{
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  "source": "vibemath",
  "problem_id": "vibemathed:k-distinct-4k-barrier",
  "title": "The 4^k Barrier for the k-Distinct Language",
  "canonical_statement": "Can the $k$-distinct language - words over $[n]$ of length at most $k$ with no repeated symbol - be recognized by an acyclic NFA of size $c^k n^{O(1)}$ for some $c < 4$? A construction of size $2^{1.96992k} n^{O(1)} < 3.918^k n^{O(1)}$ answers yes.",
  "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",
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    {
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      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-07-28T00:00:00.000Z",
      "title": "arXiv:2607.25381 - Breaking the 4^k barrier for the k-distinct language",
      "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.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:k-distinct-4k-barrier:attempt"
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      "method_family_ids": [
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  "attempts": [
    {
      "id": "vibemathed:k-distinct-4k-barrier:attempt",
      "problem_id": "vibemathed:k-distinct-4k-barrier",
      "model": "ChatGPT / Codex 5.4-5.6 Pro, Gemini 3.1 Pro",
      "provider": "OpenAI / Google DeepMind",
      "attempted_at": "2026-07-28T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The gadget-amplification framework - hashing symbols into many copies of a small local NFA gadget - was developed across ChatGPT/Codex 5.4-5.6 Pro and Gemini 3.1 Pro sessions.",
      "failed_routes": [],
      "artifacts": [],
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        },
        {
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      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "arXiv preprint with interval-arithmetic verification of the exponent and pinned verification code. Not yet peer-reviewed.",
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      "label": "arXiv:2607.25381 - Breaking the 4^k barrier for the k-distinct language",
      "url": "https://arxiv.org/abs/2607.25381",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: The 4^k Barrier for the k-Distinct Language",
      "url": "https://vibemathed.com/problem/k-distinct-4k-barrier",
      "kind": "source-assertion",
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  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
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  "generated_at": "2026-09-13T16:28:40.178Z"
}