Krauth-Mezard Storage Capacity of the Ising Perceptron
Krauth and Mezard predicted in 1989 that the storage capacity of the Ising perceptron at zero margin is an explicit constant . Ding and Sun proved the matching lower bound and Huang the upper bound, but each was conditional on a global sign condition nobody had verified. Both conditions now hold rigorously, so in probability with .
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Computation
- Field
- Probability
- Posed by
- Werner Krauth, Marc Mezard
- Year posed
- 1989
- Years open
- 37y
- Solved
- 2026-07-13
- Model
- Claude Fable 5, Codex
- Vendor
- Anthropic / OpenAI
- Collaborators
- Yitzchak Shmalo
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
the last conditional step in the Ding-Sun and Huang bounds is discharged
What the AI did
The declaration is among the most explicit in the catalog: most of the work reported was done by two AI systems. Fable 5 developed the main mathematical architecture, named as the moment-coordinate reduction, the ray majorants and their pinned identities, the bound, the sublevel-set localization and the corrected near-one chain, and wrote the bulk of the proof code and the initial manuscript. Codex ran a separate internal re-audit of the final source and proof boundary and validated the repaired verifier blocks.
Verification
A computer-assisted proof using Arb interval arithmetic. The paper is unusually candid about the failure modes: two formula-level errors, a too-tight ball rule and a sign error in a tilted constraint term, survived until a 30-digit quadrature and a finite-difference cross-check caught them, and the author notes that Arb guarantees the arithmetic and not the formulas. All reported certificates postdate both corrections. arXiv preprint, not peer-reviewed, and we have not re-run the certificates.