≤m≤n−1, via the fiber sequence relating TC, TC^-, and TP together with the standard equivalence between relative TC and relative K-theory.","Negative cyclic homology is predicted not to shift in general: for n=1 the continuous K(1)-homology of TC^-(y(1)) is nonzero, so TC^- can fail to raise height even where TP succeeds.","The theorem gives evidence for the chromatic red-shift conjecture in a setting where the input spectrum is not finitely presented, so pure fp height cannot even be defined."],"supporting_citations":[{"why":"Supplies the theorem commuting unbounded homotopy limits with Morava K-theory, used to pass from continuous to ordinary vanishing.","marker":"[5]"},{"why":"Computes the homology of topological Hochschild homology from the Hochschild homology of the homology ring, used for THH(y(n)).","marker":"[12]"},{"why":"Introduces the homological Tate spectral sequence and its differential formula, the main computational tool.","marker":"[14]"},{"why":"Provides the equivalence between relative TC and relative K-theory needed to pass from TC to K.","marker":"[16]"},{"why":"Computes y(n)'s type and supplies the localized Adams spectral sequence formalism used throughout.","marker":"[30]"},{"why":"Gives the fiber sequence relating TC, TC^-, and TP used to deduce TC vanishing.","marker":"[37]"},{"why":"Supplies the rational equivalence between K(y(n)) and K(HF2) used in the m=0 case.","marker":"[45]"}],"fun_headline_variants":["Homological trace proof: TP(y(n)) lifts chromatic height","TP(y(n)) kills first n Morava K-theories","Periodic cyclic homology red-shifts by one","Margolis homology proves TP(y(n)) red-shift","New homological proof of red-shift for TP(y(n))"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument that continuous vanishing of Morava K-theory implies ordinary vanishing depends on each spectrum TP(y(n))[i] being complete with respect to mod-2 homology in a strong technical sense; the paper never proves this, so if one of these completeness conditions fails, the main theorem does not follow from the given proof.","fun_headline_variants_meta":{"raw":{"variants":["Homological trace proof: TP(y(n)) lifts chromatic height","TP(y(n)) kills first n Morava K-theories","Periodic cyclic homology red-shifts by one","Margolis homology proves TP(y(n)) red-shift","New homological proof of red-shift for TP(y(n))"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001511,"raw_usage":{"total_tokens":6023,"prompt_tokens":880,"completion_tokens":5143,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":496,"completion_tokens_details":{"reasoning_tokens":5060}},"tokens_in":496,"tokens_out":5143,"duration_ms":37992,"temperature":1.0,"reasoning_tokens":5060,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:20:03.992145+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the missing hypothesis directly: for each n and i, check whether TP(y(n))[i] is nilpotently complete with respect to HF_2 (for instance by computing the relevant completion map or $lim^{1}$ term). A single failure would remove the bridge from continuous to ordinary Morava K-theory, leaving Theorem 1.2 without its final step; alternatively, compute K(m)_*(TP(y(n))) for m=n in a small case such as n=1 with an independent method and look for a nonvanishing class.","supporting_citations":[{"cited_title":"Commuting unbounded homotopy limits with Morava K-theory","cited_arxiv_id":"2003.03510","evidence_quote":"Supplies the theorem commuting unbounded homotopy limits with Morava K-theory, used to pass from continuous to ordinary vanishing."},{"cited_title":"Topological Hochschild homology of Z and Z/p","cited_arxiv_id":null,"evidence_quote":"Computes the homology of topological Hochschild homology from the Hochschild homology of the homology ring, used for THH(y(n))."},{"cited_title":"Differentials in the hom ological homotopy fixed point spectral sequence","cited_arxiv_id":null,"evidence_quote":"Introduces the homological Tate spectral sequence and its differential formula, the main computational tool."},{"cited_title":"The local structure of algebraic K-theory , volume 18","cited_arxiv_id":null,"evidence_quote":"Provides the equivalence between relative TC and relative K-theory needed to pass from TC to K."},{"cited_title":"The tri ple loop space approach to the telescope conjecture","cited_arxiv_id":null,"evidence_quote":"Computes y(n)'s type and supplies the localized Adams spectral sequence formalism used throughout."},{"cited_title":"On topological cycl ic homology","cited_arxiv_id":null,"evidence_quote":"Gives the fiber sequence relating TC, TC^-, and TP used to deduce TC vanishing."},{"cited_title":"Algebraic K-theory of topological spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the rational equivalence between K(y(n)) and K(HF2) used in the m=0 case."}],"review_version":1}