{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:stepsize-acceleration-lower-bound",
  "title": "Lower Bounds for Stepsize-Based Acceleration of Gradient Descent",
  "canonical_statement": "Carefully designed stepsize schedules alone accelerate plain gradient descent beyond its textbook O(1/T) rate, without momentum. Whether they can reach the optimal O(T^-2) was open. A lower bound of Omega(T^-1.9319) for last-iterate convergence under predetermined nonnegative stepsize schedules says they cannot.",
  "plain_summary": "VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "partial",
  "solution_events": [
    {
      "id": "vibemathed:stepsize-acceleration-lower-bound:event",
      "problem_id": "vibemathed:stepsize-acceleration-lower-bound",
      "type": "proved",
      "status": "partial",
      "occurred_at": "2026-08-11T00:00:00.000Z",
      "title": "arXiv",
      "summary": "Recorded as partial: the bound is Omega(T^-1.9319) against an achievable O(T^-1.2716), so it rules out reaching the optimal rate without pinning down the true one.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:stepsize-acceleration-lower-bound:attempt"
      ],
      "method_family_ids": [
        "vibemathed:stepsize-acceleration-lower-bound:method"
      ],
      "source_assertion_ids": [
        "vibemathed:stepsize-acceleration-lower-bound:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:stepsize-acceleration-lower-bound:attempt",
      "problem_id": "vibemathed:stepsize-acceleration-lower-bound",
      "model": "GPT-5.6 Sol Pro",
      "provider": "OpenAI",
      "attempted_at": "2026-08-11T00:00:00.000Z",
      "human_collaborators": [
        "Jianhao Ma",
        "Yuxin Chen"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The abstract closes with it: \"The proof was developed by GPT-5.6 Sol Pro under the authors' guidance.\" The authors added material to make the proof correct and readable, and separately used Codex to formalize the proof in Lean 4.",
      "failed_routes": [],
      "artifacts": [],
      "sources": [
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.10418",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Lower Bounds for Stepsize-Based Acceleration of Gradient Descent",
          "url": "https://vibemathed.com/problem/stepsize-acceleration-lower-bound",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:stepsize-acceleration-lower-bound:verification",
      "problem_id": "vibemathed:stepsize-acceleration-lower-bound",
      "solution_event_id": "vibemathed:stepsize-acceleration-lower-bound:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "A preprint days old. A Lean 4 formalization by Codex is linked from the paper, but it is the same pipeline that produced the proof, so it is not independent confirmation.",
      "sources": [
        {
          "label": "VibeMathed: Lower Bounds for Stepsize-Based Acceleration of Gradient Descent",
          "url": "https://vibemathed.com/problem/stepsize-acceleration-lower-bound",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv",
          "url": "https://arxiv.org/abs/2608.10418",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv",
      "url": "https://arxiv.org/abs/2608.10418",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Lower Bounds for Stepsize-Based Acceleration of Gradient Descent",
      "url": "https://vibemathed.com/problem/stepsize-acceleration-lower-bound",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "expand",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:stepsize-acceleration-lower-bound:signal:recent_partial_result",
      "problem_id": "vibemathed:stepsize-acceleration-lower-bound",
      "kind": "recent_partial_result",
      "reason": "A source-reported partial result may support bounded expansion.",
      "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"
}