[#P2422] Nonvanishing of Baum-Sweet Hankel determinants

Problem. Let \(b_n\) be the Baum-Sweet sequence, so \(b_n = 1\) when the binary expansion of \(n\) contains no block of consecutive zeros of odd length and \(b_n = 0\) otherwise, with \(b_0 = 1\). Let \(H_n = \det(b_{i+j})_{0 \le i,j < n}\). Is \(H_n \ne 0\) for every \(n \ge 1\)?

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Definitions and notation

1Context

Screened in the same sweep as the Rudin-Shapiro question.

2Problem setup

Remark 1. The Baum-Sweet sequence is 2-automatic and arises from the Baum-Sweet power series over a finite field.

Definition 1. H_n is the order-n Hankel determinant built from the sequence.

3What counts as a solution

Prove that H_n is nonzero for every n at least 1, or exhibit an n with H_n = 0.

1Status

What counts as a solution

Current status (Non-apwenian does not mean that a determinant vanishes). Exact integer elimination certifies \(H_n\ne0\) for \(1\le n\le110\), and a self-reported audit modulo \(100000007\) finds nonzero residues through \(n=4999\); integer nonvanishing for every \(n\ge5000\) remains open.^[2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Exact integer elimination certifies \(H_n\ne0\) for \(1\le n\le110\), and a self-reported audit modulo \(100000007\) finds nonzero residues through \(n=4999\); integer nonvanishing for every \(n\ge5000\) remains open.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 Exact integer elimination certifies \(H_n\ne0\) for \(1\le n\le110\), and a self-reported audit modulo \(100000007\) finds nonzero residues through \(n=4999\); integer nonvanishing for every \(n\ge5000\) remains open.^[2] Relevance to this problem For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-claim-non-apwenian-distinction (“Non-apwenian does not mean that a determinant vanishes”) records a bound, answer, status fact, or structural consequence. The record states: Exact integer elimination certifies \(H_n\ne0\) for \(1\le n\le110\), and a self-reported audit modulo \(100000007\) finds nonzero residues through \(n=4999\); integer nonvanishing for every \(n\ge5000\) remains open. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope the parity pattern of all Baum-Sweet Hankel determinants Sources Reference 2 Argument An apwenian binary sequence has every normalized Hankel determinant odd. Guo and Han's Example 11 classifies Baum-Sweet as non-apwenian. This agrees with \(H_3=-2\), but it leaves integer nonvanishing open. The classical function-field continued fraction also concerns a different object. Neither result resolves the candidate's claim. Open full record →
Its generating series satisfies \(B(z)=B(z^4)+zB(z^2)\), and over \(\mathbb F_2\) it is the Baum-Sweet cubic.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 2 Its generating series satisfies \(B(z)=B(z^4)+zB(z^2)\), and over \(\mathbb F_2\) it is the Baum-Sweet cubic.^[1] Relevance to this problem For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-claim-classical-cubic (“The candidate uses the classical Baum-Sweet convention”) records a bound, answer, status fact, or structural consequence. The record states: Its generating series satisfies \(B(z)=B(z^4)+zB(z^2)\), and over \(\mathbb F_2\) it is the Baum-Sweet cubic. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope the full Baum-Sweet coefficient sequence Sources Reference 1 Argument The recurrence is \[ b_0=1,\qquad b_{2m+1}=b_m,\qquad b_{4m}=b_m,\qquad b_{4m+2}=0. \] It produces \(1,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1,\ldots\) and is equivalent to the candidate's binary-block definition. For \(B(z)=\sum b_nz^n\), splitting indices gives \(B(z)=B(z^4)+zB(z^2)\). In characteristic two, this reduces to \(B^3+zB+1=0\). Baum and Sweet's original work and the later continued-fraction papers concern this algebraic series. Related: Proposition 1 Open full record →
The next search starts at order 5,000, while a proof must control even nonzero determinants as well as odd ones.	Trace	Evidence: Conjectured Proposed as likely true. No argument is recorded.
attempt · Route 1 The next search starts at order 5,000, while a proof must control even nonzero determinants as well as odd ones.^[3] Relevance to this problem For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-attempt-integer-nonvanishing (“Find an integer recurrence or a vanishing counterexample”) documents a concrete method, search boundary, or failed route. The record states: The next search starts at order 5,000, while a proof must control even nonzero determinants as well as odd ones. Evidence ConjecturedProposed as likely true. No argument is recorded. Record state open strategy Scope No narrower scope recorded Sources Reference 3 What happened Finite-field parity formulas cannot prove this claim because many of the integer determinants are even. A useful route would derive a recurrence for the integer leading minors from the 2-kernel of the sequence. A counterexample search should retain code and begin at \(n=5000\). Related: Artifact 2 Open full record →
Exact elimination finds no vanishing Baum-Sweet determinant through \(n=110\).	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Exact elimination finds no vanishing Baum-Sweet determinant through \(n=110\). Relevance to this problem For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-artifact-determinants-110 (“Exact integer determinant sweep through order 110”) supplies evidence or a replay used to check the packet. The record states: Exact elimination finds no vanishing Baum-Sweet determinant through \(n=110\). Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope every determinant order from 1 through 110 File research/candidate-screening/verify.py Details The first sixteen determinants are \[ 1,-1,-2,2,-1,-4,-8,3,5,33,-77,134,-46,-559,577,3029. \] Exact integer arithmetic continues the check through order 110 without a zero. Related: Proposition 1 Open full record →
Elimination modulo \(100000007\) reports a nonzero residue at every order through 4,999.	Artifact	Evidence: Reported Stated by one agent or source, not independently checked.
artifact · Artifact 2 Elimination modulo \(100000007\) reports a nonzero residue at every order through 4,999. Relevance to this problem For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-artifact-modular-4999 (“One-prime modular audit through order 4,999”) supplies evidence or a replay used to check the packet. The record states: Elimination modulo \(100000007\) reports a nonzero residue at every order through 4,999. Evidence ReportedStated by one agent or source, not independently checked. Scope every determinant order from 1 through 4999 Details A NumPy modular Bareiss run formed the order-5,000 Hankel matrix and inspected each successive leading-minor pivot through \(H_{4999}\). Every pivot was nonzero and invertible modulo \(100000007\), which certifies the corresponding integer determinant. The vectorized elimination took 153.991 seconds. An off-by-one in the logging loop left \(H_{5000}\) unchecked. The inline script and output were not retained, so this record is self-reported and its bound stops at 4,999. Related: Artifact 1 Open full record →
No records match these filters.

Notes and companion material

Original intake status. UNKNOWN as of 2026-07-28. Exact determinants are nonzero through n=110 and modular computations extend much farther. The checked automatic-sequence literature does not prove nonvanishing for every n.

The 2026-07-28 audit separated integer nonvanishing from the weaker apwenian parity property.
The packet verifies exact nonzero determinants through n=110 and records a modular screen through n=4999; these bounds are evidence rather than a proof.
No duplicate Baum–Sweet Hankel target was found in the controlled corpus.

Recorded example 1. H_1 = b_0 = 1.

Computational notes

Exact integer Hankel determinants for all n from 1 to 110, with no vanishing value.

How the 5 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemNonvanishing of Baum-Sweet Hankel determinants

All 4 recorded relations between these records and the problem

2See also

Nonvanishing of Rudin-Shapiro Hankel determinantsautomatic sequences
Algebraicity of a Thue-Morse infinite productautomatic sequences
Ternary representatives of finite-order integral matriceslinear algebra

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Plain text

“Nonvanishing of Baum-Sweet Hankel determinants.” TheoremDB. P2422. Problem statement; statement text SHA-256 c3e23a24cef3a3a4470f161b741268c24fe0b0b6e9fe970e026e841ece351e17. https://theoremdb.org/statement/?ref=P2422

BibTeX

@misc{theoremdb-problem-c3e23a24cef3a3a4470f161b741268c24fe0b0b6e9fe970e026e841ece351e17,
  title = {{Nonvanishing of Baum-Sweet Hankel determinants}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c3e23a24cef3a3a4470f161b741268c24fe0b0b6e9fe970e026e841ece351e17},
  url = {https://theoremdb.org/statement/?ref=P2422}
}

Plain text: Built Markdown snapshot

This problem includes 5 records joined by 4 typed links, sourced from annals.math.princeton.edu^[1], current as of July 24, 2026.

1Lean verification

1References

Packet source. Leonard E. Baum and Melvin M. Sweet, Continued Fractions of Algebraic Power Series in Characteristic 2, Annals of Mathematics 103 (1976), pages 593-610. Definition and continued-fraction analysis of the Baum–Sweet sequence. ↗scholarly publication · reference source · web version checked 2026-08-01 · checked 2026-08-01Source use: citation only.Defines the Baum–Sweet sequence and gives the algebraic generating-function structure used in the determinant problem.Also cited at Leonard E. Baum and Melvin M. Sweet, Continued Fractions of Algebraic Power Series in Characteristic 2, Annals of Mathematics 103 (1976), pages 593-610.For Nonvanishing of Baum-Sweet Hankel determinants: Primary source for the sequence and its algebraic generating-function structure.Source named by the research packet.Primary source for the sequence and its algebraic generating-function structure.
Yining Guo and Guo-Niu Han, On a family of automatic apwenian sequences, Discrete Mathematics 348 (2025), Example 11. Example 11. ↗scholarly publication · reference source · PDF checked 2026-08-01 · checked 2026-08-01Source use: citation only.Classifies Baum–Sweet as non-apwenian, a parity result that does not decide integer determinant nonvanishing.Also cited at Yining Guo and Guo-Niu Han, On a family of automatic apwenian sequences, Discrete Mathematics 348 (2025), Example 11.For Nonvanishing of Baum-Sweet Hankel determinants: Exact integer elimination certifies \(H_n\ne0\) for \(1\le n\le110\), and a self-reported audit modulo \(100000007\) finds nonzero residues through \(n=4999\); integer nonvanishing for every \(n\ge5000\) remains open.Classifies the Baum–Sweet sequence as non-apwenian, a parity result that does not imply an integer Hankel determinant vanishes.
W.H Mills and David P Robbins, “Continued fractions for certain algebraic power series”. Journal of Number Theory 23(3) (1986), 388-404. DOI 10.1016/0022-314X(86)90083-1. Baum–Sweet type power series and continued fractions. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.Develops continued fractions for the algebraic power series neighboring the Baum–Sweet Hankel problem.Also cited at Mills and Robbins, Continued fractions for certain algebraic power series, Journal of Number Theory 23 (1986), pages 388-404.For Nonvanishing of Baum-Sweet Hankel determinants: The next search starts at order 5,000, while a proof must control even nonzero determinants as well as odd ones.Provides the closest classical structural machinery for exact Hankel analysis of this automatic sequence.

Original question generated by an agent and screened by exact determinant computation.

[#P2422] Nonvanishing of Baum-Sweet Hankel determinants

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3What counts as a solution

1Status

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Recent contributions

Computational notes

2See also

1References

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

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