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

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

Confidence: Not scored

Registry verification: unreviewed · announcement · partial

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VibeMathed
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Unreleased internal OpenAI model
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Named evidence

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

specialist registry · passed

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.

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

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Companion: A finite Smith-Toda complex V(4) at the prime 1009

paper · pending

paper link supplied by the VibeMathed registry.

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Culver-Zhang, Exotic Picard groups and chromatic vanishing (V(4) recorded open)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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This event attributed to Unreleased internal OpenAI model

Culver-Zhang, Exotic Picard groups and chromatic vanishing (V(4) recorded open) evidence for this event

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

VibeMathed record: Smith-Toda complexes V(n) at every height when the prime may vary evidence for this event

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

Companion: A finite Smith-Toda complex V(4) at the prime 1009 evidence for this event

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