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Phys-61-A1 · revision r1 · on Phys-61

OpenAI claim summary for Phys-61

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 hard-sphere ground state is claimed to have the full scaled Bogoliubov momentum law, with depletion $\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$. 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 Bogoliubov momentum law for the dilute hard-sphere ground state
  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 Bogoliubov momentum law for the dilute hard-sphere ground state

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The hard-sphere ground state is claimed to have the full scaled Bogoliubov momentum law, with depletion $\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$.

Mathematical statement of the claimed result

Result 1.
For the three-dimensional periodic hard-sphere Bose gas of exclusion distance $a>0$, let $\mathcal G_{N,L,a}$ be all trace-one states supported on the lowest eigenspace, and $n_{\Gamma,L}(k)$ the occupation of $L^{-3/2}e^{ik\cdot x}$. Define \[ g(t)=\frac{|t|^2+1}{2\sqrt{|t|^4+2|t|^2}}-\frac12,\qquad d\nu_{\mathrm{Bog}}(t)=\frac{(8\pi)^{3/2}}{(2\pi)^3}g(t)\,dt. \] For each bounded continuous real $f$ and $\varepsilon>0$, the source gives $\rho_0(a,f,\varepsilon)>0$. If $0<\rho<\rho_0$, $\eta=\rho a^3$, and $N_j,L_j\to\infty$, $N_j/L_j^3\to\rho$, then \[ \limsup_j\sup_{\Gamma\in\mathcal G_{N_j,L_j,a}} \left|\frac{\sum_{k\in(2\pi/L_j)\mathbb Z^3,\ k\ne0} f(k/\sqrt{8\pi\rho a})n_{\Gamma,L_j}(k)}{N_j\sqrt\eta} -\int f\,d\nu_{\mathrm{Bog}}\right|\le\varepsilon. \] The measure has total mass $8/(3\sqrt\pi)$.

Outline of the argument

Local energy estimates keep excitations instead of discarding them. One-body covariance and short-trajectory certificates are compared through smoothed deletion measures, while random encounter routes connect local momentum data to the thermodynamic occupation measure before the dilute limit.

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