The source explicitly says not to factor the 80 old defect-count-25 norms or reproduce the unconstrained 277-row table except as a regression test. Sparse proof-producing marginal elimination, a constrained 16th/32nd-root gate, an integral Pell/Riccati non-lift theorem, and a uniform witness-graph or direct spectral obstruction remain proposed routes.
Route status · Narrowed routeCombinatorics and number theory · binary sequences · autocorrelation · cyclotomic constraints
Barker Sequence Conjecture
Collaboration betaA Barker sequence is a row of plus and minus signs whose shifted copies stay almost perfectly uncorrelated. The packet reports strong restrictions on any sequence longer than 13, but it does not report a proof that none exists.
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Known results and sources 
Research problem
Exact mathematical statement
For a binary word , define the aperiodic autocorrelation
A Barker sequence satisfies for every . The conjectural target is that no Barker sequence has length .
The submitted source reports packet-scoped reductions, proofs, and exact computations, but explicitly claims no complete contradiction.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Barker Sequence Conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
At equality 27, the current work reports one aggregate tuple with three positive and 24 negative defects plus forced rank and residue structure.
Evidence posture · Source-reported route statement · dependencies incompleteWork mapped so far
Barker Sequence Conjecture in numbers
896retained lines of mathematical investigation896 in the current working snapshot
- Argument development
- 760 · 85%
- Explored or eliminated routes
- 29 · 3%
- Computational analysis
- 32 · 4%
- Open obligations
- 42 · 5%
- Definitions and setup
- 33 · 4%
6selected mapped statements1routes investigated6open questions6contribution-ready tasks
How this is measured
This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
Recommended next task
Eliminate or finitely classify ordered BK-B27 lifts satisfying all exact prime-square inversion marginals, higher odd moments, distinctness, ordering, and forced residues modulo four.
Suggested move: Model two or three selected prime-square marginals jointly through sparse Chinese-remainder cells and require a separately checkable unsatisfiable core or complete residue-orbit list.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
Visible working map
Research route map
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
Scroll horizontally to explore the route
Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
Explored alternatives
Other routes
The source explicitly says not to factor the 80 old defect-count-25 norms or reproduce the unconstrained 277-row table except as a regression test. Sparse proof-producing marginal elimination, a constrained 16th/32nd-root gate, an integral Pell/Riccati non-lift theorem, and a uniform witness-graph or direct spectral obstruction remain proposed routes.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Eliminate or finitely classify ordered BK-B27 lifts satisfying all exact prime-square inversion marginals, higher odd moments, distinctness, ordering, and forced residues modulo four.Suggested move: Model two or three selected prime-square marginals jointly through sparse Chinese-remainder cells and require a separately checkable unsatisfiable core or complete residue-orbit list.
Ready to work on
Show that no marginal-compatible signed defect polynomial lifts to an actual Littlewood spectral factor through the current work's integral Pell or Riccati interfaces.Suggested move: Derive one additional endpoint, parity, or modular congruence from the integral factorization and test it only on BK-B27-compatible marginal patterns.
Ready to work on
Exact marginal gate
The immediate distinguished open task is to combine six exact inversion systems with higher moments, strict ordering, distinctness, and forced residues modulo four.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Binary spectral lift
Even a signed defect polynomial satisfying the local marginals must still be shown unable to arise from a compatible Littlewood spectral factor.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Uniform all-candidate theorem
The full conjecture remains blocked by the absence of a theorem covering every admissible arithmetic parameter, whether or not its semiprimitive witness graph is dense.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Replace the distinguished-candidate calculation by a theorem that covers every arithmetically admissible u, including sparse semiprimitive witness graphs.Suggested move: Analyze the directed witness graph dichotomy: derive exact lifting from dense incoming witnesses or an independent number-theoretic contradiction from sparse coverage.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen conjecture
Open. A Barker sequence is a finite {-1,1} sequence whose nonzero-shift aperiodic autocorrelations all have magnitude at most 1, and the conjecture asserts that none has length greater than 13. The odd-length case is settled; the remaining even-length case above 4 is open. No Barker sequence exists for 13 < n <= 4 x 10^33.
[3][4][5]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Peer reviewedAnti-field-descent restrictions, combined with earlier work, exclude Barker sequences for every length 13 < n <= 4 x 10^33.[4][5] Peer reviewedTuryn and Storer published the odd-length classification; Schmidt and Willms later supplied a corrected, simpler proof establishing that an odd Barker sequence has length in {3,5,7,11,13}.[2][3] Historical sourceBarker studied binary sequences satisfying the stricter requirement that each nonzero aperiodic autocorrelation belongs to {0,-1}; later work adopted the modern magnitude-at-most-one definition.[1][5]
Mathematical neighborhood
Related results and reusable starting points
Current focusBarker sequence conjecture
The weak Barker sequence conjecture asks only that there be finitely many Barker sequences, whereas the strong conjecture asserts that the known examples through length 13 are all of them.
[5]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetNo reviewed formal statement or formal proof artifact for the full conjecture was identified.
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
4 standing statements2 proposed statements6 open questions1 narrowed routes
- theorem candidate
1 of 6 1 - reduction
3 of 6 3 - lemma
2 of 6 2
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
- retained route statementCan a binary Barker sequence have length greater than 13?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementEven-length arithmetic formintermediate
- retained route statementDistinguished defect lower boundintermediate
- retained route statementBK-B27 equality structureintermediate
- Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
- Useful failureUnconstrained legacy higher-norm enumerationreported failure
- Research targetEliminate or finitely classify ordered BK-B27 lifts satisfying all exact prime-square inversion marginals, higher odd moments, distinctness, ordering, and forced residues modulo four.open
- Research targetShow that no marginal-compatible signed defect polynomial lifts to an actual Littlewood spectral factor through the current work's integral Pell or Riccati interfaces.open
- Research targetReplace the distinguished-candidate calculation by a theorem that covers every arithmetically admissible u, including sparse semiprimitive witness graphs.open
- Research targetExact marginal gateopen
- Research targetBinary spectral liftopen
- Research targetUniform all-candidate theoremopen
- Narrowed routeUnconstrained legacy higher-norm enumerationThe source explicitly says not to factor the 80 old defect-count-25 norms or reproduce the unconstrained 277-row table except as a regression test. Sparse proof-producing marginal elimination, a constrained 16th/32nd-root gate, an integral Pell/Riccati non-lift theorem, and a uniform witness-graph or direct spectral obstruction remain proposed routes.
How to interpret these counts
A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.
Research outlook
Conditions that would advance the current route
1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
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Prepared starting pointEliminate or finitely classify ordered BK-B27 lifts satisfying all exact prime-square inversion marginals, higher odd moments, distinctness, ordering, and forced residues modulo four.
Barker Sequence Conjecture · ready to start
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Research contextPrepared context for any AI agent
A Barker sequence is a row of plus and minus signs whose shifted copies stay almost perfectly uncorrelated. The current work reports strong restrictions on any sequence longer than 13, but it does not report a proof that none exists.
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references6 cited works · next context review by Nov 29, 2026
The mathematical context was checked on Aug 29, 2026. Status can be refreshed sooner after a material result or claim.
- 1Group synchronizing of binary digital systemsoriginal source · R. H. Barker · Communication Theory · 1953 · accessed Aug 29, 2026
- 2On binary sequencespeer reviewed result · R. Turyn, J. Storer · Proceedings of the American Mathematical Society · 1961 · DOI 10.1090/S0002-9939-1961-0125026-2 · MR MR0125026 · accessed Aug 29, 2026
- 3Barker sequences of odd lengthpeer reviewed result · Kai-Uwe Schmidt, Jurgen Willms · Designs, Codes and Cryptography · 2016 · ARXIV 1501.06035 · DOI 10.1007/s10623-015-0104-4 · accessed Aug 29, 2026
- 4The anti-field-descent methodpeer reviewed result · Ka Hin Leung, Bernhard Schmidt · Journal of Combinatorial Theory, Series A · 2016 · DOI 10.1016/j.jcta.2015.11.005 · accessed Aug 29, 2026
- 5A Note on Barker Sequences and the L1-norm of Littlewood Polynomialspeer reviewed result · Gang Yu · Comptes Rendus Mathematique · 2023 · DOI 10.5802/crmath.428 · accessed Aug 29, 2026
- 6Formalize the Barker-sequence length conjecturemaintained problem list · Google DeepMind formal-conjectures contributors · Google DeepMind · 2026-08-06 · accessed Aug 29, 2026
Important qualifications
- The 1953 Barker source studied the stricter condition that every nonzero autocorrelation lies in {0,-1}; the modern {-1,0,1} formulation and no-length-above-13 conjecture should not be attributed verbatim to that paper.
- The 2026 formal-conjectures issue corroborates that the problem is still tracked as open but is not mathematical proof authority.
- Enumerated survivors of known arithmetic tests above the proved range do not establish a stronger exhaustive lower bound.
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