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Phys-4-A3 · revision r1 · on Phys-4

OpenAI claim summary for Phys-4

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 at unsolvedmath.com: 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 each lattice dimension $d\ge3$, sufficiently small fixed iid uniform potential width gives a nonempty deterministic spectral window with nonzero purely absolutely continuous spectrum. This covers the requested suitable-width Anderson assertion. It does not add a transport or diffusion conclusion.

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

Manuscript

Contents
  1. Extended States in the Anderson 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

Extended States in the Anderson Model

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. For uniform iid Anderson site disorder in every fixed dimension at least three, sufficiently small fixed disorder is claimed to give purely absolutely continuous spectrum on a nonempty open interval.

Mathematical statement of the claimed result

Result 1.
For independent $V_x$ uniformly distributed on $[-1,1]$, consider \[ (H_{\lambda,\omega}\psi)(x)=-\sum_{j=1}^d[\psi(x+e_j)+\psi(x-e_j)]+\lambda V_x\psi(x) \] on $\ell^2(\mathbb Z^d)$. The claimed fixed constant $e_2\in(0,1/100)$ and thresholds $\lambda_d>0$ give, for each $d\ge3$ and $0<\lambda<\lambda_d$, almost surely \[ I_d\subset\sigma(H_{\lambda,\omega}),\qquad \chi_{I_d}(H_{\lambda,\omega})\ell^2(\mathbb Z^d)\subset\mathcal H_{\mathrm{ac}}(H_{\lambda,\omega}), \] where \[ I_3=(-6+e_2/2,-6+e_2),\qquad I_d=-2d+(1/200,1/100)\quad(d\ge4). \] The projection is nonzero; the assertion is purely absolutely continuous spectrum on these windows, for this uniform-distribution model.

Outline of the argument

A finite-volume matrix-state certificate is propagated through controlled ordinary sweeps. Local trials and clearing of pending components maintain the comparison forms needed for resolvent bounds, which identify an interval with nonzero absolutely continuous weight.

Formal verification

For family 261, the selected planar Anderson statement constructs the self-adjoint operator and gives its almost-sure spectral interval. Spectral type, including the claimed pure-point conclusion, is not contained in 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