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Phys-58-A1 · revision r1 · on Phys-58

OpenAI claim summary for Phys-58

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.

The same leading depletion coefficient is claimed for every fixed bounded nonnegative radial finite-range potential of positive scattering length, uniformly over ground-state density matrices. 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. The universal quantum-depletion constant for fixed bounded repulsions
  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

The universal quantum-depletion constant for fixed bounded repulsions

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The same leading depletion coefficient is claimed for every fixed bounded nonnegative radial finite-range potential of positive scattering length, uniformly over ground-state density matrices.

Mathematical statement of the claimed result

Result 1.
Let $v\ge0$ be bounded, measurable, radial, compactly supported, and nonzero on a positive-measure set. Its scattering length $a_v>0$ is determined by $(-\Delta+v/2)f=0$, with $f(x)=1-a_v/|x|$ outside the support. For the periodic symmetric Hamiltonian $-\sum_i\Delta_i+\sum_{i<j}v_L(x_i-x_j)$, let $\mathcal G^v_{N,L}$ contain all ground-space trace-one states and $B_\Gamma=N^{-1}\langle L^{-3/2},\gamma^{(1)}_\Gamma L^{-3/2}\rangle$. For every $\varepsilon>0$, some $\rho_0(v,\varepsilon)>0$ satisfies \[ 0<\rho<\rho_0,\ N_k,L_k\to\infty,\ N_k/L_k^3\to\rho \ \Longrightarrow\ \limsup_k\sup_{\Gamma\in\mathcal G^v_{N_k,L_k}} \left|\frac{1-B_\Gamma}{\sqrt{\rho a_v^3}}-\frac8{3\sqrt\pi}\right|\le\varepsilon. \] Volume and particle number tend to infinity while density remains fixed.

Outline of the argument

Local density tails and ground-state path bounds control occupations at the relevant scattering scale. Spatial comparison transports these local estimates to the constant orbital, identifying every thermodynamic accumulation value before density tends to zero.

Formal verification

For family 267, the selected statement is a density-uniform hard-sphere ground-state condensate lower bound for pure and mixed ground states. Positive-temperature condensation and quantum-depletion asymptotics are beyond it. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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