Result 270, Mathematical physics

Threshold and positive-energy bound states of the BFSS matrix model

Proves that the undeformed relative SU(N)\mathrm{SU}(N) BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For SU(2)\mathrm{SU}(2), a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper's exclusion of additional bound states at N = 2.

Proof

The bigger picture

Why it matters

A quantum model can have just one bound state at zero energy yet infinitely many at higher energies. Two unreviewed manuscripts report this contrast for the BFSS matrix model, challenging an earlier claim about its bound states.

What changes?

The relative model removes center-of-mass motion; normalizable states have finite total probability. For undeformed SU(N) BFSS matrix quantum mechanics, the first manuscript claims exactly one normalizable zero-energy state for every finite matrix size N at least 2. For SU(2) only, the companion claims infinitely many normalizable positive-energy eigenstates, with energies tending to infinity. Its Hamiltonian, the energy operator, is defined by closing the gauge-invariant supercharge form.

What does that help mathematicians do?

Together, these claims distinguish uniqueness at zero energy from uniqueness among all bound states. At N = 2, the reported zero-energy state would coexist with infinitely many normalizable states at higher energies. This would rule out the original BFSS paper's exclusion of those additional states. Researchers describing the model's bound states would therefore need more than its zero-energy sector, even at this smallest covered matrix size.

Are there practical applications?

The immediate value is foundational mathematical physics. If established, these results would give further work on BFSS definite constraints: preserve the unique zero-energy state at every finite N at least 2 and account for infinitely many higher-energy bound states at N = 2. The sources demonstrate no computational method or practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The unique threshold bound state of the SU(N) BFSS model

September 24, 2026 48 pages

For every finite N ≥ 2, we prove that undeformed SU(N)\mathop{\mathrm{SU}}\nolimits (N) BFSS matrix quantum mechanics has exactly one normalizable zero-energy state after removing the center of mass. This establishes the threshold-bound-state conjecture.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026,
  author = {{OpenAI}},
  title = {{The unique threshold bound state of the $\mathrm{SU}(N)$ BFSS model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf}{OAI:The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026}},
  year = {2026}
}

Positive eigenvalues of the relative SU(2) BFSS Hamiltonian

October 5, 2026 9 pages

We prove that the relative SU(2)\mathop{\mathrm{SU}}\nolimits (2) BFSS Hamiltonian has infinitely many positive eigenvalues tending to infinity, with square-integrable eigenvectors. The operator is defined by closing the gauge-invariant supercharge form. This refutes, at N = 2, the exclusion of normalizable positive-energy states stated in the original BFSS paper.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026,
  author = {{OpenAI}},
  title = {{Positive eigenvalues of the relative $\mathrm{SU}(2)$ BFSS Hamiltonian}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf}{OAI:Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.