Unbounded positive point spectrum of relative SU(2) BFSS

Unbounded positive point spectrum of relative SU$2$ BFSS

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The relative $SU(2)$ BFSS operator is claimed to have infinitely many normalizable positive-energy eigenstates with eigenvalues tending to infinity.

Mathematical statement of the claimed result

Result 1.

On $\mathcal H=[L^2(\mathbb R^{27};\mathcal F)]^{\operatorname{SU}(2)}$, use color basis structure constants $f_{abc}=\sqrt2\varepsilon_{abc}$, Clifford generators $\{\theta_\alpha^a,\theta_\beta^b\}=\delta_{\alpha\beta}\delta_{ab}$, and nine real gamma matrices. The relative BFSS Hamiltonian is defined by closing the invariant-core form \[ q(\Psi)=\frac1{16}\sum_{\alpha=1}^{16}\|Q_\alpha\Psi\|^2,\quad Q_\alpha=-i\gamma^i_{\alpha\beta}\theta_\beta^a\partial_{ia} +\tfrac12 f_{abc}x_i^bx_j^c\gamma^{ij}_{\alpha\beta}\theta_\beta^a, \quad\gamma^{ij}=\tfrac12[\gamma^i,\gamma^j]. \] The claim supplies orthonormal $\Psi_j\in\operatorname{Dom}H$ and $E_j>0$ with \[ H\Psi_j=E_j\Psi_j,\qquad E_j\to\infty. \] The positive point spectrum of this specified relative operator is therefore infinite. The nine $\gamma^i$ are real symmetric $16\times16$ matrices with $\{\gamma^i,\gamma^j\}=2\delta_{ij}I$ and $\gamma^{ij}=\tfrac12[\gamma^i,\gamma^j]$. The fermion space $\mathcal F$ is an irreducible complex Clifford module for the $48$ self-adjoint generators, so $\dim\mathcal F=2^{24}$.

Outline of the argument

The gauge-invariant supercharge form first defines the relative operator. Selected $\operatorname{Spin}(9)$ symmetry sectors exclude the lowest transverse oscillator levels; the remaining oscillator energy confines those sectors, giving compact spectral pieces with normalizable positive-energy eigenstates despite escape channels in other sectors.

Formal verification

Family 270 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

• OpenAI, Positive eigenvalues of the relative SU $2$ BFSS Hamiltonian, 2026-10-05; citation key OAI:Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI catalogue entry.