Finite Smith–Toda complexes at every height

What OpenAI claims

For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology BP_*/(p,v₁,…,v_n) and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes V(4) at p = 1009.

Papers (2)

1. Finite Smith–Toda Complexes at Varying Primes

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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 BP_*/(p,v₁,…,v_n) with its canonical comodule structure and generator in degree zero. Each listed generator is killed to its first power.

2. A finite Smith–Toda complex V(4) at the prime 1009

   23 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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,v₁,v₂,v₃,v₄), with its canonical comodule structure and generator in degree zero.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.