{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:ising-perceptron-storage-capacity",
  "title": "Krauth-Mezard Storage Capacity of the Ising Perceptron",
  "canonical_statement": "Krauth and Mezard predicted in 1989 that the storage capacity of the Ising perceptron at zero margin is an explicit constant $\\alpha_\\star \\approx 0.8330786$. 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 $M_N/N \\to \\alpha_\\star$ in probability with $\\alpha_\\star \\in [0.833078599, 0.833078600]$.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:ising-perceptron-storage-capacity:event",
      "problem_id": "vibemathed:ising-perceptron-storage-capacity",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-13T00:00:00.000Z",
      "title": "arXiv:2607.11010 - The storage capacity of the Ising perceptron: verification of the outstanding numerical conditions",
      "summary": "the last conditional step in the Ding-Sun and Huang bounds is discharged",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:ising-perceptron-storage-capacity:attempt"
      ],
      "method_family_ids": [
        "vibemathed:ising-perceptron-storage-capacity:method"
      ],
      "source_assertion_ids": [
        "vibemathed:ising-perceptron-storage-capacity:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:ising-perceptron-storage-capacity:attempt",
      "problem_id": "vibemathed:ising-perceptron-storage-capacity",
      "model": "Claude Fable 5, Codex",
      "provider": "Anthropic / OpenAI",
      "attempted_at": "2026-07-13T00:00:00.000Z",
      "human_collaborators": [
        "Yitzchak Shmalo"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "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.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv:2607.11010 - The storage capacity of the Ising perceptron: verification of the outstanding numerical conditions",
          "url": "https://arxiv.org/abs/2607.11010",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Krauth-Mezard Storage Capacity of the Ising Perceptron",
          "url": "https://vibemathed.com/problem/ising-perceptron-storage-capacity",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:ising-perceptron-storage-capacity:verification",
      "problem_id": "vibemathed:ising-perceptron-storage-capacity",
      "solution_event_id": "vibemathed:ising-perceptron-storage-capacity:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "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.",
      "sources": [
        {
          "label": "VibeMathed: Krauth-Mezard Storage Capacity of the Ising Perceptron",
          "url": "https://vibemathed.com/problem/ising-perceptron-storage-capacity",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2607.11010 - The storage capacity of the Ising perceptron: verification of the outstanding numerical conditions",
          "url": "https://arxiv.org/abs/2607.11010",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2607.11010 - The storage capacity of the Ising perceptron: verification of the outstanding numerical conditions",
      "url": "https://arxiv.org/abs/2607.11010",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Krauth-Mezard Storage Capacity of the Ising Perceptron",
      "url": "https://vibemathed.com/problem/ising-perceptron-storage-capacity",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:ising-perceptron-storage-capacity:signal:watch_only",
      "problem_id": "vibemathed:ising-perceptron-storage-capacity",
      "kind": "watch_only",
      "reason": "Current public evidence does not meet the default replay threshold.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}