---
title: "Finite Smith–Toda complexes at every height"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 308; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "topology"
  - "unformalized"
seed_rank: 2163
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Finite Smith–Toda Complexes at Varying Primes"
    url: "https://github.com/openai/math/blob/main/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "A finite Smith–Toda complex V(4) at the prime 1009"
    url: "https://github.com/openai/math/blob/main/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Finite Smith–Toda complexes at every height

## One-sentence takeaway

OpenAI's result family 308 (Topology) claims: For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology $BP_*/(p,v_1,\ldots,v_n)$ and the canonical comodule structure.

## Why it matters here

Topology results are background for knot, surface and configuration-space reasoning in geometry code. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes $V(4)$ at p = 1009.
- *Finite Smith–Toda Complexes at Varying Primes*: For every nonnegative integer n, we construct a Smith–Toda complex $V(n)$ at some prime p depending on n. It is a finite p-local spectrum whose Brown–Peterson homology is $\mathrm{BP}_*/(p,v_1,\ldots,v_n)$ with its canonical comodule structure and generator in degree zero. Each listed generator is killed to its first power.
- *A finite Smith–Toda complex V(4) at the prime 1009*: We construct a Smith–Toda complex $V(4)$ at the prime 1009. It is an ordinary finite 1009-local spectrum whose Brown–Peterson homology is $BP_*/(1009,v_1,v_2,v_3,v_4)$, with its canonical comodule structure and generator in degree zero.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Finite Smith–Toda Complexes at Varying Primes](https://github.com/openai/math/blob/main/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf)
- Manuscript: [A finite Smith–Toda complex V(4) at the prime 1009](https://github.com/openai/math/blob/main/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math