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Singer's algebraic transfer is not injective at rank 6, degree 36, giving a counterexample to a 1989 conjecture.

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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.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection A credible computational counterexample to Singer's conjecture that currently hinges on an unavailable codebase; referee it, but only after the artifacts ship. the 3 major comments →

arxiv 2509.09455 v11 pith:WJUMPWMB submitted 2025-09-11 math.AT math.RAmath.RT

Geometric realization via unoriented bordism and a counterexample to Singer's conjecture for the sixth algebraic transfer

Dang Vo Phuc This is my paper
classification math.AT math.RAmath.RT MSC 55Q4555S1055S0555T1568W30
keywords Singer algebraic transferSteenrod algebrahit problemKameko homomorphismgeneral linear group invariantsmod-2 cohomologyunoriented bordismcomputer-assisted proof
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

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.

Core claim

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

What carries the argument

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

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Section 3, Note 3.5, Data Availability] 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.
  2. [Abstract and Section 1] 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.
  3. [Remark 3.4] 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.
minor comments (5)
  1. [Section 3, Remark 3.1] 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.
  2. [Section 3, table after kernel computation] 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.
  3. [Section 1, Remark 1.5] 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.
  4. [Data Availability] 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.
  5. [Section 1, spike count discussion] 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.

Circularity Check

0 steps flagged

No significant circularity: the counterexample is an explicit F2-linear algebra computation cross-checked against external results; self-citations are supporting inputs, not definitional substitutes.

full rationale

The derivation chain for Theorems 1.2–1.4 is computational rather than definitional. The claimed invariant space [(QP_6)_36]^{GL(6)} is obtained by: (i) building admissible bases via streaming hit elimination, (ii) forming the Kameko bit-matrix and computing its nullspace (dimension 12,390, split as 2,725+111+1,085+6,495+1,974), (iii) solving weightwise Σ_6- and GL(6)-invariance equations (ρ_j−Id)f≡0, j=1,…,6, and (iv) reporting the two-dimensional solution space spanned by ζ_1 and ζ_2. The final assertion 'ρ_i(ζ_1)≡ζ_1 and ρ_i(ζ_2)≡ζ_2' is a check of the computed basis, not the source of the dimension. No fitted parameter is renamed as a prediction: ζ_1 and ζ_2 are not inputs to the algorithm; they are outputs of the nullspace/invariance solver. The Kameko decomposition dim(QP_6)_36 = dim Ker((˜Sq^0_*)(6,36)) + dim(QP_6)_15 is a mathematical identity (surjectivity of the Kameko map), not a self-imported conclusion. The inputs dim(QP_6)_15 = 2,184 (from [28]) and the target-degree invariant ξ (Proposition 3.2, computed with the author's earlier algorithm [31]) are supporting; Remark 3.4 reports β = 0, i.e., the target-seed library contributes nothing to the final invariants, so the conclusion does not reduce to those seeds. The codomain dimension dim Ext^{6,42}_A(F_2,F_2) = 1 is taken from independent external sources (Bruner [2], Chen [5], Lin [15]) and is not derived from the paper's domain computation. The algorithm is benchmarked against genuinely external results: Nguyen Sum's q=5, n=35 invariant dimension and basis, and Mothebe's spike counts, including an independent correction of the q=11, n=1013 value. The main limitation is reproducibility, not circularity: Note 3.5 states the OSCAR code is 'available upon request' and the detailed outputs are hosted on unversioned Google Drive links, and the last invariance check is described as 'direct manual verification with computer assistance.' A bug in the bit-packed nullspace solver, Kameko row map, or ρ_j-invariance systems could change the dimension and invalidate the counterexample. But that is a correctness/reproducibility risk, not a circular reduction of the conclusion to its inputs. The several self-citations by the author are numerous but not load-bearing in a definitional sense: the central computation is presented in the paper with explicit data (weight blocks, dimensions, the polynomials ζ_1 and ζ_2, and the q=5, n=35 cross-check), and the conclusion does not follow from any self-cited uniquene

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No fitted parameters appear in this work: it is a deterministic computation, not an empirical fit. The numbers the central claim rests on are (i) the known target dimension dim Ext^{6,42}=1 from external cohomology computations, (ii) dim(QP6)15=2184 imported from the author's prior paper [28], and (iii) the new algorithm's own outputs, which are the theorem's content rather than inputs. Axioms are the standard structural facts of the hit problem plus these two imported dimensions. No invented entities are introduced; 'AlgebraicTransfer.jl' is advertised in the abstract but is a software artifact, not a physical postulate, and no code is actually provided.

axioms (6)
  • standard math Kameko homomorphism (˜Sq0*)_(q,2n+q): (QP_q)_{2n+q} -> (QP_q)_n is surjective, so dim(QP_q)_{2n+q} = dim Ker + dim(QP_q)_n.
    Well-known result (Kameko 1990), invoked in Remark 3.1 to reduce dim(QP6)36 to the kernel dimension plus dim(QP6)15.
  • standard math GL(q) is generated by the adjacent transpositions ρ_1,...,ρ_{q-1} plus the transvection ρ_q, and invariance can be checked on these generators.
    Invoked in Definition 2.5 and used throughout Section 3 for the Σ6/GL(6)-invariant computations.
  • standard math The weight-vector decomposition (QP_q)_n ≅ ⊕_{deg ω = n} QP_q(ω), with admissible monomial bases from Walker-Wood.
    Invoked in Section 2 (after Definition 2.4) citing [38] for the admissible-basis description and the weight decomposition.
  • domain assumption dim(QP6)15 = 2184, taken from the author's own prior computation [28].
    Load-bearing input in Remark 3.1; not re-derived in this paper and not independently reproduced elsewhere yet.
  • domain assumption dim Ext^{6,42}_A(F2,F2) = 1, cited from Bruner [2], Chen [5], Lin [15].
    The counterexample needs the codomain to be 1-dimensional; two of the three cited sources are unpublished preprints (2012 and 2023).
  • ad hoc to paper The new algorithm's outputs at (q,n)=(6,36): Kameko-kernel weight-block dimensions summing to 12,390, and a 2-dimensional GL(6)-invariant subspace with basis [ζ1],[ζ2].
    These are the asserted results of the unshipped OSCAR code, stated in Section 3; they are the computational heart of the proof and are not independently verifiable from the manuscript alone.

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Cite this review

Pith. "Pith review of Geometric realization via unoriented bordism and a counterexample to Singer's conjecture for the sixth algebraic transfer." pith.science (2026). https://pith.science/paper/WJUMPWMB

@misc{pith2026250909455,
  author       = {Pith},
  title        = {Pith review of: Geometric realization via unoriented bordism and a counterexample to Singer's conjecture for the sixth algebraic transfer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WJUMPWMB}},
  note         = {Machine review of arXiv:2509.09455}
}
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abstract

Let $\mathscr{A}$ be the mod-2 Steenrod algebra acting in the usual way on $P_q = \mathbb{F}_2[x_1, \ldots, x_q]$, and let $QP_q = \mathbb{F}_2 \otimes_{\mathscr{A}} P_q$. Singer's algebraic transfer $Tr_q$ sends the dual of $[(QP_q)_n]^{GL(q, \mathbb{F}_2)}$ to $\operatorname{Ext}_{\mathscr{A}}^{q,q+n}(\mathbb{F}_2,\mathbb{F}_2)$; Singer conjectured that $Tr_q$ is always injective. We disprove this nearly forty-year-old conjecture at rank $q=6$, degree $n=36$. Verifying this requires computing $[(QP_6)_{36}]^{GL(6, \mathbb{F}_2)}$ exactly; to handle the resulting combinatorial complexity, we build a new Julia package \texttt{AlgebraicTransfer.jl}, coupling modular invariant theory with bit-level linear algebra over $\mathbb{F}_2$ via Steenrod-hit reductions and Kameko homomorphisms. We prove this source space is two-dimensional, strictly exceeding the known one-dimensional target $\operatorname{Ext}_{\mathscr{A}}^{6,42}(\mathbb{F}_2,\mathbb{F}_2)$, so $Tr_6$ is not injective. We also interpret the transfer kernel geometrically via unoriented bordism: $Tr_q$ factors through bordism classes over $B(\mathbb{Z}/2)^q$ whose Thom images are primitive, characterized by the vanishing of all mixed Wu numbers. Thom's representability theorem guarantees closed $36$-manifolds realizing the homological duals of the source generators, yet we show standard models -- the indecomposable Milnor hypersurface $H_{4,33}$, projective products, and Dold manifolds -- cannot represent them. We further interpret the inverse Kameko map via Thom spaces of universal real line bundles. Validated by recovering classical Dickson invariant dimensions, this work delivers both a counterexample to Singer's conjecture and a scalable methodology for the Peterson hit problem.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Local Parity and Systematic Peterson Counterexamples in the Motivic Hit Problem

    math.AT 2026-01 conditional novelty 6.0

    Odd-parity sums of monotone translates of Kameko's monomial z_k are never hit by the motivic Steenrod algebra, yielding infinite Peterson-type counterexamples.

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.