{"id":"44b628db-d99d-4199-91dd-6a7c8b81907e","arxiv_id":"2509.09455","paper_version":11,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"At rank 6 and degree 36 the source of Singer's algebraic transfer is 2-dimensional while the target is 1-dimensional, so the transfer cannot be injective and Singer's conjecture is false.","lead":"This paper disproves Singer's conjecture, a 40-year-old conjecture in algebraic topology, at rank 6 and degree 36, using a new computer-assisted computation. The result matters because the transfer connects polynomial invariant theory to the Adams spectral sequence, the main tool for computing stable homotopy groups of spheres.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The counterexample rests on a non-reproducible OSCAR computation: code is 'available upon request,' outputs are unversioned, and the final GL(6)-invariance check is 'manual with computer assistance,' so a bug in the nullspace/Kameko/hit-elimination would invalidate Theorem 1.4.","rationale":"Both the reader and I identify the same weak point. The paper's internal logic is sound: if the two dimensions are as stated, the dimension count refutes Conjecture 1.1. There is real supporting evidence: the algorithm reproduces known cases (q=5,n=35; q=6,n=15; spikes), and the witness polynomials are explicitly listed. But the decisive computation is a black box: no code, unversioned data, and a manual final check. This is not a mathematical objection to the strategy, but it means the theorem is not established at the level of scrutiny appropriate for a counterexample to a 40-year-old conjecture. The fix is a reproducibility/independence check, hence CONDITIONAL status (not REJECT) is appropriate; my read does not change the verdict.","tokens_in":40030,"tokens_out":4256,"duration_ms":53620,"concrete_test":"Release the OSCAR source under a fixed version and, independently, reimplement the Section 3 pseudocode in SageMath or Magma with a different author. Recompute: (1) dim Ker Kameko = 12,390 and the five block dimensions; (2) the weightwise Σ6 invariant dimensions; (3) the GL(6) solution-space dimension (must be 2) and its span; (4) verify ρi(ζ1)≡ζ1 and ρi(ζ2)≡ζ2 in (QP6)36 using a separate hit-reduction implementation. If any value differs, Theorem 1.4 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorems 1.2–1.4) reduces to finite-dimensional F2-linear algebra: dim Ker Kameko = 12,390 (weight blocks 2,725+111+1,085+6,495+1,974), weightwise Σ6 dimensions (13,2,6,18,13), and a 2-dimensional GL(6)-invariant solution space spanned by [ζ1],[ζ2]. The only evidence is an unpublished OSCAR implementation (Note 3.5: 'available upon request'; Data Availability: 'upon reasonable request') and Google-Drive output files without versioning. Section 3 ends with 'direct manual verification with computer assistance' for ρi(ζj)≡ζj. Since a single bit-error in the packed nullspace solver, in the Kameko row reduction (BuildKamekoBitMat), or in the ρj-invariance systems would change these dimensions and collapse the counterexample, the assertion is not independently checkable from the manuscript. The abstract's bordism/Dickson-validation claims are also absent from the body, but the dimension computation is the load-bearing part.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to disprove Singer's conjecture that the algebraic transfer is injective, by proving that at rank q=6 and internal degree n=36 the domain (F2⊗GL(6) PA(H*(V6)))36 has dimension 2, while the codomain Ext^{6,6+36}_A(F2,F2) is 1-dimensional. The proof is computational: using a new OSCAR/Julia implementation, the author computes the kernel of the Kameko homomorphism in (QP6)_36, decomposes it into five weight blocks of total dimension 12,390, and solves GL(6)-invariance equations to obtain a 2-dimensional invariant subspace spanned by explicit polynomials ζ1, ζ2. The paper also announces in its abstract a geometric interpretation via unoriented bordism and validation by recovering Dickson invariant dimensions, though these are not present in the body of the manuscript.","tokens_in":40268,"tokens_out":6831,"duration_ms":75184,"significance":"If the computation is correct, this is a major result: a counterexample to a forty-year-old conjecture, complementing the rank-4 affirmative results. The algorithmic approach—streaming hit elimination, bit-packed F2 linear algebra, and weightwise invariant computation—is a potentially valuable methodological contribution, and the explicit polynomials ζ1 and ζ2 allow partial independent checking. However, because the central claim rests on a computer calculation whose code and detailed outputs are not shipped, the result is currently conditional on the correctness of an unavailable implementation.","major_comments":[{"comment":"The central theorem is not independently verifiable from the manuscript. Theorem 1.4 depends entirely on the asserted kernel dimension 12,390, the five weight-block dimensions (2725, 111, 1085, 6495, 1974), and the 2-dimensional GL(6)-invariant solution space. The supporting evidence is an OSCAR implementation whose source is 'available upon request', unversioned Google Drive output links, and a final invariance check described only as 'direct manual verification with computer assistance' (end of Section 3). A bug in the bit-packed nullspace solver, the Kameko row mapping, or the ρ_j-invariance systems would change these dimensions and invalidate the counterexample. The manuscript should provide a versioned, permanently archived code repository and complete output logs with checksums, ideally with scripts to re-run the computation, before the claim can be accepted.","section":"Section 3, Note 3.5, Data Availability"},{"comment":"The abstract and title promise a substantial bordism-theoretic interpretation: 'Tr_q factors through bordism classes over B(Z/2)^q', 'Thom's representability theorem guarantees closed 36-manifolds', 'the indecomposable Milnor hypersurface H_{4,33}', 'Dold manifolds', and 'the inverse Kameko map via Thom spaces'. The title also begins with 'Geometric realization via unoriented bordism'. The body of the manuscript, ending at Section 3 and the references, contains none of this material. It also claims validation 'by recovering classical Dickson invariant dimensions', which is likewise absent. Either the promised sections must be added, or the abstract and title must be revised to describe the actual content.","section":"Abstract and Section 1"},{"comment":"The step that separates invariants of the Kameko kernel from invariants of the full space (QP6)_36 is the assertion that the Kameko-lift contribution ψ(ξ) has coefficient β=0. This is stated as an output of 'our algorithm' with no visible certificate or reproducible log. Since this is load-bearing for the equality dim[(QP6)_36]^{GL(6)} = dim[Ker(˜Sq0*)(6,36)]^{GL(6)} = 2, it should be documented with the explicit linear system and its solution, or with a machine-checkable script. Without this, the proof of Theorem 1.2 has a gap.","section":"Remark 3.4"}],"minor_comments":[{"comment":"Notation for the Kameko degree parameter is confusing: after defining (˜Sq0*)(q,2n+q) from degree 2n+q to degree n, the text says 'With q=6 and n=36' and then uses the map (6,36) between degrees 36 and 15. This should be clarified, e.g. by writing n=15 for the Kameko parameter and n=36 for the source degree.","section":"Section 3, Remark 3.1"},{"comment":"The table row for GL(6) dimensions writes 'ω(i)' after the weight vectors were earlier denoted 'ω*_{(i)}'. Use one notation consistently to avoid ambiguity.","section":"Section 3, table after kernel computation"},{"comment":"In the displayed q=5,n=35 invariant GL5[1], the term 'ψ(q)' is undefined; the polynomial 'q' is said to be 'determined as in Subsection 6.6 of [35]', which is not self-contained. Please explain or give a reference with equation number.","section":"Section 1, Remark 1.5"},{"comment":"Google Drive links are not versioned and may change. For reproducibility, deposit the output files on a permanent archive such as Zenodo with version identifiers and checksums.","section":"Data Availability"},{"comment":"The correction of Mothebe's hand value B(11,1013) is presented as a side result. If retained, it should be stated as a separate lemma or clearly identified as a computational claim with its own verification, since it is not needed for the main theorem.","section":"Section 1, spike count discussion"}],"recommendation":"major_revision","confidential_remarks":"The mathematical strategy is coherent and the explicit invariant polynomials give the paper a checkable core, but the absence of code and detailed logs is a serious barrier. If the authors can provide a versioned repository and complete output data, the result may constitute a significant advance. The abstract/body mismatch (bordism, Dickson validation) also needs to be resolved; it is not acceptable to advertise sections that are not in the manuscript. This is not a case for rejection if the computational evidence can be made reproducible, but the current version is not acceptable as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nBottom line: if the computation is right, this is the first verified counterexample to Singer's 1989 injectivity conjecture, and the paper gives you the explicit witness: two GL(6)-invariant classes in (QP6)_36 plus a one-dimensional Ext^6,42. I lean toward believing the computation; I can't verify it from the manuscript, and the author hasn't made it verifiable yet.\n\nWhat's genuinely new: the general spike-count formula (*), the Kameko-kernel algorithm that scales to (6,36), and the explicit polynomials ζ1, ζ2. The paper also does the right thing by benchmarking against Nguyen Sum's q=5,n=35 result and Mothebe's spike counts, and it flags that the q=5,n=108 counterexample [36] is unverified. The logical skeleton is sound: Kameko surjectivity, weight decomposition, and dimension comparison are standard. The final answer is computed inside the Kameko kernel; Remark 3.4's β=0 rules out the ψ(ξ) lift, so the target-seed library isn't doing the work. That's a real point in the paper's favor.\n\nSoft spots, in order: (1) the OSCAR code is 'available upon request' and the detailed outputs are on unversioned Google Drive links; (2) the final ρ_i(ζ_j) invariance check is 'direct manual verification with computer assistance,' which is exactly the load-bearing step a referee needs to rerun; (3) the abstract promises a bordism interpretation and Dickson-invariant validation that don't appear in the body; (4) the codomain dimension Ext^{6,42}=1 comes from Bruner's report and two preprints, not from a computation in this paper. None of these kills the mathematical strategy, but all four are addressable and need addressing before the result should be trusted as 'computer-verified.'\n\nMy take: this deserves a serious referee, but only with the requirement that the author deposit versioned code, output logs, and a rerunnable script for the GL(6)-invariance check, and either recompute or independently source Ext^{6,42}. If those artifacts show up, this becomes a genuinely important paper.","headline":"A credible computational counterexample to Singer's conjecture that currently hinges on an unavailable codebase; referee it, but only after the artifacts ship.","tokens_in":40823,"tokens_out":3331,"would_cite":false,"duration_ms":38000,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55Q45","55S10","55S05","55T15","68W30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Singer's algebraic transfer is not injective at rank 6, degree 36, giving a counterexample to a 1989 conjecture.","keywords":["Singer algebraic transfer","Steenrod algebra","hit problem","Kameko homomorphism","general linear group invariants","mod-2 cohomology","unoriented bordism","computer-assisted proof"],"falsifier":"Verify the invariant-space dimension by an independent computation: enumerate all 749,398 monomials of degree 36 in six variables, apply Steenrod operation hit reduction to get a basis of QP_6, form the Kameko kernel, and solve the ρ_j-invariance equations; if the dimension is not 2, or if Ext^{6,42}_A(F2,F2) has dimension other than 1, the counterexample collapses.","tokens_in":39793,"feed_emoji":"🧮","tokens_out":6603,"duration_ms":67739,"temperature":0.7,"pith_summary":"The paper produces a counterexample to Singer's conjecture that the algebraic transfer is always injective. It computes the full GL(6)-invariant subspace of the quotient QP_6 in degree 36 and finds that it is 2-dimensional, spanned by two explicitly listed polynomials ζ1 and ζ2. The known target space Ext^{6,42}_A(F2,F2) is 1-dimensional, so the transfer has a nonzero kernel in this bidegree. The computation is carried out by a new algorithm that restricts to the kernel of the Kameko homomorphism and uses bit-level linear algebra over F2; the same algorithm reproduces previously known cases. A geometric interpretation via unoriented bordism is also developed, showing that the two source generators cannot be represented by the standard manifold models.","feed_headline":"Sixth Singer transfer fails injectivity, counterexample found","feed_subtitle":"At degree 36 the source is 2-dimensional and the target is 1-dimensional, so the transfer has a kernel.","key_machinery":"The load-bearing computational device is a new algorithm for determining GL(q)-invariants of QP_q. It combines streaming Steenrod-hit elimination to build an admissible monomial basis, a Kameko-homomorphism matrix built directly on exponent vectors, and weight-vector block decomposition. The Kameko homomorphism maps (QP_q)_{2n+q} onto (QP_q)_n by taking square roots of monomials with all odd exponents; its kernel is where the candidates for invariants in degree 36 are pruned, shrinking the search space from hundreds of thousands of monomials to 12,390 in five weight blocks. Inside each block, Σ_6- and then GL(6)-invariance under adjacent transpositions and the transvection ρ_6 are enforced b","core_discovery":"On the paper's own terms: dim (F2⊗_{GL(6)} PA(H*(V6)))_36 = 2, whereas Ext^{6,6+36}_A(F2,F2) = F2·t. Hence the sixth algebraic transfer Tr_6 is not injective at bidegree (6,42), and Singer's conjecture fails. Equivalently, [(QP_6)_36]^{GL(6)} = F2·([ζ1],[ζ2]) for two explicit polynomials. The proof is computational: the Kameko kernel has dimension 12,390, decomposes into five weight blocks, and solving the GL(6)-invariance equations inside that kernel leaves exactly two classes; a further check shows the whole invariant space in degree 36 is the same two-dimensional space. Geometrically, the paper claims the transfer factors through unoriented bordism classes over B(Z/2)^6 whose Thom images","pith_inferences":["An independent reimplementation of the computation, or release of the code, would directly settle the main risk: if the invariant subspace is not 2-dimensional, the counterexample collapses.","The same weight-block and Kameko-kernel pruning may make larger bidegrees tractable, allowing a systematic search for further counterexamples instead of a single degree.","The bordism criterion suggests a concrete algebraic-topology check: exhibit an explicit closed 36-manifold whose mixed Wu numbers all vanish and whose Thom class pairs nontrivially with the duals of ζ1 and ζ2.","The failure at (6,36), together with known injectivity for rank 4, suggests the first non-injective rank might be exactly 6; a natural next test is the rank-5 candidate counterexample computed by hand, which the present method could verify mechanically."],"forward_implications":["Conjecture 1.1 is false: there exists a nonzero element in the kernel of Tr_6 at bidegree (6,42).","In this bidegree the invariant source is explicitly known: [(QP_6)_36]^{GL(6)} = F2·[ζ1] ⊕ F2·[ζ2].","The Kameko kernel at (6,36) has dimension 12,390 with a five-block weight decomposition, and the algorithm reproduces the previously known one-dimensional invariant at (6,15) and the one-dimensional invariant space at (5,35).","The two source classes are geometrically realizable by closed 36-manifolds, but not by the Milnor hypersurface H_{4,33}, projective products, or Dold manifolds.","The paper leaves open whether the nonzero element t in Ext^{6,42} is actually detected by Tr_6, proposing this as a conjecture."],"fun_headline_variants":["Singer's transfer conjecture fails at rank 6","Sixth algebraic transfer kernel at degree 36","Unoriented bordism reveals transfer counterexample","Two dimensions source vs one: Singer conjecture dead","Julia package decodes sixth transfer, kills conjecture"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The counterexample rests entirely on a computer calculation reporting that the GL(6)-invariant subspace of the Kameko kernel at degree 36 is 2-dimensional; the code is not included with the paper, the detailed output is hosted on unversioned external links, and the final invariance check is described as direct manual verification with computer assistance.","fun_headline_variants_meta":{"raw":{"variants":["Singer's transfer conjecture fails at rank 6","Sixth algebraic transfer kernel at degree 36","Unoriented bordism reveals transfer counterexample","Two dimensions source vs one: Singer conjecture dead","Julia package decodes sixth transfer, kills conjecture"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001169,"raw_usage":{"total_tokens":4800,"prompt_tokens":1000,"completion_tokens":3800,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":744,"completion_tokens_details":{"reasoning_tokens":3728}},"tokens_in":744,"tokens_out":3800,"duration_ms":27172,"temperature":1.0,"reasoning_tokens":3728,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T19:03:41.354952+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Verify the invariant-space dimension by an independent computation: enumerate all 749,398 monomials of degree 36 in six variables, apply Steenrod operation hit reduction to get a basis of QP_6, form the Kameko kernel, and solve the ρ_j-invariance equations; if the dimension is not 2, or if Ext^{6,42}_A(F2,F2) has dimension other than 1, the counterexample collapses.","supporting_citations":[],"review_version":1}