Smith-Toda complexes V(n) at every height when the prime may vary

At a prime $p$ , a Smith-Toda complex $V(n)$ is a finite $p$ -local spectrum whose Brown-Peterson homology is $BP_*/(p,v_1,\ldots,v_n)$ as a $BP_*BP$ -comodule, each generator killed to its first power. Adams, Smith and Toda constructed $V(1)$ , $V(2)$ , $V(3)$ for $p>2,4,6$ ; Nave proved that $V((p+1)/2)$ does not exist for $p\ge7$ ; and Culver-Zhang (2023 version) recorded the existence of $V(4)$ as open at every prime. Hopkins-Smith periodicity realizes suitable higher powers, not first powers. The Smith-Toda realization problem asks for which $n$ and $p$ the complex $V(n)$ exists. For every height $n$ , does $V(n)$ exist at some prime $p$ ?

Result: Proved(see note)
Status: Partial result
AI contribution: AI-discovered
Method: Construction
Field: Stable homotopy theory; chromatic homotopy
Posed by: The Smith-Toda realization problem (Smith 1970, Toda 1971); V(4) recorded open by Culver and Zhang (2023 arXiv version)
Year posed: 1971
Years open: 55y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 30 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every $n\ge0$ there are a prime $p$ and a finite $p$ -local spectrum $X$ with $BP_*X\cong BP_*/(p,v_1,\ldots,v_n)$ as comodules with the canonical coaction and generator in degree zero. The companion constructs $V(4)$ at $p=1009$ , answering the existence question Culver-Zhang recorded as open at every prime. Partial for the realization problem: no explicit or effective prime for $n\ge5$ , no statement that $V(n)$ exists for all large $p$ , no multiplicative structure, and nothing on the gap between Nave's nonexistence range and the constructed cases.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. A companion of the same date constructs V(4) explicitly at p = 1009 by a direct obstruction computation.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript and the companion's main theorem were read against the realization problem. The principal theorem gives, for each $n$ , a prime depending on $n$ (no bound on it), via an ultraproduct of $p$ -local categories and descent to one prime; the companion gives $V(4)$ at $p=1009$ . Neither proof was refereed here. The paper states that it does not claim one prime for all heights, a bound on the prime, or a ring structure. No Lean formalization exists for this family.

Smith-Toda complexes V(n) at every height when the prime may vary

What was actually shown

What the AI did

Verification

Sources

Discussion