# Counterexamples to finite generation at chromatic height two > A paper claims to disprove the Hahn–Wilson conjecture at height two, for every large enough prime. Plain-English explainer with sources. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-319/ Last updated: 2026-10-14 **A paper claims to disprove the Hahn–Wilson conjecture at height two, for every large enough prime.** **Trust:** Claimed, not checked · **Papers:** 1 · **First paper:** September 26, 2026 · **Field:** Topology · **Entry:** 319 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. The Hahn–Wilson conjecture says a simple algebraic test picks out exactly which infinite spectra can be built from BP⟨n⟩ in finitely many steps. The paper claims a counterexample at height two for every large enough prime. Its spectrum passes the algebraic test but cannot be built from BP⟨2⟩ that way. It also passes two localization tests that were proposed as a possible extra condition. The paper says the problem lies in how the local pieces are glued. ## At a glance - **First step:** 2021 (Jeremy Hahn and Dylan Wilson, conjecture later written down by Lee and Pstrągowski) - **Significance:** Solid. It claims to refute a recent conjecture describing which infinite spectra have finite constructions from truncated Brown–Peterson spectra. - **Trust:** Claimed, not checked - **Field:** Topology (1 paper) ## The explanation Fix the prime p = 11. A standard spectrum called BP⟨2⟩ has homotopy built from two variables, v₁ and v₂. Their degrees are 2(p − 1) and 2(p² − 1). For p = 11 that gives degrees 20 and 240. So the coefficients form a polynomial ring in two variables, sitting in degrees 20 and 240. This spectrum is the basic building block in this story. Some background first. A spectrum is the stable version of a space, the object studied in stable homotopy theory. A thick subcategory is everything you can build from a starting object. You may take finite sums, shifts, cones of maps and retracts, a finite number of times. The question is which infinite spectra you can build from BP⟨n⟩ this way. The Hahn–Wilson conjecture gives a cohomological answer. It says finite presentation over the Steenrod algebra, plus a height bound, should pick out exactly these spectra[1]. The Steenrod algebra is a standard ring of operations on mod-p cohomology. Finitely presented means described by finitely many generators and relations. Such a spectrum is called fp. Mahowald and Rezk proved an equivalent test. Smashing with some finite spectrum must give only finitely many nonzero homotopy groups, each finite[2]. The height is a level in chromatic homotopy theory. Lee and Pstrągowski wrote the conjecture down and report learning it from Dylan Wilson in 2021[3]. Some cases were known. Lee and Pstrągowski proved the conjecture at height one[4]. A related claim, the telescope conjecture, compares two ways to localize a spectrum. Burklund, Hahn, Levy and Schlank disproved it at every prime and every height of at least two[5]. But those examples fail the fp finiteness test. So they do not give an fp counterexample[6]. The paper claims to disprove the Hahn–Wilson conjecture at height two[7]. For each large enough prime, it builds a spectrum X of exact fp-type two. Yet X cannot be built from BP⟨2⟩ in finitely many steps. The result holds for sufficiently large primes[8]. The spectrum X also passes both localization comparisons. Lee and Pstrągowski had suggested such a hypothesis might be enough. The paper says this extra hypothesis does not suffice at height two[9]. Why is X hard to catch? Its local pieces, seen one height at a time, keep strong finiteness properties. The obstruction lies in how those local pieces are joined[10]. So any local test will pass. Only the global gluing shows that X is not finitely built. The construction uses chromatic algebraicity. At large primes, spectra can be modelled by sheaves on a stack of formal group laws. The paper uses Pstrągowski's comparison, available at height two for p > 8[11]. Its own operator needs larger primes still[12]. It builds the example by gluing a complete part to an open part, through a carefully chosen map on the overlap. How sure is this? No Lean check covers this paper. No specialist review has been published. The proof relies on deep comparison theorems, so the counterexample remains a claim. **What it doesn't show.** This does not cover small primes, and it does not address the conjecture at height three or above. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Solid if correct. No Lean check and no specialist review; the counterexample depends on deep comparison theorems and holds only for large primes. ## Words used - **Spectrum:** The stable version of a space; the basic object of stable homotopy theory. - **BP⟨n⟩:** A truncated Brown–Peterson spectrum; its coefficients are polynomials in v₁ to vₙ over the p-adic integers. - **Thick subcategory:** Everything you can build from given objects using finite sums, shifts, cones and retracts, finitely many times. - **Steenrod algebra:** A standard ring of operations that acts on mod-p cohomology of every space or spectrum. - **fp spectrum:** A spectrum whose mod-p cohomology is finitely presented over the Steenrod algebra. - **fp-type:** A height attached to an fp spectrum, measured by which finite spectra make its homotopy finite. - **Height:** A level in chromatic homotopy theory; higher heights see finer information. - **Telescope conjecture:** A prediction that two ways of localizing at a height agree; it is now known to fail at heights two and above. ## The original paper - [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf) - September 26, 2026 ## Quoted lines from the paper (12) [1] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > The Hahn--Wilson conjecture proposes a cohomological answer: finite presentation over the Steenrod algebra, together with a chromatic height bound, should characterize precisely these finite constructions. [2] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > Mahowald and Rezk prove that this is equivalent to requiring $\pi_*(V\wedge X)$ to be finite in total for some nonzero finite $p$-local spectrum $V$. [3] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > Its written formulation appears in Lee--Pstr\k{a}gowski, who report learning it from Dylan Wilson in 2021. [4] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > The cases of fp-type at most $-1$ and $0$ are classical, and Lee--Pstr\k{a}gowski prove the global generation conjecture at height one. [5] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > Burklund, Hahn, Levy, and Schlank subsequently disproved the unrestricted telescope conjecture at every prime and every height at least two. [6] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > Thus those counterexamples do not supply the fp counterexample studied here. [7] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > We disprove the Hahn--Wilson conjecture at height two. [8] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > The theorem refutes the universally quantified conjecture (…); its prime range is sufficiently large primes. [9] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > Theorem … shows that this additional hypothesis does not suffice at height two. [10] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > Their chromatic localizations retain strong finiteness properties; the obstruction lies in how the local pieces are joined. [11] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > We choose Pstr\k{a}gowski's homology-compatible triangulated comparison, which at height two is available for $p>8$. [12] From [Counterexamples to the Hahn-Wilson conjecture at height two](https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf): > This bound concerns the comparison; the operator constructed below requires larger primes. ## Other sources - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” ## Related - [Chromatic splitting: filtrations and counterexamples](https://closedform.nishachay.tech/e/openai-math-2026-318/) - Topology - [Cyclic length and chromatic fixed-point loss](https://closedform.nishachay.tech/e/openai-math-2026-314/) - Topology - [Failure of rational injectivity for maximal coarse assembly](https://closedform.nishachay.tech/e/openai-math-2026-307/) - Topology - [QAOA attains the SK optimum in the thermodynamic-first limit](https://closedform.nishachay.tech/e/openai-math-2026-281/) - Mathematical physics - [Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample](https://closedform.nishachay.tech/e/openai-math-2026-353/) - Differential geometry - [Topology](https://closedform.nishachay.tech/science/mathematics/topology/) - Field · 18 results - [Graph theory](https://closedform.nishachay.tech/topic/graph-theory/) - Topic · 40 results - [Open since the 2020s](https://closedform.nishachay.tech/open-since/2020s/) - Era · 9 results - [Claimed, not yet checked](https://closedform.nishachay.tech/trust/claimed/) - Trust · 130 results - [Solid results](https://closedform.nishachay.tech/significance/solid/) - Significance · 50 results - [Report an error](https://github.com/nishachay/closed-form/issues/new?template=report-error.yml&labels=entry-error&title=%5Bentry%20319%5D%20)