Simon's Problem 2: pure-point spectrum for the two-dimensional Anderson model at every disorder

For each fixed $h>0$ , the operator $H_v=\Delta+v$ on $\ell^2(\mathbb{Z}^2)$ with iid $v_x$ uniform on $[-h,h]$ almost surely has pure-point spectral type with a complete orthonormal eigenbasis, spectrum $[-4-h,4+h]$ and dense eigenvalues. The null set may depend on $h$ . Not shown: one event for all $h$ , bounds uniform in energy, exponential decay of eigenfunctions, dynamical localization, or other single-site laws such as Bernoulli.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Simon's Problem 2: pure-point spectrum for the two-dimensional Anderson model at every disorder · 2D Anderson localization (Simon's Problem 2)

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Attribution

VibeMathed
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Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Simon's Problem 2: pure-point spectrum for the two-dimensional Anderson model at every disorder

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the main theorem was read against Simon's Problem 2 as the manuscript quotes it; Simon's text itself was not opened. The proof was not refereed. Lean: lean/docs/261.md describes a comparator challenge (PlanarAndersonSpectrum.lean) that states only that the spectrum is [-4-h, 4+h], explicitly not the pure-point claim; it is not in formalization.yaml and its solution module is absent at the pinned commit, so it counts for nothing. Scope stated by the paper: uniform single-site law only; no dynamical localization or eigenfunction decay is claimed.

Source ↗

Same family: Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Simon's Problem 1: absolutely continuous spectrum for the weakly disordered Anderson model in dimension three and higher

witness · pending

paper link supplied by the VibeMathed registry.

Source ↗

Lean scope note for family 261 (supporting spectrum statement only)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean scope note for family 261 (supporting spectrum statement only)

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

Simon's Problem 2: pure-point spectrum for the two-dimensional Anderson model at every disorder parent of this event

Same family: Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder evidence for this event

This event attributed to Unreleased internal OpenAI model

Lean scope note for family 261 (supporting spectrum statement only) evidence for this event

VibeMathed record: Simon's Problem 2: pure-point spectrum for the two-dimensional Anderson model at every disorder evidence for this event

Simon's Problem 1: absolutely continuous spectrum for the weakly disordered Anderson model in dimension three and higher evidence for this event

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