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The BFSS exclusion of positive-energy normalizable states in the relative SU(2) model

For the relative SU(2)\mathrm{SU}(2) BFSS Hamiltonian (supersymmetric quantum mechanics of three 99-vectors with quartic potential ∑a<b∣xa∧xb∣2\sum_{a<b}|x^a\wedge x^b|^2 and sixteen fermions per color), de Wit, Luscher and Nicolai (1989) proved the spectrum is [0,∞)[0,\infty) and left open whether normalizable eigenstates lie inside it. Banks, Fischler, Shenker and Susskind (1997, Section 5, after Equation (5.4)) stated that besides the zero-energy threshold states 'No other normalizable bound states can occur'. Danielsson, Ferretti and Sundborg had found metastable excited states in a Born-Oppenheimer approximation. Does the relative SU(2)\mathrm{SU}(2) Hamiltonian have normalizable eigenstates of positive energy?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Supersymmetric matrix quantum mechanics; embedded eigenvalues
Posed by
T. Banks, W. Fischler, S. H. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D (1997), Section 5; open question in de Wit, Luscher and Nicolai (1989)
Year posed
1997
Years open
29y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
25 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: the relative SU(2)\mathrm{SU}(2) BFSS Hamiltonian has infinitely many positive eigenvalues Ej→∞E_j\to\infty with square-integrable eigenvectors, so σp(H)∩(0,∞)\sigma_p(H)\cap(0,\infty) is infinite. This contradicts, for this exact operator at N=2N=2, the BFSS paper's sentence excluding other normalizable bound states. It concerns only N=2N=2, says nothing about N≥3N\ge3, and does not bear on the zero-energy state (the companion entry) or on large-NN matrix theory.

What the AI did

Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. No Lean formalization accompanies it.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against the BFSS sentence quoted in the manuscript. Theorem 1.1 gives an orthonormal sequence of eigenvectors in the domain of HH with eigenvalues Ej>0E_j>0, Ej→∞E_j\to\infty, for the closed gauge-invariant supercharge form at N=2N=2. The mechanism is symmetry: Spin(9)\mathrm{Spin}(9) sectors with large second highest-weight coordinate exclude low transverse oscillator levels, so those sectors are confined. The manuscript says the large-NN matrix-theory conjecture is a separate question. Not refereed here. No Lean formalization.

Sources

Changelog1 change

Discussion