{"id":"d844ff72-e537-4e9d-9fd9-cf1bdbd9a97d","arxiv_id":"2504.14383","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New w1-periodic families in C-motivic stable homotopy and in the cohomology of the moduli stack of formal group laws have eta-exponents governed by 2-adic valuations.","lead":"This paper finds new infinite families of nonzero elements in an algebraic approximation to stable homotopy, with torsion exponents following a number-theoretic pattern. It uses the C-motivic Adams spectral sequence and the Burklund-Xu spectral sequence to prove the families and to translate them to the Adams-Novikov E2-page.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The permanent cycle Δ₁h₁²i₁ in §7.5 rests on external machine data and an asserted collapse in Proposition 7.25; an error in stems 100–107 would remove the η-coexponent 12 entries from Theorem 4.7(5).","rationale":"I read the paper as making two kinds of claims: the existence of the w₁-periodic families, which is largely an induction from known base cases with Chow-degree arguments that are sound; and the refined η-coexponents, which require new permanent-cycle computations. The first three permanent-cycle families (h₁h₃gₖ, i₁g₂ₖ, and the infinite d₂ family on h₃gₖ) have base cases in the published range [IWX23] or in the independent S/τ charts, and their induction via Moss convergence is coherent. The genuinely new base case is Δ₁h₁²i₁ in the 107-stem. Everything downstream of Proposition 7.25 inherits its reliability, so this is the load-bearing point. The reader's weakest assumption identifies exactly this spot, including the terse gap in Proposition 7.25. I do not see a separate internal flaw in the Toda-bracket or Burklund–Xu arguments that would be more serious. The abstract-level statement with an unknown '?' entry is a presentation problem, not a mathematical collapse, since the precise theorems are guarded. Because the concern is real but conditional on external data that could in principle be checked, the appropriate verdict remains CONDITIONAL; the reader's verdict does not need to change.","tokens_in":37647,"tokens_out":17320,"duration_ms":145346,"concrete_test":"Recompute the C-motivic Adams E₂-page in stems 100–107, filtrations 13–22, directly from the cobar complex of the C-motivic dual Steenrod algebra using an independent implementation (e.g., a fresh minimal-resolution or May-spectral-sequence computation in Sage/Python), and recompute the S/τ charts in the same range. Then verify the following explicitly: (1) the only possible values of d_r({13-300}) for r = 2,...,7 are exactly those listed in Lemmas 7.10–7.24; (2) every E₂-class in degree (106, f ≥ 21) is killed before E₅, so the final assertion of Proposition 7.25 is true; and (3) the S/τ differentials cited in Lemmas 7.10, 7.12, 7.14, 7.19, 7.20, and 7.21 are present with the stated targets. If any discrepancy appears, rerun Lemmas 7.22–7.24 and Proposition 7.26 with the corrected data.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The deepest new numerical content is the η-coexponent table (Theorem 4.7(5), Table 3), especially the value 12 in coweights 32k+48. That value is built on Proposition 7.26, whose base case is Proposition 7.25, the claim that {13-300} = Δ₁h₁²i₁ in degree (107, 13, 60) is a permanent cycle. The proof of Proposition 7.25 is a chain of lemmas that exclude potential Adams differentials by inspecting the file Adams-motivic-E2-machine.csv and the S/τ charts of [IWX22a]. Each lemma either states that a certain element is the 'only possible' target or source, or compares against a differential in the S/τ Adams spectral sequence. No independent verification of this dataset, or of the completeness of the 'by inspection' lists, is given in the manuscript. A single omitted class in stems 100–107 — for example, an additional E₂ class in degree (106, ≥21) surviving to E₅, or a missing S/τ differential used in Lemmas 7.10–7.24 — would make d_r({13-300}) nonzero for some r and destroy the permanent cycle, collapsing the δ and γ submodule claims derived from it. Proposition 7.25 also contains a terse final assertion: after survival to E₈ is shown, 'for higher differentials, all possible non-zero values disappear already in the E₅-page' is stated without listing those possible values or the differentials that kill them. This is a gap in the written proof, even if the underlying computational data is correct. The concern is not internal inconsistency; it is that the central new coexponent rests on computational artifacts that the paper does not independently certify.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies 2-primary periodic families in the C-motivic stable homotopy groups and, via the Betti/τ-localization comparison, in the classical Adams–Novikov E2-page, i.e. the cohomology of the moduli stack of 1-dimensional formal group laws. The main objects are w1-periodic elements detected by the classes w_1^{4k}/h_1 and w_1^{4k+2}, and the paper computes, in many coweights, the smallest N such that η^N (respectively α_1^N) annihilates the element. The precise statements are Theorem 4.7 and Theorem 4.11 for C-motivic homotopy, and Corollaries 8.1 and 8.3 for the Adams–Novikov E2-page. The technical core is a detailed analysis of the C-motivic Burklund–Xu spectral sequence in Chow degree one, an infinite family of Adams d2 differentials d2(h3g_k) = h0h2·h2g_k, and a specific permanent cycle Δ1h1^2i1 in the C-motivic 107-stem. The paper also contains a self-contained computation of the Chow-degree-one part of the cohomology of C-motivic A(2).","tokens_in":38005,"tokens_out":2844,"duration_ms":26225,"significance":"If the main claims are correct, this is a substantial contribution: it identifies new number-theoretic periodic structure in the Adams–Novikov E2-page that is distinct from v_n-periodicity, and it provides the first systematic computation of η-coexponents beyond Andrews's η-torsion families. The Burklund–Xu spectral sequence arguments are well chosen and the paper gives a clear conceptual framework, including explicit Massey products and Toda-bracket inductions. The authors are appropriately careful in the precise theorems, distinguishing what is proved from what is conjectural in the introductory sequence (1.1). The machine-assisted nature of some parts is normal for the field, but the manuscript would be strengthened by making the relevant machine-data checks auditable.","major_comments":[{"comment":"The advertised Theorem 1.2(4) and Corollary 1.5(4) claim that for all n≥2 the nth term of sequence (1.1) gives the smallest N such that η^N (resp. α_1^N) annihilates the element. But sequence (1.1) contains an explicit unknown value at the 24th entry, and the paper itself states in §1.4 that the first unknown η-exponent occurs in coweight 95 = 4·24−1. The precise statements in Theorem 4.7(5) and Remark 4.10 are correctly hedged, so the abstract and the unqualified item (4) overstate what is proved. This mismatch should be fixed in the abstract, Theorem 1.2, Corollary 1.5, and any summary statements that repeat the claim.","section":"§1.4, Theorem 1.2(4), Corollary 1.5(4)"},{"comment":"Proposition 7.25 is the base case for the permanent cycle Δ1h1^2i1g^{4k}, which in turn supports the η-coexponent 12 entries in Table 3 and Theorem 4.7(5). The proof shows survival to the E8-page by a chain of lemmas, but the final sentence states: 'For higher differentials, all possible non-zero values disappear already in the E5-page' without listing those possible values or the differentials that kill them. Since this assertion is load-bearing for the coexponent 12, the proof is incomplete as written; the authors should provide the explicit list of possible higher differentials and the relevant vanishing arguments, or refer to a precise dataset/lemma that supplies them.","section":"§7.5, Proposition 7.25"},{"comment":"The proof of Proposition 7.25 relies on machine-generated data in the file Adams-motivic-E2-machine.csv from [IWX22b] and on the S/τ charts of [IWX22a]. Several lemmas are justified by 'by inspection' of this external dataset, and Lemma 7.10, Lemma 7.12, and Lemma 7.21 compare against S/τ differentials. If the dataset omits a class in stems 100–107 or the S/τ charts contain an error, the permanent cycle and the coexponent 12 results would collapse. The manuscript does not provide an independent verification of the relevant part of the dataset. I do not regard this as internal inconsistency, but because it is a correctness-risk at the central new numerical content, the authors should either include the relevant excerpts of the machine data as an appendix or state explicitly the exact subset of the external data on which each lemma depends.","section":"§7.5, Lemmas 7.10–7.24"}],"minor_comments":[{"comment":"The caption of Figure 6 says 'Some w1-periodic h1-submodules in the classical Adams E∞-page', but the surrounding text in §4.3 and the chart itself concern the C-motivic Adams E∞-page; this should be corrected.","section":"Figure 6 caption"},{"comment":"The sentence 'There are no possible non-zero values for d4 ({13-300}) or d5 ({13-300})' is stated without explanation; adding a one-line degree or filtration reason would help the reader verify the claim.","section":"§7.5 between Lemma 7.11 and Lemma 7.12"},{"comment":"The equation w_1^{2^j-2}·η_j = η^{2^j-2}·η_{j+1} is striking and would benefit from a small explanatory sentence about the indexing of η_j, since the notation η_j is not defined in this section and the reader must infer it from later references to Mahowald elements.","section":"§1.6, Equation (1.6)"},{"comment":"In Table 3, the row for coweight 2^j+3 lists the η-coexponent as 2^{j+2}−1, while the text of Theorem 4.7(4) writes the generator as η^{2^{j+2}−2}·η_{j+4}; the apparent off-by-one between the exponent of η and the coexponent should be clarified, since the generator may be divisible by η once more than the coexponent indicates.","section":"§4.3, Table 3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central computational strategy is credible and the precise theorems are carefully hedged, but the abstract and introductory theorem overclaim in a way that is visible to any reader, and Proposition 7.25 contains a genuine proof gap at the base of the most interesting new numerical value. These are fixable within the scope of a revision: restate the main claims to exclude the unknown entry, and provide the missing enumeration in Proposition 7.25. I do not see grounds for rejection, but the revision needs to be checked carefully."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the genuinely new part is part (4) of Theorem 4.7/4.11 — the infinite family d2(h3gk)=h0h2*h2gk, the permanent cycles i1g2k and Δ1h1²i1, and the eta-exponents (and alpha1-coexponents) with a 2-adic valuation pattern. Existence of one element in each coweight was Andrews's. The paper is honest about this split and reproves the existence parts independently.\n\nWhat it does well: the exposition is careful, and the precise theorems are more guarded than the abstract. The main arguments via Toda brackets and Moss convergence look structurally sound, and the target coexponents are genuinely outputs of spectral sequence computations, not inputs fed back into the machinery. Base cases use previous charts [IWX23], which is normal and not circular.\n\nThe soft spots are real but not fatal. First, abstract-level Theorem 1.2(4) and Corollary 1.5(4) claim that every entry of sequence (1.1) gives a value, but the sequence contains an unknown '?' at the 24th entry; as written those statements are false. The precise statements in Theorem 4.7/4.11 and the remarks handle this correctly. That's a fixable wording problem.\n\nSecond, the deepest new numerical content is the coexponent 12 in coweights 32k+48, built on the permanent cycle Δ1h1²i1 in degree (107,13,60). Proposition 7.25 ends with a terse assertion that for higher differentials all possible non-zero values 'disappear already in the E5-page', without listing those values, and the lemmas lean on machine-generated Adams E2 data and S/tau charts [IWX22a,b]. If those charts have an omission in stems 100-107, this part collapses. That is a genuine condition, not a rhetorical one. It is also addressable: a short table of the surviving E5 classes or a script that checks the differentials would fill the gap.\n\nBottom line: the structural work is credible and the central construction holds up on reading. The reader's CONDITIONAL verdict is about right. I'd send this to a serious referee, with the abstract and Section 7.5 flagged for revision. For a stable homotopy reading group, it is worth a session, but expect the discussion to concentrate on Proposition 7.25.","headline":"Genuinely new coexponent computations with a real but fixable dependence on machine charts; deserves refereeing despite the over-strong abstract.","tokens_in":38612,"tokens_out":2252,"would_cite":true,"duration_ms":21713,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55T15","14F42","55Q51"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every coweight $4n-1$ there is a non-zero $\\eta$-indivisible class whose $\\eta$-exponent follows a 2-adic pattern.","keywords":["Adams-Novikov spectral sequence","C-motivic stable homotopy","w1-periodicity","eta-torsion","Burklund-Xu spectral sequence","moduli stack of formal group laws","Mahowald elements","2-adic valuation"],"falsifier":"Recompute the $\\mathbb{C}$-motivic Adams differentials in stems 100--107 and check directly whether $d_6(\\{13\\text{-}300\\})=\\{19\\text{-}449\\}$ or any other differential hits $\\{13\\text{-}300\\}$; if one does, $\\Delta_1 h_1^2 i_1$ is not a permanent cycle and the coexponent claims at coweights 47 and above fail.","tokens_in":37449,"feed_emoji":"","tokens_out":12136,"duration_ms":98749,"temperature":0.7,"pith_summary":"This paper establishes an infinite family of non-trivial classes in $\\mathbb{C}$-motivic stable homotopy: in every coweight $4n-1$ there is a class not divisible by $\\eta$, and in many cases the paper determines exactly the smallest power of $\\eta$ that annihilates it. The resulting exponents form the sequence $\\infty, 3, 2, 7, 2, 6, 2, 15, \\dots$, whose entries appear to depend only on the 2-adic valuation of the index. Because the $\\mathbb{C}$-motivic sphere's $S/\\tau$ quotient computes the classical Adams--Novikov $E_2$-page, the same argument produces non-zero elements in the cohomology of the moduli stack of 1-dimensional formal group laws in degrees $(20n-2, 4n)$, annihilated by $\\alpha_1$, with the same $\\alpha_1$-exponents. The pattern is a new algebraic periodicity, distinct from the familiar $v_n$-periodicities, and it explains why number-theoretic structure can appear in this cohomology.","feed_headline":"A 2-adic pattern rules new η-torsion families","feed_subtitle":"New classes in C-motivic homotopy and the Adams-Novikov E2-page have annihilation exponents repeating by powers of two.","key_machinery":"The central mechanism is the $\\mathbb{C}$-motivic Burklund--Xu spectral sequence, a Cartan--Eilenberg-type spectral sequence that computes the $\\mathbb{C}$-motivic Adams $E_2$-page in Chow degree one, i.e. in the grading $s+f-2w=1$. There it collapses for degree reasons, and the paper uses it to prove that the infinite family of products $h_0h_2\\cdot h_2g^k$ is non-zero for $k\\ge 1$. Recursive Massey products then yield an infinite family of Adams differentials $d_2(h_3g^k)=h_0h_2\\cdot h_2g^k$ and permanent cycles $h_1h_3g^k$, $i_1g^{2k}$, and $\\Delta_1 h_1^2 i_1 g^{4k}$; a finite check near the 107-stem shows that $\\Delta_1 h_1^2 i_1$ is a permanent cycle. Moss convergence converts these algebraic statements into Toda brackets and then into $\\eta$-divisibility statements in homotopy, while a Chow-degree argument rules out hidden $\\eta$-extensions.","core_discovery":"On the paper's own terms, the discovery is a new algebraic periodicity, distinct from the $v_n$-periodicities, that appears in the cohomology of the moduli stack of 1-dimensional formal group laws and in $\\mathbb{C}$-motivic stable homotopy. Precisely, for every $n\\ge 1$ there is a non-zero $\\mathbb{C}$-motivic class of coweight $4n-1$ that is not divisible by $\\eta$; for $n=2^j$ it is the Mahowald element $\\eta_{j+3}$, and for $n\\ge 2$ the $n$th term of the displayed sequence is the smallest $N$ such that $\\eta^N$ annihilates it. Translating through the $S/\\tau$ quotient gives the Adams--Novikov counterpart: non-zero $\\alpha_1$-torsion elements in degrees $(20n-2,4n)$ whose $\\alpha_1$-coexponents are given by the same sequence, beginning $\\infty, 3, 2, 7, 2, 6, 2, 15, \\dots$.","pith_inferences":["If the observed dependence on 2-adic valuations holds in general, the unknown 24th entry of sequence (1.1), in coweight 95, is determined by $v_2(24)=3$; computing that single $\\eta$-exponent would directly test the pattern.","The same Burklund--Xu method, applied one Chow degree higher, should produce parallel families of $\\alpha_1$-coexponent classes in the Adams--Novikov $E_2$-page beyond the range treated here; the paper's charts leave concrete candidates in stems beyond 107.","The close analogy with $v_1$-periodicity suggests that the $\\alpha_1$-exponents in the moduli stack cohomology may have a purely number-theoretic description, perhaps tied to denominators of Bernoulli-related invariants; such a description is not proved in the paper."],"forward_implications":["The classical Adams--Novikov $E_2$-page contains infinitely many $\\alpha_1$-torsion classes in degrees $(20n-2,4n)$, not divisible by $\\alpha_1$, with $\\alpha_1$-coexponents given by the sequence (1.1).","$\\mathbb{C}$-motivic stable homotopy contains non-zero $\\eta$-indivisible classes in every coweight $4n-1$, and in coweights $4k+3$ there are additional non-zero classes detected by $w_1^{4n+2}$ with $\\eta$-exponent 2.","The Mahowald $\\eta_j$ classes are the first members of $w_1^{2^j-2}$-periodic families whose second members are $\\eta$-multiples of $\\eta_{j+1}$, mirroring the classical relation $v_1^{2^n-1}\\rho_{2^n-1}=2\\rho_{2^{n+1}-1}$.","The infinite family of Adams differentials $d_2(h_3g^k)=h_0h_2\\cdot h_2g^k$ shows that the elements $h_3g^k$ cannot detect homotopy classes, while their $\\eta$-multiples $h_1h_3g^k$ are permanent cycles.","The permanent cycle $\\Delta_1 h_1^2 i_1$ in the 107-stem generates an infinite $w_1$-periodic family $\\Delta_1 h_1^2 i_1 g^{4k}$, so the $\\eta$-exponent results extend to all $k$ by Toda brackets."],"supporting_citations":[{"why":"Identified the $w_1$-periodic $\\eta$-torsion elements in each coweight and the $w_1^4$ self-map on $S/\\eta$ that the new families extend.","marker":"[And18]"},{"why":"Introduces the Burklund--Xu spectral sequence and the differential formulas used throughout the paper.","marker":"[BX25]"},{"why":"Supplies the $\\mathbb{C}$-motivic Burklund--Xu linearity and differential results used in Chow degree one.","marker":"[Ben24]"},{"why":"Provides the isomorphism between the Adams--Novikov $E_2$-page and the homotopy of $S/\\tau$, the bridge to formal group law cohomology.","marker":"[Isa19]"},{"why":"Provides the $\\mathbb{C}$-motivic Adams charts for the cofiber of $\\tau$, used to rule out Adams differentials on $\\{13\\text{-}300\\}$ by inclusion comparison.","marker":"[IWX22a]"},{"why":"Provides the machine-generated $\\mathbb{C}$-motivic Adams $E_2$ data containing $\\{13\\text{-}300\\}$ and the low-dimensional charts used for base cases.","marker":"[IWX22b]"},{"why":"Gives the explicit low-dimensional stable homotopy computations used as base cases for differential and Toda bracket inductions.","marker":"[IWX23]"},{"why":"Gives the Toda bracket convergence theorem used to detect the permanent-cycle families in homotopy.","marker":"[BK21]"},{"why":"Gives the Moss convergence theorem used to pass from Massey products in the Adams $E_2$-page to Toda brackets in homotopy.","marker":"[Mos70]"}],"fun_headline_variants":["Power-of-two periodicity in formal group cohomology","η-torsion exponents obey 2-adic periodicity","New periodicity beyond v_n: exponents are 2-powers","Moduli stack cohomology: 2-adic torsion pattern","Exotic periodicity in stable homotopy: powers of two"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The 107-stem permanent cycle $\\Delta_1 h_1^2 i_1$ is certified by machine-generated $\\mathbb{C}$-motivic Adams $E_2$ data and $S/\\tau$ charts that the paper does not independently re-verify; an error in those charts would break the $\\eta$-coexponent values built on it.","fun_headline_variants_meta":{"raw":{"variants":["Power-of-two periodicity in formal group cohomology","η-torsion exponents obey 2-adic periodicity","New periodicity beyond v_n: exponents are 2-powers","Moduli stack cohomology: 2-adic torsion pattern","Exotic periodicity in stable homotopy: powers of two"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001116,"raw_usage":{"total_tokens":4612,"prompt_tokens":879,"completion_tokens":3733,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":3645}},"tokens_in":495,"tokens_out":3733,"duration_ms":27333,"temperature":1.0,"reasoning_tokens":3645,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:50:44.365506+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the $\\mathbb{C}$-motivic Adams differentials in stems 100--107 and check directly whether $d_6(\\{13\\text{-}300\\})=\\{19\\text{-}449\\}$ or any other differential hits $\\{13\\text{-}300\\}$; if one does, $\\Delta_1 h_1^2 i_1$ is not a permanent cycle and the coexponent claims at coweights 47 and above fail.","supporting_citations":[],"review_version":1}