{"id":"63ee53c6-f4e3-445f-9a7f-61270aefe874","arxiv_id":"2607.18670","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New explicit formulas for the additive C_2 power operations on the indecomposables of KU_0(BU), E_2[BU], and E_2[BSU] modulo decomposables, derived from the level-structure interpretation of power operations.","lead":"This paper develops an algebro-geometric recipe for computing cyclic power operations on the homology of BU and BU×Z, then works out explicit p=2 formulas for K-theory and height-2 Morava E-theory. It makes homology-side power operations more accessible and offers evidence about finite generation of Δ-modules in chromatic homotopy theory.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.2 rests on the unproved Schumann identity Fact 5.2; if its conventions or proof are wrong, the central formulas for E_2[BU]/[BSU] and the Δ-module conclusion do not follow.","rationale":"The reader's weakest-assumption analysis correctly identifies Fact 5.2 as the load-bearing point. The paper is otherwise careful and self-consistent: the KU computation has independent support, the internal checks of the mod(2,a) table are consistent with formulas (1)-(3), and the heuristic MP-level-structures remark is not needed for the finite-height theorems. The sole serious risk is that the entire height-2 calculation is anchored to a computational identity quoted from a thesis, with the author explicitly disclaiming a theoretical proof. Because the identity is used algebraically at the exact step where the Waring expansion is applied, any error in that identity would propagate through all subsequent formulas and the BSU/Δ-module consequences. A direct symbolic verification of Fact 5.2 in the stated curve and coordinate conventions would settle the matter. This does not change the reader's CONDITIONAL verdict: the paper should be accepted only after such verification, and our proposed check is precisely the condition that needs to be met.","tokens_in":17951,"tokens_out":17658,"duration_ms":147834,"concrete_test":"Use a computer algebra system to verify Fact 5.2 directly in Rezk's model. Work in the ring R = Z2[[a]][d]/(d^3-ad-2)[[u,v]]/(v^2+auv+v-u^3). Let P=(u,v) and Q=(d,-d^3). Compute the u-coordinate of P+Q using the elliptic curve addition law for the curve v^2+auv+v=u^3 (with u=X/Y, v=Z/Y), reducing modulo the relation d^3-ad-2, and check that the result equals (d-u)/(1+d^2u) as a formal power series in u. If the equality holds, the derivation of Theorem 5.2 is supported; if it fails or requires extra assumptions, the central formulas do not follow from the given argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central load-bearing step is the use of Schumann's Fact 5.2: for the universal 2-torsion point Q with coordinate d and a point P with coordinate u on Rezk's supersingular curve, the coordinate of P+Q is w=(d-u)/(1+d^2u). This identity is inserted immediately before the Waring-formula expansion: it gives u+w=d+d^2u', and without that exact form the expansions of u^i+w^i do not reduce to powers of u'. The paper does not prove Fact 5.2 and states, 'We are not currently aware of a more theoretical way to prove this surprising identity.' It is quoted from Schumann's thesis as a computational result, but the computation is not reproduced and the precise curve/coordinate conventions are not pinned down. If the identity has a sign error, a rescaling mismatch, or a hidden hypothesis about the formal group or the deformation parameter a, then all three formulas (1)-(3) and Corollary 5.1 change, and the asserted F2⊗Δ M structure and the speculation about finite generation collapse. The height-1 case is independently checked against Reeker, but the height-2 case has no such external check. Thus the central theorem is only as secure as this unverified computational identity.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an algebro-geometric framework, based on level structures and the work of Ando–Hopkins–Strickland and Rezk, for computing cyclic power operations on the homology of BU, BU×Z, and related E∞ spaces. The main application is explicit computation, at p=2, of the additive C2 power operation on the indecomposables of KU0(BU) and (E2)0(BU), and subsequent formulas for (E2)0(BSU). The central height-2 result, Theorem 5.2, expresses θ(b_n), Q1(b_n), Q2(b_n) as explicit finite sums, from which a computation of F2⊗Δ M for M = Q((E2)0(BSU)) is derived.","tokens_in":18214,"tokens_out":52502,"duration_ms":423917,"significance":"If correct, this paper would provide one of the few explicit computations of power operations on homology in the Morava E-theory setting, complementing Rezk's and Zhu's cohomological calculations. The derivation is genuinely synthetic: it uses known external inputs (Quillen's formula, Rezk's model, the elliptic curve group law) rather than fitted parameters, and the height-1 formulas are independently checked against Reeker's results. The algebro-geometric interpretation of operations on homology is a useful conceptual contribution. However, the central height-2 computation rests on an unproved and, as stated, apparently inconsistent identity from Schumann's thesis, so the significance of the paper as a whole is currently conditional.","major_comments":[{"comment":"The proof of Theorem 5.2 hinges on the identity w=u(P+Q)=(d-u)/(1+d^2u). This is load-bearing: it is used to derive u+w=d+d^2u', which is essential for the Waring-formula expansion. As stated, however, the identity is inconsistent with the coordinate conventions. In any formal group with coordinate u, the series u(P+Q)=F(u,d) has linear coefficient 1 in u. The displayed rational function has linear coefficient -(1+d^3)=-(ad+3), which is not 1 in Z2[[a]][d]/(d^3-ad-2). For example, at a=0 the coefficient is 3, not 1. Thus the quoted formula cannot be the coordinate of P+Q in the coordinate u=X/Y described in the paper. This indicates a sign or coordinate mismatch in the citation to Schumann. The paper states that no more theoretical proof is known, but the computational identity is not reproduced and no alternative verification is supplied. The author must either prove the identity in the","section":"§5.2, Fact 5.2"},{"comment":"Even assuming the identity, the passage from u+w=d+d^2u' to the displayed formulas is rapid. The treatment of the boundary case n+i-3j≤0 is correct in substance, but the verification that the only nonzero contribution is the j=n term of [b_{2n}]P(b_n) is compressed. Please expand this step so that a reader can check the indexing without reconstructing the argument. This is secondary to the Fact 5.2 issue, but it would improve confidence in the final formulas.","section":"§5.2, proof of Theorem 5.2 after Fact 5.2"},{"comment":"The claim that F2 ⊗Δ M ≅ F2{d2}⊕F2{d3} is presented with only a sketch: 'it is not too hard to see' and 'from these observations and the commutation relations of [Rez08], it follows.' This is a concrete stated result, not merely a suggestion. Please provide a more detailed derivation, including the relevant commutation relations from [Rez08] and the verification that d2 and d3 do not vanish in the tensor product. The current level of detail is insufficient for a published claim.","section":"Remark 5.1"}],"minor_comments":[{"comment":"The notation '2t' in Corollary 5.1 is evidently meant to be '2^t' (powers of two). Several such superscripts appear to be lost in the typesetting, making the formulas hard to read.","section":"Throughout"},{"comment":"The table displaying the mod (2,a) reductions is hard to parse; the alignment of the columns and the entries is ambiguous. Please ensure the superscripts and row breaks are clear.","section":"Introduction, Table 1"},{"comment":"The phrase 'level structures can be understood through algebraic geometry' is intuitive, but the paper would benefit from a precise statement of which of the two quotient rings (by [p](z) vs ⟨p⟩(z)) is used for the target of the additive operation. The distinction is mentioned but could be spelled out.","section":"§2.2"},{"comment":"The diagrams in the proof of Theorem 3.1 are informative but somewhat informal. For instance, the pullback diagram in rings would be clearer with the maps explicitly labeled by the universal property being used.","section":"§3"}],"recommendation":"major_revision","confidential_remarks":"The paper's framework and height-1 results are solid, and the height-2 formulas are presented with full explicitness. However, the central identity Fact 5.2 appears to be internally inconsistent with the stated coordinate system, and it is not proved in the manuscript. This is a load-bearing issue: if the identity is wrong or misquoted, Theorem 5.2 and the subsequent BSU conclusions collapse. The author should be asked to provide a corrected and fully verified version of this identity, or to supply a proof from the elliptic curve group law in the exact conventions used. If such a proof cannot be supplied, the height-2 section should be removed or substantially revised. I recommend major revision rather than immediate rejection because the error may be a fixable sign/coordinate issue, and the rest of the paper has genuine value."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Candid take: the paper gives genuinely new explicit formulas for the additive C2 power operation on E_2[BU] and E_2[BSU] indecomposables at p=2, and the height-1 formula for KU[BU] is integral and consistent with Reeker's mod-2,4,16 computations. The derivations are mostly clean: the reduction mod decomposables is careful, the Waring steps are valid with stated conventions, and the D_m recurrence for d^3=ad+2 checks out. Section 3's level-structure perspective is mostly recollection of AHS04 but usefully packaged for homology of BU and BU×Z. The real soft spot is exactly the one flagged in the stress test: the proof of Theorem 5.2 depends on Schumann's Fact 5.2, the coordinate formula w=(d-u)/(1+d^2 u) for P+Q on Rezk's curve, quoted from a thesis with no proof and no independent verification. That identity is used to get u+w=d+d^2 u', which is necessary for the Waring expansion to produce the stated formulas (1)-(3). If it has a sign error or hidden hypotheses, the central height-2 formulas and the BSU corollary do not follow. The paper is honest about this, but the honesty doesn't replace a proof. The height-1 case has an external check; the height-2 case does not. Minor: Remark 2.1 admits the M P framework is heuristic, but it's not essential. The finite-generation/K(2)-local finite-cell speculation is clearly labeled as future work. Recommended for peer review. A referee should be explicitly asked to verify or prove Schumann's identity. If it holds, this is a solid computational contribution. If not, the main new results are unsupported. I would not desk-reject.","headline":"Useful new computations in power operations, but the height-2 formulas rest on one unproved identity from Schumann's thesis; refereeing should require verification.","tokens_in":783,"tokens_out":1498,"would_cite":true,"duration_ms":31291,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14L05","55N20","55N22","55P43","55S12"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives explicit finite-sum formulas for the additive C2 power operation on the E-homology of BU and BSU for complex K-theory and height-2 Morava E-theory, and shows that at height 2 the indecomposables of E2[BSU] form a module t","keywords":["power operations","Morava E-theory","level structures","formal groups","BU homology","BSU","chromatic homotopy theory","cyclic operations"],"falsifier":"Evaluate the identity w=(d-u)/(1+d^2 u) numerically or symbolically for a few points P and Q on the universal deformation of the supersingular curve, using the elliptic curve group law, and check that the result satisfies the curve's equation and the 2-torsion condition; a single counterexample would invalidate the central formulas and the finite-generation conclusion.","tokens_in":17786,"feed_emoji":"🔄","tokens_out":5138,"duration_ms":127909,"temperature":0.7,"pith_summary":"This paper works out explicit formulas for the additive C2 power operation on the homology of the infinite loop spaces BU, BU×Z, and BSU, for two complex-oriented theories: complex K-theory (height 1) and a height-2 Morava E-theory at p=2. The driving idea is to read power operations through level structures on formal groups: the power operation pulls back a universal map to the multiplicative group, and the answer becomes a product over the level structure. For K-theory the paper recovers the known operation θ, and for height 2 it derives closed formulas for the three component operations Q0, Q1, Q2 on the indecomposables of E2[BU]. Applying these to BSU, the paper finds that after killing the ideal (θ,Q1,Q2,2,a) the module of indecomposables is F2{d2}⊕F2{d3}, a structural contrast with the height-1 case and evidence that the module is finitely generated.","feed_headline":"Formulas compute C2 power operations on BU homology at heights 1 and 2","feed_subtitle":"Explicit sums for θ, Q1, Q2 on E2[BU] and E2[BSU] suggest a finite module structure, unlike K-theory.","key_machinery":"The key objects are level C2 structures on formal groups: a chosen point of order 2 whose coordinate z satisfies the 2-series relation. On an E∞ ring E, the additive C2 power operation factors through E0(BC2)/I_tr, which is isomorphic to the scheme of level-2 structures on the formal group of E. The computational engine is Theorem 3.3: the power operation on the universal pointed map h(x)=1+b1x+b2x2+… to G_m is the product ∏_{a∈C2} h(x+_G x(ℓ(a))) / h(x(ℓ(a))). Expanding this identity and matching coefficients, using a standard combinatorial identity for power sums and, at height 2, a computationally verified identity for the coordinate of the sum of a point with the universal 2-torsion poin","core_discovery":"The central claim is that the additive C2 power operation on E2[BU] decomposes uniquely as P(x)=Q0(x)+Q1(x)d+Q2(x)d2, and the paper proves explicit finite-sum formulas for θ(bn), Q1(bn), and Q2(bn) modulo decomposables, in terms of the generators bi and the parameters a and 2 (Theorem 5.2, equations (1)–(3)). The same computation yields formulas for E2[BSU] (Corollary 5.1). A striking consequence is that for the module of indecomposables M of E2[BSU], the quotient F2 ⊗Δ M is F2{d2}⊕F2{d3}; the paper interprets this as evidence that M is finitely generated as a Δ-module, and speculates that E2[BSU] is K(2)-locally finite-celled as an E∞ E2-algebra, in contrast to the height-1 case.","pith_inferences":["The same level-structure method should extend to other heights and primes (e.g., height 2 at p=3 using different elliptic curve models) to produce analogous explicit formulas, potentially revealing a pattern in how the operations depend on the formal group law.","The apparent mod-2 splitting of M (where Q1(dn)=Q2(dn)=0 for n a power of 2) may indicate an internal periodicity in the Δ-module structure, possibly reflecting a cellular filtration on BSU.","If finite generation of M can be lifted to an explicit resolution, it might yield a new family of K(2)-local finite-celled E∞ algebras, with consequences for the structure of K(2)-local homotopy theory.","The formulas depend polynomially on the parameter a, so specializing a to specific values (e.g., a=0) could provide a quick check of the computations and illuminate the transition between different formal group laws."],"forward_implications":["For complex K-theory, the formula θ(bn)=b2n+b2n+1 mod 2 (and the full integral formula) determines the additive operation on indecomposables, agreeing with prior results in the literature.","At height 2, the explicit formulas for θ, Q1, and Q2 allow one to compute the Δ-module structure on the indecomposables of E2[BU] and E2[BSU] at p=2, giving a concrete starting point for studying the module of operations.","The collapse F2 ⊗Δ M ≅ F2{d2}⊕F2{d3} suggests M is finitely generated as a Δ-module; if true, E2[BSU] would be K(2)-locally finite-celled, contrasting with the non-finite behavior of KU[BSU].","The method shows how level structures on formal groups can be used to compute power operations on homology of E∞ spaces without rederiving Quillen's universal calculation from scratch."],"fun_headline_variants":["Explicit C2 power operations on BU homology at p=2","Finite sums for Δ on BU indecomposables at height 2","Level structures yield Δ formulas on BU E-homology","Height-2 power operations: explicit sums on BU","C2 actions on BU: explicit Δ, finite module evidence"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The height-2 computation rests on a single computationally verified identity for the coordinate of the sum of a point with the universal order-2 point on the supersingular curve; if that identity is wrong or fails under hidden hypotheses about the formal group, the formulas (1)–(3) and their consequences do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Explicit C2 power operations on BU homology at p=2","Finite sums for Δ on BU indecomposables at height 2","Level structures yield Δ formulas on BU E-homology","Height-2 power operations: explicit sums on BU","C2 actions on BU: explicit Δ, finite module evidence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000336,"raw_usage":{"total_tokens":1697,"prompt_tokens":740,"completion_tokens":957,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":484,"completion_tokens_details":{"reasoning_tokens":880}},"tokens_in":484,"tokens_out":957,"duration_ms":11306,"temperature":1.0,"reasoning_tokens":880,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T14:40:50.458132+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the identity w=(d-u)/(1+d^2 u) numerically or symbolically for a few points P and Q on the universal deformation of the supersingular curve, using the elliptic curve group law, and check that the result satisfies the curve's equation and the 2-torsion condition; a single counterexample would invalidate the central formulas and the finite-generation conclusion.","supporting_citations":[],"review_version":1}