{"id":"17e92eb4-c248-4ad0-b467-dace746feaf6","arxiv_id":"2607.21567","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For connective Morava K-theory, the paper determines algebraic K-theory group cardinalities in all degrees outside two congruence classes over finite fields, and proves even-degree groups vanish over algebraically closed fields.","lead":"This paper computes the sizes of many integral algebraic K-theory groups of connective Morava K-theory, a family of spectra that play the role of prime fields in chromatic homotopy theory. It introduces an \"orbit filtration\" on topological cyclic homology and uses it to prove even-degree vanishing and explicit cardinality formulas.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 5.20 is the load-bearing point: its proof has an unjustified induction step, and Theorem A's even vanishing rests on it. If the lemma is false, the central claim collapses; if the proof can be repaired, the paper is likely sound.","rationale":"I read the full argument in good faith. The main theorems are carefully stated, the excluded congruence classes are handled honestly, and the undetermined m(r)/A_r factors appear only in those excluded degrees. I found no evidence of circularity, normalization fitting, or internal inconsistency. The orbit filtration and the Bockstein arguments are intricate but internally coherent; the reliance on [AKHW24] is a black-box input, but that is not itself a defect in this paper. The single place where the proof is thinner than the claim is Lemma 5.20, exactly as the reader identified. The concern is not that I have a counterexample — the lemma is probably true — but that the induction as written has an unverified substitution step, and the lemma is structurally necessary for Theorem A and Corollary C. If the lemma is false or the induction cannot be repaired, the even-vanishing result collapses. If it is repaired, the central argument appears sound. Since the reader's CONDITIONAL verdict already reflects this uncertainty, no adjustment is needed.","tokens_in":54116,"tokens_out":22284,"duration_ms":219350,"concrete_test":"Give a complete proof of Lemma 5.20, for example by homogenizing the system to P^k and using that the leading forms X_i^p have no common zero at Z=0, which would imply the affine intersection is nonempty over an algebraically closed field. To test the paper's specific induction, write out the generic k=3, p=2 case and verify that substituting the solution for x_{k−1} preserves the induction template; alternatively, run a computer algebra check on random instances over finite fields of order p^N for N≥2 as a sanity check, though this would not replace a proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim for algebraically closed coefficients depends on Lemma 5.23, which proves surjectivity of W(F)⊗M → W(F)⊗N via F⊗φ − id⊗c. Lemma 5.23 reduces to Lemma 5.19, whose proof reduces surjectivity of H to solving x_i^p − f_i(x) = 0 for arbitrary degree-one polynomials f_i with arbitrary constant terms. This is exactly what Theorem A uses to make φ−can surjective in the orbit-filtration fiber sequence (5.16); if Lemma 5.20 fails, π_{2k}Krel(SW(F)⊗A) need not vanish and Corollary C collapses. The proof of Lemma 5.20 is a terse induction whose key WLOG step is not actually demonstrated: after solving (5.22) for x_{k−1} when f_k depends nontrivially on x_{k−1}, substituting the resulting linear expression into (5.21) for i = k−1 introduces x_i^p terms for i < k−1, and the paper does not explain how these are absorbed into the induction template (5.21)–(5.22). The lemma is likely true — the homogenized equations have no common zero at Z = 0, so a Bézout/elimination argument should give existence — but the argument as written does not establish it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a new 'orbit filtration' on topological cyclic homology, derived from the May filtration on THH, and uses it to compute cardinalities of the integral algebraic K-theory groups of connective Morava K-theory. For a finite field F_q, Theorem F determines |K_t(k(n)_{F_q})| in all degrees not congruent to 0 or 1 modulo 2p-2, leaving undetermined p-groups only in the excluded congruence classes. For algebraically closed fields, Theorem A and Corollary C give even vanishing and countably infinite odd K-theory groups for k(n) over \\bar F_p. The paper also proves analogous results for truncated Witt vectors W(F)/p^n and for certain motivic pullback squares.","tokens_in":54440,"tokens_out":16289,"duration_ms":150620,"significance":"If correct, this is the first infinite-family integral algebraic K-theory computation for a non-Eilenberg-MacLane ring spectrum, and the orbit filtration is a promising new structural tool. The paper is explicit about which quantities are determined and which are left as undetermined p-groups, and it gives concrete falsifiable cardinality formulas. The main theorems are substantial, but the proof as written contains a load-bearing gap in Lemma 5.20 and a central dependency on an unpublished preprint [AKHW24].","major_comments":[{"comment":"The proof of Lemma 5.20 is incomplete in the induction step. After solving (5.22) for x_{k−1} when f_k depends nontrivially on x_{k−1}, substituting into the equations (5.21) for i<k−1 introduces terms of the form c_i g_k(x_k^p) in addition to the original g_i(x_k^p). The induction hypothesis requires deg(g_i)<deg(g_k) for all i<k−1, but the new coefficients may have degree exactly deg(g_k), so the induction template (5.21)–(5.22) is not satisfied. The sentence 'After this, one uses (5.21) for i<k−1 to deduce...' does not explain how these terms are absorbed. Since Lemma 5.20 is exactly what makes W(F)⊗M→W(F)⊗N surjective in Lemma 5.23, and that surjectivity is the key step in the even-vanishing claim of Theorem A, this is a load-bearing gap. The lemma is plausible (homogenizing the equations shows no common zero at Z=0, so the associated polynomial map is finite and surjective), but the","section":"§5.2, Lemma 5.20"},{"comment":"Theorem F's even-degree K-theory bounds for k(n)_{F_q} are derived from the mod (p,...,v_{n+1}) syntomic cohomology computation of k(n)_F, stated as Theorem 8.13 and attributed to [AKHW24], an unpublished preprint co-authored by the first author. This is a substantial external dependency for a core advertised result. The manuscript should either reproduce the proof of the imported theorem or state explicitly that Theorem F is conditional on the correctness of [AKHW24]. As written, the reader cannot verify a load-bearing input from within the paper.","section":"§8.2–8.4, Theorem 8.13 / Theorem 8.33"}],"minor_comments":[{"comment":"The notation F_p is used in Theorem A and Section 5 where A has π_*A≅F_p[x_{2m}]. If F_p denotes the prime field, Proposition 4.39 applies to A; if it denotes the algebraic closure, Proposition 4.39 does not. Please clarify the convention explicitly.","section":"Notation 1.4 and §5"},{"comment":"The theorem phrase 'W_n denotes the truncated big Witt vectors of length n' is confusing because Notation 1.4 uses W_n(F) for p-typical Witt vectors and a separate symbol for big Witt vectors. Please disambiguate.","section":"Theorem F statement"},{"comment":"Minor typos: 'Neverless' in Remark 9.2; missing closing parenthesis in π_{2k}(Σ(THH(K(1))^{hS1}) in Proposition 9.1.","section":"Remark 9.2 / Proposition 9.1"},{"comment":"The construction Q_n is defined for N-filtered spectra, but later used for Z-filtered objects in Lemma 4.23 and elsewhere. The comparison via Remark 2.2 is mentioned only parenthetically; please make the convention explicit where Q_n is applied to Z-graded objects.","section":"§4.2, Lemma 4.23"}],"recommendation":"major_revision","confidential_remarks":"The main technical issue is the gap in Lemma 5.20; if it can be repaired, the paper is likely sound. The dependency of Theorem F on [AKHW24], a preprint by the first author, is not circular in a strict logical sense, but the editor may wish to require that the imported computation be made available or independently verified. The upcoming work of Bayındır–Land–Tamme–Speirs is cited as already knowing Theorem 2.10; please consider priority issues if relevant."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"If you work in algebraic K-theory or chromatic homotopy, read this paper. It delivers the first integral algebraic K-theory computation for a non-Eilenberg-MacLane ring spectrum in infinitely many degrees: cardinalities of K_*(k(n)) away from two exceptional residue classes mod 2p-2, plus even vanishing over algebraically closed fields. The orbit filtration on TC is a genuinely new tool, and the paper is candid about what it does not determine (the m(r) and A_r factors in the excluded degrees). The proofs are detailed and cross-checked through multiple spectral sequences; the main structure looks sound to me.\n\nWhat I want to flag: Lemma 5.20, the algebraic lemma that carries the even-vanishing proof, is proven in a very compressed induction. The stress-test note worries that the WLOG step and the substitution into the i=k-1 equation are unjustified. I think the argument is repairable—use the i<k-1 equations to replace the x_i^p terms that appear after solving for x_{k-1}, and the system then matches the induction template—but the text does not say that. A referee should ask for a fuller proof. That is the paper's softest spot, and it is a soft spot in exposition, not a detected false claim.\n\nThe other caveat is the dependency of Theorem F on [AKHW24], a preprint co-authored by the first author. That is a real external input for the syntomic cohomology computation. The paper is upfront about it, and it also notes that Theorem 2.10 was independently known to Land–Tamme–Speirs. I would not call this circularity; it is a normal preprint-to-preprint dependency, but the finite-field results are conditional on [AKHW24] being refereed. The undetermined m(r) and A_r terms are explicitly confined to the excluded degrees, and the p∤m assumption is flagged as removable in upcoming work.\n\nThis paper is for specialists in algebraic K-theory and chromatic homotopy theory. It deserves a serious referee. If I were handling it, I would send it out with a request to expand Lemma 5.20 and to double-check the import from [AKHW24]. I expect the main results to survive. I'd cite it, and I'd bring it to the reading group.","headline":"Real result, honest paper: first integral K-theory computations for a non-Eilenberg-MacLane ring spectrum in infinite families, but the proof of Lemma 5.20 is too terse and the finite-field theorem leans on [AKHW24], a preprint by the first author.","tokens_in":54947,"tokens_out":11360,"would_cite":true,"duration_ms":97389,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["19D55","55P43","55N22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The integral algebraic K-theory groups of connective Morava K-theory have their cardinalities determined in every degree not congruent to 0 or 1 modulo 2p−2.","keywords":["algebraic K-theory","Morava K-theory","topological cyclic homology","topological Hochschild homology","orbit filtration","May filtration","Witt vectors","cardinality"],"falsifier":"Test Lemma 5.20 for $p=3$, $k=2$, $f_1=x_2$, $f_2=2x_1+1$: if $x_1^3-x_2=0$ and $x_2^3-2x_1-1=0$ has no solution in the algebraic closure of $\\mathbb{F}_3$, the lemma is false. More generally, search for random linear $f_i$ over $\\mathbb{F}_{3^N}$ for increasing $N$; any system with no solution in any finite extension would falsify the lemma. A direct check of the predicted vanishing, e.g. computing $K_2(k(1)_{\\bar{\\mathbb{F}}_3})$ via the orbit spectral sequence and finding a nonzero group, would also falsify the central even-vanishing claim.","tokens_in":54011,"feed_emoji":"🧮","tokens_out":11940,"duration_ms":101946,"temperature":0.7,"texified_at":"2026-08-05T21:39:34.164791+00:00","pith_summary":"The paper determines the sizes of the integral algebraic K-theory groups of connective Morava K-theory $k(n)$ in every degree not congruent to 0 or 1 modulo $2p-2$. Over a finite field $\\mathbb{F}_q$, the odd-degree orders are $|W_{\\lfloor r/(p^n-1)\\rfloor}(\\mathbb{F}_q)|(q^{r+1}-1)$, and the even groups in the resolved range are either 0 or cyclic p-groups $\\mathbb{Z}/p^{m(r)}$. After base change to the algebraic closure $\\bar{\\mathbb{F}}_p$, all even groups vanish and the positive odd groups are countably infinite with bounded p-torsion. The proof runs through a new orbit filtration on topological cyclic homology, induced by the May filtration on topological Hochschild homology, which makes the Frobenius correction map finite enough to count. This matters as the first infinite-family integral K-theory computation for a ring spectrum that is not an Eilenberg–MacLane spectrum.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":7339,"prompt_tokens":994,"completion_tokens":6345,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":994,"completion_tokens_details":{"reasoning_tokens":5325}},"feed_headline":"Morava K-theory's K-groups counted in all but two degree classes","feed_subtitle":"Odd groups follow Witt-vector formulas; even groups vanish over algebraically closed fields.","key_machinery":"The orbit filtration on topological cyclic homology: starting from the multiplicative Whitehead filtration $\\tau_{\\ge \\bullet} A$, the May filtration on THH is promoted to a filtration of cyclotomic spectra by postcomposing the twisted Frobenius with the weight-decreasing map $R_p \\Rightarrow \\mathrm{id}$; applying TC levelwise yields the orbit filtration $\\mathrm{fil}^*_{\\mathrm{orb}} TC(A)$. Its associated graded is $\\Sigma TC^+(\\pi_* A)$, and a quotient $Q_n$ of the filtration recovers TC in degrees $\\le n$ while having finite homotopy groups when $\\pi_* A \\cong \\mathbb{F}_q[x_{2m}]$. This finiteness, together with the odd-concentration of $TC^-/TP$ for formal polynomial DGAs (Theorem 2.10, which identifies $\\pi_{2r+1} TC(F[x_{2m}])$ with $W_{\\lfloor r/m \\rfloor}(F)$), is what allows cardinality counting,","core_discovery":"The central claim is that the integral algebraic K-theory of connective Morava K-theory, previously known only in low degrees or with finite coefficients, satisfies a sharp numerical law in nearly all degrees. For finite fields $\\mathbb{F}_q$, Theorem F gives isomorphisms $K_{2r}(k(n)_{\\mathbb{F}_q}) \\cong 0$ or $\\mathbb{Z}/p^{m(r)}$ in the resolved even cases and cardinalities $|K_{2r+1}(k(n)_{\\mathbb{F}_q})| = |W_{\\lfloor r/(p^n-1)\\rfloor}(\\mathbb{F}_q)|(q^{r+1}-1)$ in the resolved odd cases, leaving only two p-power ambiguities in periodic families. For algebraically closed fields, Corollary C states $K_{2k}(k(n)_{\\bar{\\mathbb{F}}_p}) = 0$ and $K_0$ and $K_{2k-1}$ are countably infinite with bounded p-torsion. The paper derives these from a theorem about $E_1$-rings with hom","pith_inferences":["The orbit filtration should apply to other connective complex-oriented theories whose associated graded homology is polynomial, such as truncated Brown–Peterson spectra; if the Frobenius surjectivity lemma extends, even vanishing and Witt-vector cardinalities would follow without new homotopy-group computations.","The undetermined p-powers m(r) and A_r likely arise from v_n-Bockstein differentials in Adams weight 2; resolving the differentials d_1(v_{n+1}^j ∂λ_{n+1} ε_i) (i ≤ n) would make Theorem F completely explicit.","The algebraically closed case suggests a general principle: for p-complete ring spectra with polynomial F̄_p-homology, even integral K-theory is controlled by the surjectivity of F⊗φ − id⊗c on finite quotient modules; this may unify known even-vanishing theorems for rings such as Z/p^n.","The countably infinite odd K-groups with bounded p-torsion over F̄_p are a new chromatic phenomenon: unlike finite-field cases, the integral K-groups are infinite yet have only bounded torsion in each degree."],"forward_implications":["For every finite field F_q, the full cardinality of K_t(k(n)_{F_q}) is now determined for all t not congruent to 0 or 1 mod 2p−2; the two pending p-powers m(r) and |A_r| are the only gaps in the resolved period.","Base-changing k(n) to F̄_p forces all even K-groups to vanish, so the chromatic analogue of Quillen's computation for F_p has the same even parity.","The quotient |K_{2r+1}(k(n)_{F_q})|/|K_{2r}(k(n)_{F_q})| equals |W_{⌊r/(p^n−1)⌋}(F_q)|, a clean ratio law that relates to Lichtenbaum-style quotient formulas for number fields.","For W(F)/p^n with F algebraically closed, relative K-theory vanishes in even degrees and is infinite in odd degrees; over F̄_p these odd groups are countably infinite with bounded p-torsion.","Relative K-theory of the arithmetic coordinate axis W(F̄_p) ×_{F̄_p} W(F̄_p) (the Burnside-ring base change) is trivial in odd degrees ≥1 and infinite in even degrees ≥1."],"fun_headline_variants":["Morava K-theory K-groups: even vanish, odd cardinalities pinned","All but two degree classes: K-groups of Morava K-theory counted","K-theory cardinalities for Morava K-theory and Witt vectors","Even K-groups vanish for Morava K-theory over algebraic closure","Integral K-groups of Morava K-theory resolved in all but p-powers"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is Lemma 5.20: over an algebraically closed field of characteristic $p$, any system $x_i^p - f_i(x_1,\\ldots,x_k)=0$ with $f_i$ linear has a solution; the proof given is a terse induction with a 'without loss of generality' step, and if that lemma fails, the surjectivity of the Frobenius correction—and with it the even-vanishing theorem—collapses.","fun_headline_variants_meta":{"raw":{"variants":["Morava K-theory K-groups: even vanish, odd cardinalities pinned","All but two degree classes: K-groups of Morava K-theory counted","K-theory cardinalities for Morava K-theory and Witt vectors","Even K-groups vanish for Morava K-theory over algebraic closure","Integral K-groups of Morava K-theory resolved in all but p-powers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000231,"raw_usage":{"total_tokens":1370,"prompt_tokens":841,"completion_tokens":529,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":585,"completion_tokens_details":{"reasoning_tokens":425}},"tokens_in":585,"tokens_out":529,"duration_ms":4983,"temperature":1.0,"reasoning_tokens":425,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:04:29.462707+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test Lemma 5.20 for $p=3$, $k=2$, $f_1=x_2$, $f_2=2x_1+1$: if $x_1^3-x_2=0$ and $x_2^3-2x_1-1=0$ has no solution in the algebraic closure of $\\mathbb{F}_3$, the lemma is false. More generally, search for random linear $f_i$ over $\\mathbb{F}_{3^N}$ for increasing $N$; any system with no solution in any finite extension would falsify the lemma. A direct check of the predicted vanishing, e.g. computing $K_2(k(1)_{\\bar{\\mathbb{F}}_3})$ via the orbit spectral sequence and finding a nonzero group, would also falsify the central even-vanishing claim.","supporting_citations":[],"review_version":1}