Current status (The minimum peak lies between 1064^(1/4) and 7.7174713). Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713.[1]
1Packet records
3 records
Record
Kind
Assessment
Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713.
Result
Evidence: Reproduced
A replay is recorded for this computation. Inspect the replay record for its procedure and scope.
claim · Computation 1
Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713.[1]
Relevance to this problem
For Flattest 32-term Littlewood polynomial on the unit circle, record l32peak-claim-certified-interval (“The minimum peak lies between 1064^(1/4) and 7.7174713”) records a bound, answer, status fact, or structural consequence. The record states: Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713.
Evidence
ReproducedA replay is recorded for this computation. Inspect the replay record for its procedure and scope.
Record state
established
Scope
all 32-term Littlewood polynomials on the complex unit circle
Write
\[
C_k=\sum_{j=0}^{31-k}\varepsilon_j\varepsilon_{j+k}.
\]
The normalized fourth moment is
\[
\|P\|_4^4=32^2+2\sum_{k=1}^{31}C_k^2.
\]
For odd \(k\), the integer \(C_k\) is odd. For even \(k\), it is even. The parity floor is therefore \(\sum C_k^2\geq16\). Equality would make every odd-shift correlation equal to \(\pm1\) and every even-shift correlation zero, which is a Barker sequence of length 32. Turyn proved that an even Barker length greater than four must have the form \(4u^2\). Since 32 has no such form, equality is impossible. The next possible increase in the integer energy is four, so
\[
\sum_{k=1}^{31}C_k^2\geq20.
\]
Since \(\|P\|_\infty\geq\|P\|_4\), every polynomial in the candidate family satisfies
\[
\|P\|_\infty\geq1064^{1/4}=5.7113057054\ldots.
\]
The interval artifact proves that the displayed 32-sign polynomial has circle maximum below \(7.7174713\). Hence the present certified result is
\[
\boxed{1064^{1/4}\leq\min_P\|P\|_\infty<7.7174713}.
\]
The exact minimum remains open in this record.
Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.
Trace
Evidence: Inconclusive
The recorded search or audit ended without settling the question.
attempt · Route 1
Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.[2][3][1]
Relevance to this problem
For Flattest 32-term Littlewood polynomial on the unit circle, record l32peak-attempt-symmetry-and-literature-audit (“Symmetry leaves 536,887,296 reversal orbits”) documents a concrete method, search boundary, or failed route. The record states: Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.
Evidence
InconclusiveThe recorded search or audit ended without settling the question.
Scope
symmetries and primary literature relevant to the exact 32-term Littlewood circle-peak problem
Global sign change, substitution \(z\mapsto-z\), and coefficient reversal preserve the circle peak. Global sign and alternation select a unique representative with \(\varepsilon_0=\varepsilon_{31}=1\), leaving \(2^{30}\) sequences. Reversal has \(2^{15}\) fixed representatives in this normalization. Burnside's lemma therefore leaves
\[
\frac{2^{30}+2^{15}}2=536{,}887{,}296
\]
reversal orbits.
This count was checked by the artifact, though those orbits were not enumerated. The 500 seeded one-flip descents recorded in the candidate found the displayed incumbent. They carry no exclusion force.
Balister, Bollobás, Morris, Sahasrabudhe, and Tiba prove the existence of uniformly flat Littlewood polynomials at every length, with absolute constants. Erdélyi surveys the sup-norm flatness problem and the Rudin-Shapiro construction. Turyn's even-Barker restriction supplies the finite lower bound used here. Focused searches for fixed-degree tables, length 32, degree 31, minimum circle maximum, and Littlewood sup norm found no primary-source table giving this exact optimum. A complete proof can continue with the 536,887,296 normalized reversal orbits, using interval lower bounds to discard every orbit below the incumbent.
A standard-library Python verifier encloses every grid value and every intervening arc using integer interval arithmetic.
Artifact
Evidence: Executable material
Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1
A standard-library Python verifier encloses every grid value and every intervening arc using integer interval arithmetic.
Relevance to this problem
For Flattest 32-term Littlewood polynomial on the unit circle, record l32peak-artifact-fixed-point-circle-bound (“Exact fixed-point interval certificate for the incumbent”) supplies evidence or a replay used to check the packet. The record states: A standard-library Python verifier encloses every grid value and every intervening arc using integer interval arithmetic.
Evidence
Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay.
Record state
available
Scope
rigorous full-circle peak bounds for the displayed 32-term Littlewood polynomial
Entry point
join source_lines with newline and run with python3
Runtime
CPython 3.9 or later, standard library only
Details
The coefficient sequence, in increasing exponent order, is
\[
(1,-1,1,1,-1,1,1,-1,-1,1,-1,-1,1,-1,1,-1,
1,1,-1,-1,-1,-1,-1,-1,1,1,-1,-1,-1,1,1,1).
\]
For its aperiodic correlations \(C_k\), the verifier uses the exact identity
\[
F(t)=|P(e^{it})|^2=32+2\sum_{k=1}^{31}C_k\cos(kt).
\]
Cosines on a \(2^{18}\)-point grid are enclosed at scale \(10^{30}\). The program starts from explicit rational bounds on \(\pi\), reduces every angle to \([0,\pi/2]\), and evaluates 24 terms of the alternating cosine series with outward integer rounding. The next term bounds the remainder. Sines are obtained by a quarter-turn of the same cosine table.
For a grid point \(t_m\), Taylor's theorem gives
\[
F(t)\leq F(t_m)+|F'(t_m)|h+\tfrac12Bh^2,
\qquad h=\pi/2^{18},
\]
where the exact coefficient bound is
\[
B=2\sum_{k=1}^{31}k^2|C_k|=21264.
\]
Every point of the circle is within \(h\) of a grid point. The resulting all-cell comparison proves
\[
7.7174710<\max_{|z|=1}|P(z)|<7.7174713.
\]
A separate 100-decimal Newton evaluation places the apparent maximizing angle at \(0.9689875113845837543182660733966\ldots\) and the peak at \(7.717471112388378843091998621392\ldots\), consistent with the certified enclosure. The integer certificate supplies the stated bound without relying on that Newton calculation.
Original intake status. UNKNOWN as of 2026-07-25. Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713. The checked sources do not settle the full acceptance condition.
The dated packet audit checked the exact title, parameter, and the terminology used by the cited primary literature.
The strongest recorded neighboring result is: Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713.
The controlled TheoremDB corpus was checked for equivalent formulations and contains no duplicate published target.
Recorded example 1. An incumbent sign vector is [1,-1,1,1,-1,1,1,-1,-1,1,-1,-1,1,-1,1,-1,1,1,-1,-1,-1,-1,-1,-1,1,1,-1,-1,-1,1,1,1].
Computational notes
Five hundred seeded one-flip descents evaluated on a 32768-point FFT grid found maximum 7.717469866649506 for the displayed signs. The derivative bound adds at most 0.047553404424455 between grid points, and direct coefficient squaring verified L2 norm sqrt(32).
How the 3 records connectThe overview places each record once. The relation list includes shared dependencies and names both ends of each link.
ProblemFlattest 32-term Littlewood polynomial on the unit circle
Contribute to this problemCite this problem statement
Cite the original sources separately.
Plain text
“Flattest 32-term Littlewood polynomial on the unit circle.” TheoremDB. P2690. Problem statement; statement text SHA-256 e4f8cd6f5384d1073dbe19c7f9f3f58053bfd98d3de7e5407b55bec7f02e4da9. https://theoremdb.org/statement/?ref=P2690
BibTeX
@misc{theoremdb-problem-e4f8cd6f5384d1073dbe19c7f9f3f58053bfd98d3de7e5407b55bec7f02e4da9,
title = {{Flattest 32-term Littlewood polynomial on the unit circle}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 e4f8cd6f5384d1073dbe19c7f9f3f58053bfd98d3de7e5407b55bec7f02e4da9},
url = {https://theoremdb.org/statement/?ref=P2690}
}
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1References
Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, Flat Littlewood polynomials exist, Annals of Mathematics 192 (2020), 977-1004; Tamás Erdélyi, On the sup norm of Littlewood polynomials with Mahler measure one on the unit circle; R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256. R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256; exact fourth-moment calculation and interval certificate l32peak-artifact-fixed-point-circle-bound; Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, Flat Littlewood polynomials exist, Annals of Mathematics 192 (2020), 977-1004; Tamás Erdélyi, On the sup norm of Littlewood polynomials with Mahler measure one on the unit circle; R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.The minimum peak lies between 1064^(1/4) and 7.7174713. Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713. Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.Also cited at R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256; exact fourth-moment calculation and interval certificate l32peak-artifact-fixed-point-circle-bound.For Flattest 32-term Littlewood polynomial on the unit circle: The minimum peak lies between 1064^(1/4) and 7.7174713. Autocorrelation parity and Turyn's restriction on even Barker lengths give the lower bound; an explicit polynomial has a rigorously certified peak below 7.7174713. Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.
Packet source. Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba, Flat Littlewood polynomials exist, Annals of Mathematics 192 (2020), 977-1004; Tamás Erdélyi, On the sup norm of Littlewood polynomials with Mahler measure one on the unit circle; R. J. Turyn, On Barker Codes of Even Length, Proceedings of the IEEE 51 (1963), 1256. ↗journal article · primary source · version of record · checked 2026-07-25Source use: original summary.Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.For Flattest 32-term Littlewood polynomial on the unit circle: Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.Source named by the research packet.
Tamás Erdélyi, On the sup norm of Littlewood polynomials with Mahler measure one on the unit circle. people.tamu.edu checked 2026-08-01. Author-hosted manuscript. ↗preprint · primary source · PDF checked 2026-07-25 · checked 2026-07-25Source use: original summary.Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.For Flattest 32-term Littlewood polynomial on the unit circle: Symmetry leaves 536,887,296 reversal orbits. Standard symmetries sharply state the remaining computation, while the located primary sources give general flatness results rather than the length-32 optimum.
Original CC0 fixed-length flat-polynomial optimization.
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