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Phys-45-A1 · revision r1 · on Phys-45

OpenAI claim summary for Phys-45

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

For every finite $N\geq2$, the relative undeformed $SU(N)$ BFSS Hamiltonian is claimed to have exactly one square-integrable zero-energy state. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Exactly one threshold bound state in every relative SU(N) BFSS model
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Exactly one threshold bound state in every relative SU(N) BFSS model

OpenAI's claim

OpenAI presents a proof in the selected manuscript. For every finite $N\geq2$, the relative undeformed $SU(N)$ BFSS Hamiltonian is claimed to have exactly one square-integrable zero-energy state.

Mathematical statement of the claimed result

Result 1.
For each $N\ge2$, use the gauge-invariant space $\mathcal H_N=[L^2(\mathbb R^{9(N^2-1)})\otimes\mathcal F_N]^{\operatorname{SU}(N)}$, with $\operatorname{Tr}(T_AT_B)=\delta_{AB}$, $[T_A,T_B]=if_{ABC}T_C$, and $\{\theta_\alpha^A,\theta_\beta^B\}=\delta_{\alpha\beta}\delta_{AB}$. The relative Hamiltonian closes the smooth invariant-core form \[ q_N(\Psi)=\frac1{16}\sum_\alpha\|Q_\alpha\Psi\|^2,\quad Q_\alpha=-i\partial_{iA}\gamma^i_{\alpha\beta}\theta_\beta^A +\tfrac12 f_{ABC}x_i^Bx_j^C\gamma^{ij}_{\alpha\beta}\theta_\beta^A. \] The asserted threshold-state count is \[ \dim\ker H_N=1. \] That kernel is $\operatorname{Spin}(9)$-invariant and even for the exterior grading whose polarization is $\gamma^1\gamma^2\gamma^3$, with the specified empty vector chosen even. 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 space $\mathcal F_N$ is the irreducible complex Clifford module for $16(N^2-1)$ self-adjoint fermions, of dimension $2^{8(N^2-1)}$. The polarization's empty vector is the minimizer of $\tfrac i2\theta^{\mathsf T}(\gamma^1\gamma^2\gamma^3)\theta$; it is declared even, fixing the grading's sign.

Outline of the argument

A massive deformation has a computable Fredholm index. Corrected profiles near separated centers and exterior estimates control the undeformed limit, while massive tightness prevents loss of states and the profile exhaustion fixes the kernel count after center-of-mass removal.

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