{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:separation-of-ordinary-and-strong-kreiss-constants",
  "title": "Separation Between the Ordinary and Strong Kreiss Constants",
  "canonical_statement": "Question 6.1 of Chalmoukis, Tsikalas and Yakubovich asks how far the Power boundedness constant $P(T)$ of a matrix can exceed its ordinary Kreiss constant $K(T)$. Answered more strongly: for every $K > 1$ there are matrices whose Cayley transforms satisfy $K(C_h(A_{n,h})) \\le K$ while the strong Kreiss constant satisfies $K_s(C_h(A_{n,h})) \\ge \\tfrac{1}{2}Cn^{\\alpha_K}$ with $\\alpha_K = (K-1)/(C+K-1)$. Since $P(T)\\geq K_s(T)$, this solves the question. Moreover, since the Kreiss matrix theorem gives $K_s(T) \\le P(T) \\le edK(T)$ in dimension $d$, the exponent $\\alpha < 1$ is optimal up to an arbitrarily small power loss.",
  "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:separation-of-ordinary-and-strong-kreiss-constants:event",
      "problem_id": "vibemathed:separation-of-ordinary-and-strong-kreiss-constants",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-19T00:00:00.000Z",
      "title": "arXiv:2608.06272 - A solution to the inverse generator problem and related questions",
      "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_assisted",
      "attempt_ids": [
        "vibemathed:separation-of-ordinary-and-strong-kreiss-constants:attempt"
      ],
      "method_family_ids": [
        "vibemathed:separation-of-ordinary-and-strong-kreiss-constants:method"
      ],
      "source_assertion_ids": [
        "vibemathed:separation-of-ordinary-and-strong-kreiss-constants:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:separation-of-ordinary-and-strong-kreiss-constants:attempt",
      "problem_id": "vibemathed:separation-of-ordinary-and-strong-kreiss-constants",
      "model": "ChatGPT 5.6 Pro, Claude Fable",
      "provider": "OpenAI",
      "attempted_at": "2026-08-19T00:00:00.000Z",
      "human_collaborators": [
        "Emiel Lorist",
        "Martin Meyries",
        "Mark Veraar"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "The same disclosure as the inverse generator entry, since both fall to one paper: ChatGPT 5.6 Pro explored Schauder basis counterexamples, assisted the adaptation of Ansorena's work that produces the explicit basis in Proposition 2.1, and helped optimize the explicit constants. The formalization in Lean 4 was done using\nClaude Fable by Anthropic. Note the boundary honestly - the disclosure names Proposition 2.1 and Theorem 1.1, not Theorem 1.4. Proposition 2.1 is the finite-dimensional construction every result in the paper is deduced from, including this one, so the model is in the loop for the machinery rather than for this theorem's derivation.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Chalmoukis, Tsikalas and Yakubovich, Operators with small Kreiss constants (Question 6.1)",
          "url": "https://arxiv.org/abs/2512.10025",
          "kind": "problem-record",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "arXiv:2608.06272 - A solution to the inverse generator problem and related questions",
          "url": "https://arxiv.org/abs/2608.06272",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Separation Between the Ordinary and Strong Kreiss Constants",
          "url": "https://vibemathed.com/problem/separation-of-ordinary-and-strong-kreiss-constants",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:separation-of-ordinary-and-strong-kreiss-constants:verification",
      "problem_id": "vibemathed:separation-of-ordinary-and-strong-kreiss-constants",
      "solution_event_id": "vibemathed:separation-of-ordinary-and-strong-kreiss-constants:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Checked by this site on 22 August 2026 against the v2 PDF (arXiv:2608.06272v2, 19 Aug): the paper states \"We furthermore note that Theorem 1.4(i) solves [6, Question 6.1]\" and gives the explicit constants quoted in the statement, and reference [6] is Chalmoukis, Tsikalas and Yakubovich, arXiv:2512.10025. The mathematics of Theorem 1.4 was not checked here, though a curator numerical check of Proposition 2.1 - the construction it is deduced from - was carried out for the sibling entry and confirmed its bounds up to n = 256. \nA lean certificate of Theorem 1.1 was added in v3 of the paper (arXiv:2608.06272v2, 27 Aug)",
      "sources": [
        {
          "label": "VibeMathed: Separation Between the Ordinary and Strong Kreiss Constants",
          "url": "https://vibemathed.com/problem/separation-of-ordinary-and-strong-kreiss-constants",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "arXiv:2608.06272 - A solution to the inverse generator problem and related questions",
          "url": "https://arxiv.org/abs/2608.06272",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "arXiv:2608.06272 - A solution to the inverse generator problem and related questions",
      "url": "https://arxiv.org/abs/2608.06272",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Separation Between the Ordinary and Strong Kreiss Constants",
      "url": "https://vibemathed.com/problem/separation-of-ordinary-and-strong-kreiss-constants",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:separation-of-ordinary-and-strong-kreiss-constants:signal:watch_only",
      "problem_id": "vibemathed:separation-of-ordinary-and-strong-kreiss-constants",
      "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"
}