trailThe jump from two to three dimensions is the hardest step in resolving the Kakeya conjecture in all dimensions.
Framework carries over· for
Extending the Wang–Zahl method to Kakeya in dimensions four and higher faces primarily technical rather than fundamental obstacles.
atomic
The Wang–Zahl proof of the three-dimensional Kakeya conjecture can be adapted to prove the conjecture in higher dimensions.
Every set of delta-tubes pointing in delta-separated directions satisfies the polynomial Wolff axioms in every dimension.
Higher-dimensional algebraic obstruction· against
The convex Wolff axioms version of the Kakeya conjecture fails in dimensions four and above.
atomic
assumed
The Kakeya set conjecture remains open in dimensions four and higher.
assumed
The three-dimensional Kakeya conjecture has been proved: every three-dimensional Kakeya set has dimension three.
claim · evaluativeA judgment of worth or quality against some standard: good, fair, effective.constitution →claim page ↗︎
The jump from two to three dimensions is the hardest step in resolving the Kakeya conjecture in all dimensions.
↑SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →
Nothing in the graph builds on this claim yet.
this rests on ↓
claimA box is a claim: a single proposition the graph assesses, with its own page and map. Click any claim to centre the map on it.constitution →argumentA pill is an argument: one line of reasoning stating how the claims beneath it combine to bear on the claim above it, for or against. Arguments are not destinations; click their claims to explore.constitution →✓verifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →↑supportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →⇄contestedCredible evidence or argument exists on multiple sides.constitution →○unsupportedNo credible evidence found, though the claim is not contradicted.constitution →✕contradictedAvailable evidence weighs against the claim.constitution →?unknownInsufficient information to assess.constitution →◌unassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →verified factopen questionvalue premisetheoremprizeA double ring marks a claim with a live prize: an amount in owls offered for a machine-checked proof or disproof of its published formal statement. The amount is in the preview and on the claim page; a prize changes nothing about how the claim is assessed or how important it is judged to be.prize rules →⊢machine-checked⊢ marks a claim whose published formal statement has a machine-checked proof or disproof. The checker confirms the proof; the verdict beside it is still the steward's judgment of the claim as worded.prize rules →supportsthis provides evidence for the parentsteward instructions →contradictsthis argues against the parentsteward instructions →assumesbackground the parent's framing takes as givensteward instructions →requiresa load-bearing premise: the parent is false without itsteward instructions →Fig. Detail falls off with distance; every claim is an address.