OpenAI · 308-01 · 2026-09-23 · Topology

Finite Smith–Toda Complexes at Varying Primes

preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026

Family 308. Smith–Toda complexes at every height with varying primes.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

For every $n \geq 0$ , constructs a finite Smith–Toda spectrum at a prime $p$ depending on $n$ , with Brown–Peterson homology $B P_{*} / (p, v_{1}, \dots, v_{n})$ 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 $p = 1009$ .

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 $BP_{*} / (p, v_{1}, \dots, v_{n})$ with its canonical comodule structure and generator in degree zero. Each listed generator is killed to its first power.

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.