{"id":"f217960e-9739-42f0-b320-1eaf989c4df6","arxiv_id":"2504.16220","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The C-motivic Steenrod algebra cohomology contains an indecomposable family e0 g_{2^n-1} in degrees (2^n-1)(20,4,12)+(17,4,10), with h1-localization to v_n and sparse w1-periodicity.","lead":"This paper proves that certain exotic elements, named e0 g_k, do exist in the cohomology of the C-motivic Steenrod algebra exactly when k is one less than a power of two. It settles a conjecture by Hieu Thai and connects these elements to h1-localization and to the modular forms spectrum mmf.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.5's permanence proof only rules out d1 targets (v1-intercept -4); any hypothetical d_r with r>1 would land below -4, outside Lemma 2.4's classification, and the needed 'no higher differentials in Chow degree one' assertion is not proved or cited in this text.","rationale":"I read the paper in good faith. The main construction is coherent: Lemma 2.1, Proposition 2.2, Lemma 2.4, Lemma 2.5, and Proposition 2.7 form a credible BXSS argument, and the later results Propositions 2.8, 2.10, and 2.11 follow if the uniqueness and permanence step holds. The reader's conditional verdict is reasonable because [IKL+25] is an unpublished same-author companion and Lemma 2.3's finite computation is not shown. My stress-test sharpens this to an internal point: in Lemma 2.5, the possible target of a differential is computed as though every differential had length 1. In a spectral sequence one must also rule out d_r for r>1; those targets have v1-intercept less than −4 and therefore are not classified by Lemma 2.4. Unless this fact is explicitly supplied, likely from [IKL+25, Section 5.2], the proof of permanence is incomplete. This does not make me doubt the theorem; it makes me agree that the paper should remain conditional on the missing verification. I therefore recommend no change to the reader's verdict.","tokens_in":7229,"tokens_out":18952,"duration_ms":180938,"concrete_test":"Check the Burklund-Xu differentials in Chow degree one for the degrees (2^n−1, 2^{n−1}) with n=3,4,5, including all r>1: compute or look up whether each potential target (2^n−2, 2^{n−1}+r) is nonzero in the relevant E_r-page. If [IKL+25, Section 5.2] already contains a proof that d_r = 0 for r>1 in Chow degree one, cite it and the concern dissolves. Alternatively, independently re-derive the full list of E1-elements with v1-intercept at most −6 in these degrees and check that none can be a target of any d_r; for example, verify whether any d2 from q_3·h0^3 at (7,4) to degree (6,6) exists.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.7 (existence and uniqueness of x_n) rests on Lemma 2.5, which asserts that q_n·h0^{2^{n-1}-1} is a permanent cycle. The proof of Lemma 2.5 considers a hypothetical differential and fixes its target degree as (2^n−2, 2^{n−1}+1), the case of a d1, whose v1-intercept is −4; Lemma 2.4 then classifies all elements with intercept −4. But in the Burklund-Xu spectral sequence, as in any spectral sequence in this grading, a differential d_r with r>1 from the same source lands in (2^n−2, 2^{n−1}+r), with v1-intercept −2−2r, which is at most −6. Such targets are not covered by Lemma 2.4. The text neither proves that d_r = 0 for r>1 in the Chow-degree-one BXSS nor cites [IKL+25, Section 5.2] for that fact. If higher differentials occur, q_n·h0^{...} may fail to survive, and the uniqueness and detection arguments in Proposition 2.7 collapse. This is a distinct and more concrete gap than the general dependence on [IKL+25]: it is a missing step inside the proof, not merely an imported classification. The 'not hit' half of Lemma 2.5 is handled by h0-localization, but the 'no outgoing differentials' half is incomplete as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a family of elements in the tri-graded cohomology of the C-motivic Steenrod algebra, also viewed as the C-motivic Adams E2-page. It proves that for every n ≥ 3 there is a unique non-zero element x_n in degree (2^{n-3}-1)(20,4,12)+(17,4,10), that x_n is h1-periodic and multiplicatively indecomposable, that it maps under h1-localization to v_n (up to h1^{±1}-multiples), and that it maps to the product e0g_{2^{n-3}-1} in the cohomology of C-motivic A(2). This answers Thai's conjecture for k of the form 2^m-1. The main tool is the Burklund-Xu spectral sequence in Chow degree one, with heavy reliance on the companion paper [IKL+25] for the structure of the E1-page, classifications of elements with specified v1-intercepts, and notation. The paper also describes a sparse w1-periodicity relation among the x_n and discusses implications for the algebraic mmf-Hurewicz image.","tokens_in":7582,"tokens_out":8301,"duration_ms":74159,"significance":"If the proof is correct, the result settles a conjecture of Thai in the positive direction and exhibits a genuinely new kind of sparse periodic family in C-motivic stable homotopy, one that has no analogue in classical or C-motivic v1-periodicity. The connection to h1-localization gives explicit preimages of the generators v_n, and the map to H***A(2) gives concrete information about the algebraic mmf-Hurewicz image. The argument is well structured and the stated detection properties are concrete and checkable; the negative result of Thai and classical comparison provide external anchors. However, the proof depends substantially on an unpublished companion paper [IKL+25], and one step inside the permanence argument (Lemma 2.5) omits a necessary discussion of higher differentials. Neither issue appears fatal if addressed, but both must be resolved before the central claim is fully established.","major_comments":[{"comment":"The proof that q_n·h0^{2^{n-1}-1} is a permanent cycle only considers targets of a d1-differential. A differential d_r with r>1 from the same source would land in degree (2^n-2, 2^{n-1}+r), whose v1-intercept is -2-2r ≤ -6. Such targets are not covered by Lemma 2.4, which classifies only elements with v1-intercept -4. The text neither proves that all higher differentials in the Burklund-Xu spectral sequence in Chow degree one vanish nor cites a statement from [IKL+25, Section 5.2] to that effect. Since Lemma 2.5 is the key permanence step in Proposition 2.7, this is a load-bearing gap: without it, the existence and uniqueness of x_n are not established.","section":"Section 2, Lemma 2.5"},{"comment":"The proof of Lemma 2.3 asserts that 'the result follows from explicit low-dimensional computations in degrees up to (11,7)' but those computations are not displayed. This classification of elements with v1-intercept -3 is used in Lemma 2.4, which in turn is used in Lemma 2.5. To make the paper verifiable, the low-dimensional computation should be tabulated, or a precise published reference should be supplied, or the computation should be given in an appendix.","section":"Section 2, Lemma 2.3"},{"comment":"The uniqueness and detection argument depends critically on [IKL+25, Lemma 6.5] for the classification of elements with v1-intercept -1, and on [IKL+25, Section 5.2] for the entire BXSS E1-page description. The companion paper is cited as unpublished work by the same five authors and is not reproduced here. If [IKL+25] is not available to the reader or has not been accepted, the central theorem cannot be verified from the present manuscript alone. The authors should either include the necessary classifications in this paper or ensure that the companion is publicly available and give precise statements of the imported results.","section":"Section 2, Proposition 2.7 and [IKL+25]"}],"minor_comments":[{"comment":"The assertion 'For degree reasons, v_n is the only possible non-zero value' is not demonstrated. The degrees of the generators v_i in the h1-localized cohomology should be listed, or a short degree computation should be included, so that the reader can check that no other v_i has the same degree after multiplication by h1^{±1}.","section":"Section 2, Proposition 2.8"},{"comment":"The claim that e0g_{2^{n+1}-1} = w_1^{2^{n+2}} · e0g_{2^n-1} with a Massey product operator, and that these operators cannot be iterated, is stated without proof or reference. If this is a theorem of the paper, a proof or a precise citation should be given; otherwise it should be labeled as a conjecture or observation.","section":"Introduction, w1-periodicity"},{"comment":"The sentence 'For degree reasons, there are no other possible differentials' is terse. Since this lemma is used to rule out differentials hitting the permanent cycles, the degree count should be spelled out or a reference given for why d_r=0 for r≥2 in the h0-localized Burklund-Xu spectral sequence.","section":"Section 2, Lemma 2.1"},{"comment":"The parenthetical statement that [GI15, Conjecture 5.5] has been proved should include a more precise pointer, such as which combination of results establishes which part of the conjecture, so the reader can verify the stated map H***A[h1^{-1}] -> H***A(2)[h1^{-1}].","section":"Remark 2.9"}],"recommendation":"major_revision","confidential_remarks":"The paper is a nice contribution if the gaps are closed. The most important technical issue is the missing treatment of higher differentials in Lemma 2.5; this is likely fixable by adding a short argument or a precise citation to [IKL+25]. The reliance on the unpublished companion [IKL+25] should be resolved before publication, either by making the companion available or by including the needed classifications. The paper fits the journal's scope and the central conjecture is of genuine interest, but the current manuscript is not yet self-contained enough for acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The new thing here is the existence of e0 g_k in the C-motivic Steenrod algebra cohomology for k = 2^n−1, exactly Thai's conjectured case. The paper also shows the sparse w1-periodicity relation and connects the family to h1-localization and the algebraic mmf-Hurewicz image. That is a solid, useful result for people computing motivic Adams E2-pages, and the Burklund-Xu spectral sequence approach is appropriate.\n\nWhat the paper does well: it sets up the definitions carefully (the two ways to define e0 g_k), and Propositions 2.7, 2.8, 2.10, and 2.11 form a coherent chain. The h0-localization argument in Lemma 2.1 is clean, and the use of the h1-localized BXSS to rule out elements being hit is elegant.\n\nNow the soft spots, in order of seriousness.\n\nFirst, Lemma 2.5's proof as written only considers d1 targets. It computes the target degree (2^n−2, 2^{n−1}+1), notes the v1-intercept is −4, and cites Lemma 2.4. But a d_r for r>1 from the same source lands in (2^n−2, 2^{n−1}+r), with v1-intercept −2−2r, which is at most −6. Those targets are outside Lemma 2.4's classification. The paper neither proves that higher differentials are impossible in Chow-degree-one BXSS nor cites [IKL+25] for that fact. That is a missing step in the proof of permanence, not just an imported classification issue. The 'not hit' half is fine via h0-localization, but the 'no outgoing differentials' half is incomplete.\n\nSecond, the proof leans heavily on [IKL+25], a companion paper by the same five authors that is not yet published, for Lemma 6.4, Lemma 6.5, and the BXSS background. Lemma 2.3 also refers to unshown low-dimensional checks up to degree (11,7). This makes independent verification hard. It may be that [IKL+25] does contain the missing fact about higher differentials; if so, the fix is a one-line citation. But as written, the proof is conditional.\n\nThe central argument holds up if that gap is filled. I see no reason to doubt the result; the structure is coherent and the detection claims are plausible. But a referee will need to see the higher-differential issue addressed.\n\nWho is this for: specialists in motivic homotopy theory and Adams spectral sequence computations. It is not a paper that reshapes the field, but it is a real step forward for a specific subfield.\n\nRecommendation: send it to peer review. It deserves a serious referee. The authors should be asked to fix Lemma 2.5, either by citing the missing fact or proving it, and to make the dependence on the companion paper explicit. With that, it would be a solid contribution.","headline":"The paper proves a genuine conjecture and the detection theorem is likely right, but Lemma 2.5 has a real gap: it only handles d1, not higher differentials, so the proof as written is incomplete.","tokens_in":8100,"tokens_out":4842,"would_cite":true,"duration_ms":41659,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55T15","14F42"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper settles a conjecture on a sparse family in the $\\mathbb{C}$-motivic Steenrod cohomology: existence holds exactly for $k=2^m-1$, with unique indecomposable elements detected by $h_1$-localization.","keywords":["Steenrod algebra","Adams spectral sequence","C-motivic stable homotopy theory","motivic modular forms","h1-localization","Burklund-Xu spectral sequence","w1-periodicity","sparse family"],"falsifier":"Independently compute $H^{***}A$ in the first new degree $(17,4,10)$; if one finds an additional nonzero element or shows that the candidate $x_3$ is annihilated by some power of $h_1$, the uniqueness or $h_1$-periodicity claim fails. Alternatively, check the companion classification for a missing element with $v_1$-intercept $-1$ in the relevant Burklund-Xu degrees.","tokens_in":7068,"feed_emoji":"🧮","tokens_out":14699,"duration_ms":119249,"temperature":0.7,"pith_summary":"This paper proves the existence half of a characterization of a family of elements in the cohomology of the $\\mathbb{C}$-motivic Steenrod algebra, the $E_2$-page of the $\\mathbb{C}$-motivic Adams spectral sequence. In tri-degrees of the form $k(20,4,12)+(17,4,10)$, it shows that a nonzero element exists exactly when $k=2^m-1$: non-existence for other $k$ was previously known, and the new result supplies existence, uniqueness, and indecomposability. The elements are called $e_0g_k$ and are detected by permanent cycles in the Burklund-Xu spectral sequence in Chow degree one. They matter because they give explicit minimal-stem preimages of the polynomial generators $v_1^4$, $v_2$, and $v_n$ under $h_1$-localization, and because they show that the algebraic Hurewicz image of motivic modular forms contains $e_0g_k$ only for these special $k$, while $\\tau e_0g_k$ always survives.","feed_headline":"New C-motivic cohomology family exists exactly for k = 2^m - 1","feed_subtitle":"Unique indecomposable elements complete the h1-localization map and control the modular-forms Hurewicz image.","key_machinery":"The central tool is the Burklund-Xu spectral sequence in Chow degree one, a spectral sequence that computes the $\\mathbb{C}$-motivic Adams $E_2$-page in Chow degree one ($s+f-2w=1$) and whose $E_1$-page is built from elements $q_i$ times classical Steenrod-cohomology classes. Its $h_0$-localization collapses to the tail $q_2,q_3,\\dots$, with the only differential $d_1(q_1)=q_0h_0$; this controls all possible differentials. The argument classifies $E_1$-elements with $v_1$-intercept $-4$, shows that $q_n\\cdot h_0^{2^{n-1}-1}$ has no possible target and cannot be hit, and combines an imported classification of $v_1$-intercept $-1$ elements with the $h_0$-localized picture to prove uniqueness. Adams' classical vanishing-line theorem supplies the slope bound that makes the $v_1$-intercept classifications finite and explicit.","core_discovery":"For each $n\\ge 3$, the paper establishes the existence of a unique non-zero element $x_n$ in the tri-graded cohomology $H^{***}A$ of the $\\mathbb{C}$-motivic Steenrod algebra in degree $(2^{n-3}-1)(20,4,12)+(17,4,10)$. The proof shows that $x_n$ is detected by the permanent cycle $q_n\\cdot h_0^{2^{n-1}-1}$ in the Burklund-Xu spectral sequence, while its only competitor in the same degree is killed by a $d_1$ differential. The element is indecomposable, is not annihilated by any power of $h_1$, maps to $e_0g_{2^{n-3}-1}$ in the cohomology of $\\mathbb{C}$-motivic $A(2)$, and maps to $v_n$ up to $h_1$-units in $h_1$-localization. With the new name $e_0g_{2^{n-3}-1}$, these elements satisfy $e_0g_{2^{n+1}-1}=w_1^{2^{n+2}}\\cdot e_0g_{2^n-1}$ for a Massey-product operator $w_1^{2^{n+2}}\\cdot(-)$, but the operator cannot be iterated, so the family is sparse with geometrically growing degrees. The non-existence of such elements for $k$ not of the form $2^m-1$ had been shown previously, so the full statement is an if-and-only-if characterization.","pith_inferences":["A testable extension is to run the same permanent-cycle argument in higher Chow degrees; the weight shift $12k+10$ versus $12k+9$ suggests a general rule for which lifts exist, independent of the Burklund-Xu machinery.","Because the $w_1$-Massey operator cannot be iterated, the sparse family points to a broader phenomenon: periodicity operators that generate families only one step at a time, and one could look for analogues among $v_1$-periodic families in the cohomology of subalgebras $A(n)$.","The paper leaves open whether the $\\tau$-localized image of $x_n$ is exactly $e_0g_{2^{n-3}-1}$ in the classical Steenrod cohomology; the authors guess the unknown correction $y$ vanishes, which can be tested by direct computation in the first few degrees.","If minimal-stem preimages in $h_1$-localization are unique more generally, examples like the 45-stem class $M h_1$ could be organized into a sparse family rather than isolated computations."],"forward_implications":["The $h_1$-localization map is now understood on polynomial generators: $P h_1$, $c_0$, and $e_0g_{2^{n-3}-1}$ are explicit minimal-stem preimages of $v_1^4$, $v_2$, and $v_n$ up to $h_1$-units, so products of these give preimages of every element in the localization.","The algebraic mmf-Hurewicz image contains $e_0g_k$ if and only if $k=2^m-1$, while $\\tau e_0g_k$ is present for every $k$, so the image has a more intricate structure than a simple polynomial algebra.","The elements form a sparse $w_1$-periodic family: $e_0g_{2^{n+1}-1}=w_1^{2^{n+2}}\\cdot e_0g_{2^n-1}$ via a Massey product that cannot be iterated, so degree growth is geometric rather than additive.","Each $x_n$ is indecomposable and $h_1$-periodic: no power of $h_1$ annihilates it, and it is not a product of smaller elements."],"supporting_citations":[{"why":"Supplies the Burklund-Xu E1-page in Chow degree one and the classifications of elements with v1-intercepts -1 and -3 that are used to isolate the permanent cycle.","marker":"[IKL+25]"},{"why":"Establishes the differential d1(q1)=q0h0, which determines the h0-localized E-infinity page and kills the competing candidate.","marker":"[BX24]"},{"why":"Provides the periodicity theorem and the slope-1/2 vanishing line that bound the classical elements appearing in Lemmas 2.3 and 2.4.","marker":"[Ada66]"},{"why":"Gives the h1-localized cohomology H***A[h1^{-1}] = F2[h1^{±1}][v1^4, v2, ...] and the localization values on A(2) used in the detection argument.","marker":"[GI15]"},{"why":"States the original conjecture and proved non-existence of e0g_k for k not of the form 2^m-1, the other half of the characterization.","marker":"[Tha21]"},{"why":"Classifies the cohomology of C-motivic A(2) in the relevant tri-degree, allowing Theorem 2.10 to identify the image uniquely as e0g_{2^{n-3}-1}.","marker":"[Isa09]"},{"why":"Defines the C-motivic modular forms spectrum mmf and the algebraic Hurewicz image, the context for the image statements.","marker":"[GIKR22]"}],"fun_headline_variants":["C-motivic cohomology: new sparse family iff k=2^m-1","Unique C-motivic cohomology elements only when k=2^m-1","Sparse family in C-motivic Steenrod cohomology when k=2^m-1","If and only if: C-motivic cohomology family at k=2^m-1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on an unpublished companion paper's complete classification of certain boundary elements (by their v1-intercept) and on a small explicit low-degree computation that is not shown; if either contains an error, the uniqueness of the new element in each degree collapses.","fun_headline_variants_meta":{"raw":{"variants":["C-motivic cohomology: new sparse family iff k=2^m-1","Unique C-motivic cohomology elements only when k=2^m-1","Sparse family in C-motivic Steenrod cohomology when k=2^m-1","If and only if: C-motivic cohomology family at k=2^m-1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000758,"raw_usage":{"total_tokens":3382,"prompt_tokens":972,"completion_tokens":2410,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":2307}},"tokens_in":588,"tokens_out":2410,"duration_ms":13436,"temperature":1.0,"reasoning_tokens":2307,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:08:50.807449+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently compute $H^{***}A$ in the first new degree $(17,4,10)$; if one finds an additional nonzero element or shows that the candidate $x_3$ is annihilated by some power of $h_1$, the uniqueness or $h_1$-periodicity claim fails. Alternatively, check the companion classification for a missing element with $v_1$-intercept $-1$ in the relevant Burklund-Xu degrees.","supporting_citations":[],"review_version":1}