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Exotic periodic phenomena in the cohomology of the moduli stack of $1$-dimensional formal group laws

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Pith's one-line read For every coweight $4n-1$ there is a non-zero $\eta$-indivisible class whose $\eta$-exponent follows a 2-adic pattern.

desk verdict Genuinely new coexponent computations with a real but fixable dependence on machine charts; deserves refereeing despite the over-strong abstract. read the letter →

arxiv 2504.14383 v2 pith:WBOSVXQ6 submitted 2025-04-19 math.AT

Daniel C. Isaksen , Hana Jia Kong , Guchuan Li , Yangyang Ruan , Heyi Zhu This is my paper · ORCID
classification math.AT MSC 55T1514F4255Q51
keywords Adams-NovikovspectralsequenceC-motivicstablehomotopyw1-periodicityeta-torsionBurklund-XumodulistackofformalgrouplawsMahowaldelements2-adicvaluation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $\infty, 3, 2, 7, 2, 6, 2, 15, \dots$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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).

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 (3)
  1. [§1.4, Theorem 1.2(4), Corollary 1.5(4)] 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.
  2. [§7.5, Proposition 7.25] 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.
  3. [§7.5, Lemmas 7.10–7.24] 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.
minor comments (4)
  1. [Figure 6 caption] 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.
  2. [§7.5 between Lemma 7.11 and Lemma 7.12] 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.
  3. [§1.6, Equation (1.6)] 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.
  4. [§4.3, Table 3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the eta-coexponents are computed from spectral sequence differentials along an acyclic dependency graph, with machine data serving as external computational input rather than as a fitted prediction.

full rationale

The claimed coexponents are outputs, not inputs, of the computations. The dependency graph is acyclic: Theorem 4.7(1)-(4) and Theorem 4.11(1)-(3) are proved without the later permanent-cycle propositions, and those propositions (7.5, 7.8, 7.26) use only part (2) of the earlier theorems, not the coexponent conclusions they support. The base cases and machine data come from published external datasets and charts ([IWX23], [IWX22a], [IWX22b]) rather than from the target eta-coexponents or the Adams-Novikov elements being derived. The paper explicitly flags its reliance on machine-generated Adams E2 data in Section 7.5 and on S/tau charts for differential comparisons; an error in those datasets would be a correctness risk, not a circular reduction. Proposition 7.25's terse final assertion about higher differentials is a gap in the written proof, but it does not make the coexponents equal to their inputs by construction. No fitted parameter is renamed as a prediction, and no load-bearing argument reduces to a self-citation chain.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters: the paper is a pure homotopy theory computation; all numerical values are either computed via spectral sequences or imported from cited machine charts, not fitted to the target result. No new theoretical entities are introduced; the classes gamma_n and delta_n are elements of previously studied homotopy groups, not new structural inputs.

assumptions (6)
  • standard math Adams vanishing line and periodicity operator isomorphisms in H**Acl in the ranges used (Proposition 6.3; Lemma 3.3; Lemma 6.4).
    Imported from [Ada66b], [Li20], [Mayb]; used to classify Burklund-Xu E1-page elements and h0-coexponents.
  • domain assumption Burklund-Xu spectral sequence converges to the C-motivic Adams E2-page in Chow degree one, with differential formulas dr(q_i x) = q_{i-r} times Massey products.
    Used throughout Sections 5 and 6; imported from [BX25, Lemma 4.10] and [Ben24, Proposition 14]; not reproved here.
  • domain assumption Moss convergence theorem for Toda brackets in the C-motivic Adams spectral sequence.
    Used in Propositions 7.5, 7.8, 7.26 and Theorems 4.7/4.11 to identify detected elements; requires no crossing differentials, verified in each case by the negative Chow degree argument.
  • standard math Mahowald's classical eta_j families exist and h1 h_n are permanent cycles.
    Used in Proposition 4.5 and Theorem 4.7(4) to extend the eta_2^j family to C-motivic homotopy; cited to [Mah77].
  • domain assumption The machine-generated C-motivic Adams E2-page data and S/tau charts in the 100-107 stem range are complete and correct.
    Load-bearing for Section 7.5; the paper relies on the file Adams-motivic-E2-machine.csv and charts [IWX22a,b] without shipping verification code.
  • standard math Isomorphism between the homotopy of the C-motivic two-cell complex S/tau and the classical Adams-Novikov E2-page.
    Cited to [Isa19, Theorem 6.12]; used in Section 8 to translate C-motivic results into cohomology of the moduli stack.

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Pith. "Pith review of Exotic periodic phenomena in the cohomology of the moduli stack of $1$-dimensional formal group laws." pith.science (2026). https://pith.science/paper/WBOSVXQ6

@misc{pith2026250414383,
  author       = {Pith},
  title        = {Pith review of: Exotic periodic phenomena in the cohomology of the moduli stack of $1$-dimensional formal group laws},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WBOSVXQ6}},
  note         = {Machine review of arXiv:2504.14383}
}
abstract

We describe some periodic structure in the cohomology of the moduli stack of 1-dimensional formal group laws, also known as the $E_2$-page of the classical Adams--Novikov spectral sequence. This structure is distinct from the familiar $v_n$-periodicities, and it displays interesting number-theoretic properties. Our techniques involve the $\mathbb{C}$-motivic Adams spectral sequence, and we obtain analogous periodic structure in $\mathbb{C}$-motivic stable homotopy.

Figures

Figures reproduced from arXiv: 2504.14383 by the authors.

Figure 1
Figure 1. Some w1-periodic elements in the C-motivic Adams E ∞-page. Gray elements were previously studied by Andrews. Red elements are studied in this manuscript [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The classical v1-periodic motif of period (8, 4). Gray dots detect 2-torsion; red dots detect elements that are not 2-torsion. The analogy between v1-periodicity and w1-periodicity involves a degree shift in terms of Adams charts. Consequently, the classical v1-periodic motif of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. A C-motivic w1-periodic motif of period (20, 4, 12). Gray dots detect h1-torsion; red dots detect elements that are not h1- torsion. 1.8. The Burklund–Xu spectral sequence. Our main algebraic tool is the Burk￾lund–Xu spectral sequence [BX25]. This spectral sequence was originally intended to compute the E2-page of the Cartan–Eilenberg spectral sequence associated to the extension F2[ζ 2 1 , ζ2 2 , . . .] → F2[ζ1, ζ2… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Some v1-periodic h0-submodules in the classical Adams E2-page, or some w1-periodic h1-submodules in the C-motivic Adams E2-page. Numbers along the top are (stem + 1), resp., (coweight + 1). Numbers along the bottom are coexponents. The data in [PITH_FULL_IMAGE:figures…
Figure 5
Figure 5. Figure 5: Some v1-periodic h0-submodules in the classical Adams E∞-page. Numbers along the top are (stem + 1). Numbers along the bottom are coexponents. 4.3. w1-periodic homotopy. In Section 3.5, we studied some h1-submodules in the C-motivic Adams E2-page. We now push further f…
Figure 6
Figure 6. Figure 6: Some w1-periodic h1-submodules in the classical Adams E∞-page. Numbers along the top are (coweight + 1). Numbers along the bottom are h1-exponents [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: The E 1 2-page of the Burklund–Xu spectral sequence for the cohomology of the C-motivic Steenrod algebra in Chow degree one. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 2 4 6 8 10 12 14 16 1 h 2 0 h3 h 3 0 h 3 0h3 P h1 h 2 0h3 P h1 h 2 3 h4 h0h 2 3 h 7 0 h 7 0h4 P 2h1…
Figure 8
Figure 8. Figure 8: The E∞-page of the Burklund–Xu spectral sequence for the cohomology of the C-motivic Steenrod algebra in Chow degree one [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]
Figure 9
Figure 9. Figure 9: The cohomology of the C-motivic Steenrod algebra in Chow degree one through the 140-stem [PITH_FULL_IMAGE:figures/full_fig_p033_9.png]
Figure 10
Figure 10. Figure 10: The E2-page of the Burklund–Xu spectral sequence for the cohomology of C-motivic A(2) in Chow degree one [PITH_FULL_IMAGE:figures/full_fig_p043_10.png]
Figure 11
Figure 11. Figure 11: The E∞-page of the Burklund–Xu spectral sequence for the cohomology of C-motivic A(2) in Chow degree one Lemma 9.3. In the C-motivic Burklund–Xu spectral sequence for the cohomology of A(2) in Chow degree one, there are no hidden h0 extensions. The only hidden h1 exte…

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