Smith–Toda complexes at every height

Smith–Toda complexes at every height concern the realization problem in stable homotopy theory: whether, at a prime p, there is a finite p-local spectrum V(n) whose Brown–Peterson homology is the quotient of the coefficient ring by p, v₁, …, v_n, each killed to its first power, with the canonical comodule structure. Two manuscripts dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim such a complex for every n at some prime depending on n, and a V(4) at the prime 1009. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

As recounted in the introductions of the two manuscripts:
- The mod p Moore spectrum gives V(0) at every prime. Adams, Smith and Toda constructed V(1), V(2) and V(3) for p greater than 2, 4 and 6 respectively. The periodicity theorem of Hopkins and Smith realizes suitable higher powers of the generators at every height, but not first powers.
- The September 2023 version of a paper of Culver and Zhang records the existence of V(4) as open at every prime.
- Nave proved that V((p + 1)/2) does not exist for p at least 7, and that V((p − 1)/2), if it exists, is not a ring spectrum. These obstructions depend on the prime and leave room for a fixed height at large primes.
- Barthel, Schlank and Stapleton established asymptotic algebraicity in chromatically localized categories; Kato, Shimomura and Shimomura constructed at the prime 1009 a localized spectrum with the corresponding homology after inverting v₅.

The original results

Every height at a varying prime (23 September 2026). Theorem 1.1 of the manuscript states that for every n at least 0 there exist a prime p and a finite p-local spectrum whose Brown–Peterson homology is the quotient by p, v₁, …, v_n with its canonical comodule structure and generator in degree zero. The manuscript states that it does not assert that one prime works at every height and gives no explicit bound on the prime.
- Method, as described: the construction takes place in an ultraproduct over a nonprincipal ultrafilter of the categories of p-local spectra, with sphere degrees that grow with p. Successive attaching classes come from a p-fold permutation power with the affine group acting on the factors, normalized through an Euler class, universal Koszul coefficient algebras, connectivity estimates at weights growing with p, and an Amitsur completion. Detection uses the p-series of complex cobordism. Finitely many maps and homotopies then descend to a single prime.
V(4) at the prime 1009 (23 September 2026). The companion manuscript constructs a finite 1009-local spectrum whose Brown–Peterson homology is the quotient by 1009, v₁, v₂, v₃, v₄, with the canonical comodule structure and generator in degree zero.
Sources. Commit 1581d4c of the openai/math repository, folders Finite Smith–Toda complexes at varying primes and A finite Smith–Toda complex V(4) at the prime 1009.
Formalization. The collection has no Lean scope note in lean/docs/ for this family.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 17 October 2026)
2026-10-16	J. Jardine, Lattice24, referee reviews of the varying-primes and prime 1009 manuscripts in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	The referee reviews report no gap in either manuscript. The review of the varying-primes manuscript remarks that the proof works for an arbitrary nonprincipal ultrafilter, so it would imply that V(n) exists for all but finitely many primes, a much stronger statement than the theorem as written. It describes this as not an error and states that, given the strength of the result, the deep homotopical inputs need an expert check; it checked them for plausibility only.	Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.

Notes

The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.