The restriction, Bochner–Riesz, and local smoothing conjectures each imply the Kakeya conjecture.

The claim is a statement of three proven implications, so it is assessed against the mathematical literature rather than against a body of empirical evidence.

Sources read. Terence Tao's 2026 expository article for the ICM proceedings (arxiv.org/html/2608.22209, section 2) states that "many long-standing conjectures in harmonic analysis, number theory, and partial differential equations were shown to imply the Kakeya conjecture, in that any counterexample to the Kakeya conjecture could be converted to a counterexample to these latter conjectures", and lists Stein's restriction conjecture, the Bochner–Riesz conjecture, Sogge's local smoothing conjecture, and the Montgomery conjecture; it explains the wave-packet mechanism going back to Fefferman's 1971 disk multiplier counterexample and notes that the implications only partially reverse. Quanta Magazine's 2023 explainer (www.quantamagazine.org/a-tower-of-conjectures-that-rests-upon-a-needle-20230912/) names the same three conjectures, gives the order restriction, then Bochner–Riesz, then local smoothing, states that each implies the one below, and correctly adds that proving Kakeya would not automatically prove the others. Quanta's 2025 report on the Wang–Zahl proof (www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/) restates the tower framing without naming the conjectures. All three sources affirm the claim; no source read denies it, and the field's surveys treat the chain as standard.

The subclaims. Restriction implies Kakeya↗︎ follows from the fact that a restriction estimate at the conjectured exponents yields the Kakeya maximal function conjecture, which yields full dimension for Kakeya sets (Fefferman 1971; Bourgain 1991; presented as standard in Tao's 2003 Park City notes, arxiv.org/abs/math/0311181, and his 2003 survey "Some recent progress on the Restriction conjecture"). Bochner–Riesz implies restriction↗︎ is the title theorem of Tao, Duke Math. J. 96 (1999), 363–375, proved for the sphere and compact elliptic surfaces, which suffices since restriction for the sphere or paraboloid already implies Kakeya. Local smoothing implies Bochner–Riesz↗︎ is due to Sogge (Invent. Math. and J. Amer. Math. Soc., 1991), who introduced the local smoothing conjecture and showed that sharp local smoothing yields Carleson–Sjölin-type oscillatory integral bounds and hence the Bochner–Riesz conjecture. Each link is stated with the usual epsilon-loss formulations of the conjectures; none is disputed in the literature. The three subclaims are seeded and unassessed at this pass; the verdict here rests on the direct reading of Tao's exposition together with the well-known published theorems, not on the seeds.

Adversarial check. Two misreadings could make the claim look false. Reading "depend logically on" as the converse (Kakeya implies the three) would be wrong, and the canonical form now fixes the direction as each of the three implying Kakeya, which is the direction every source states. Reading "three" as exhaustive would also be wrong, since Montgomery's conjecture and others also imply Kakeya, but the claim as posed concerns the three harmonic analysis conjectures of the tower and does not assert exclusivity. Neither the Wang–Zahl proof of the three-dimensional case nor the open status of Kakeya in dimensions four and higher affects the claim, which concerns the implications and not the truth of the consequent.

Limitations. Tao's 1999 Duke paper and Sogge's 1991 papers were not opened in this pass (the PDF of Tao's 2003 Park City notes could not be parsed); the verdict relies on Tao's 2026 exposition, the field's surveys, and the well-known published record. What would change the verdict: a demonstration that one of the three links holds only under a formulation of a conjecture materially stronger than the one usually meant, which would downgrade that link to a qualified implication; nothing in the literature suggests this.