Finite Smith–Toda complexes at every height

Claims Smith–Toda complexes exist at every height if the prime may grow, with an explicit V(4) at the prime 1009.

2 papers September 23, 2026 Explained by AI

A Smith–Toda complex V(n) is a finite building block of stable homotopy theory that kills the first n+1 chromatic numbers once each. Classical work built V(1), V(2) and V(3) at large enough primes. V(4) was recorded as open at every prime. One paper claims V(n) exists for every n at some prime chosen for that n. A second paper claims an explicit V(4) at the prime 1009.

First step

1970

Larry Smith, realizing complex bordism modules; with Adams and Toda, the first complexes V(1) to V(3)

Significance

Major

It claims a classical realization problem in stable homotopy theory at every height, where earlier constructions stopped at V(3).

Trust

Claimed, not checked

Field

Topology

2 papers

The explanation

Start with the whole numbers and divide out by a prime p. The result is the numbers mod p. Topology has a version of this step for spectra, the basic objects of stable homotopy theory. At height zero the Moore spectrum kills p this waySource 1. Brown–Peterson homology, written BP, assigns to each spectrum a list of algebraic data. For the sphere it is a polynomial ring in numbers v₁, v₂, v₃ and so on. The number vᵢ sits in degree 2(pⁱ − 1). At p = 1009, v₁ sits in degree 2,016. The number v₄ sits in degree 2,072,977,845,120.

A Smith–Toda complex V(n) is a finite spectrum whose BP homology kills p, v₁, up to vₙ, each once. Killing the next number needs an actual map of spectra that multiplies by it. Algebra describes the result once that map exists, but it does not supply the mapSource 2. That is the obstacle. Adams, Smith and Toda built V(1), V(2) and V(3) when p is larger than 2, 4 and 6Source 3. Hopkins and Smith could kill higher powers of these numbers at every heightSource 4. But the question asks for first powers. A 2023 version of a paper by Culver and Zhang recorded V(4) as open at every primeSource 5. Nave proved that V((p+1)/2) does not exist for p at least 7Source 6. At p = 7 that rules out V(4), since (7 + 1)/2 = 4. At p = 1009 the barrier is V(505), far above height four.

The first paper claims that V(n) exists for every n, at a prime p that depends on nSource 7. It works with an ultraproduct, a way to average a statement over all primes at once. The ultraproduct is formed from ordinary p-local spectraSource 8. Only finitely many maps then need to come down to one actual primeSource 9. The result names no bound on the prime and no single prime for all heightsSource 10.

The second paper claims an explicit V(4) at p = 1009Source 11. It builds V(3) with a product, then shows that v₄ survives a spectral sequence, a step-by-step calculation of mapsSource 12. The large prime keeps the finite inequalities in the calculation comfortableSource 13.

How sure is this? There is no Lean check. A computer proof checker has not looked at any part of it. Both papers rely on long algebraic calculations and delicate constructions in stable homotopy theory. No specialist review of either paper has been published.

What it doesn't show. This does not show that one prime works at every height, and it does not give a bound on the prime in general.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if the proofs hold. The all-height result allows the prime to grow and gives no bound. There is no Lean check and no published specialist review.

Words used

Spectrum: The basic object of stable homotopy theory, a stabilized version of a space.
Brown–Peterson homology (BP): An invariant that turns a spectrum into algebraic data over a polynomial ring in numbers v₁, v₂ and so on.
Smith–Toda complex V(n): A finite spectrum whose BP homology kills p and v₁ up to vₙ, each to the first power.
Moore spectrum: The spectrum version of the numbers mod p, which is V(0).
Height: The index n that says how many of the numbers v₁, v₂ and so on are killed.
Ultraproduct: A way to combine one object for each prime into a single object that averages over all primes.
Spectral sequence: A step-by-step calculation that approximates an answer page by page.

The original papers (2)

Finite Smith–Toda Complexes at Varying Primes (PDF, opens in a new tab)
September 23, 2026
A finite Smith–Toda complex V(4) at the prime 1009 (PDF, opens in a new tab)
September 23, 2026

All quoted lines (13)

At height zero the Moore spectrum supplies the quotient by $p$ .
Finite Smith–Toda Complexes at Varying Primes Back to text
Regularity computes the homology of the cofiber once such a map exists; it does not supply the map.
Finite Smith–Toda Complexes at Varying Primes Back to text
The low-height constructions are due to Adams, Smith, and Toda: $V(k)$ exists for $k=1,2,3$ when $p>2k$ , respectively, as summarized in.
Finite Smith–Toda Complexes at Varying Primes Back to text
Suitable higher powers can be realized at every finite height by the periodicity methods of Hopkins–Smith.
Finite Smith–Toda Complexes at Varying Primes Back to text
In the September 2023 version of, the existence of $V(4)$ was recorded as open at every prime.
Finite Smith–Toda Complexes at Varying Primes Back to text
For $p\geq7$ , Nave proves that $V((p+1)/2)$ does not exist and that $V((p-1)/2)$ cannot be a ring spectrum if it exists.
Finite Smith–Toda Complexes at Varying Primes Back to text
For every nonnegative integer $n$ , we construct a Smith–Toda complex $V(n)$ at some prime $p$ depending on $n$ .
Finite Smith–Toda Complexes at Varying Primes Back to text
Here the ultraproduct is formed from ordinary $p$ -local spectra.
Finite Smith–Toda Complexes at Varying Primes Back to text
Only finitely many maps and homotopies must subsequently descend to an individual prime.
Finite Smith–Toda Complexes at Varying Primes Back to text
It does not assert that one prime works at every height, provide an explicit bound on the prime, or require a multiplication on the final spectrum.
Finite Smith–Toda Complexes at Varying Primes Back to text
We construct a Smith–Toda complex $V(4)$ at the prime $1009$ .
A finite Smith–Toda complex V(4) at the prime 1009 Back to text
Second, we prove that the filtration-zero class $v_4$ survives in the Adams–Novikov spectral sequence for $Y$ .
A finite Smith–Toda complex V(4) at the prime 1009 Back to text
The choice $1009$ makes the finite inequalities in our calculation generous.
A finite Smith–Toda complex V(4) at the prime 1009 Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

309The Kervaire invariant problem at the prime threeTopology
310Quillen's conjecture in rational homologyTopology
315The four-dimensional Singer conjectureTopology
316Curtis’s conjectureTopology
307Failure of rational injectivity for maximal coarse assemblyTopology

Browse Topology Open since the 1970s Claimed, not yet checked