{
  "status": "PASS_EXACT_FINITE_CHECKS",
  "scope": "Theorem4.1 prescribed stochastic insertion/mixing/deletion maps only. No LLM, human, ethical or causal-benefit inference.",
  "cases": {
    "zero_influence": {
      "result": {
        "theta": [
          "1/2",
          "1/2",
          "0"
        ],
        "qstar": null,
        "incumbent_mean": [
          "11/20",
          "9/20"
        ],
        "returned_mean": [
          "11/20",
          "9/20"
        ],
        "error": [
          "0",
          "0"
        ]
      },
      "conclusion": "Dissent need not move the endpoint when newcomer influence is zero; the positive-theta condition matters."
    },
    "connected_unequal_masses": {
      "result": {
        "theta": [
          "3/10",
          "2/5",
          "3/10"
        ],
        "qstar": [
          "2/3",
          "1/3"
        ],
        "incumbent_mean": [
          "11/20",
          "9/20"
        ],
        "returned_mean": [
          "11/20",
          "9/20"
        ],
        "error": [
          "0",
          "0"
        ]
      },
      "conclusion": "Even fully connected doubly stochastic mixing need not give q*=incumbent mean under unequal insertion masses."
    },
    "stay_not_monotone": {
      "results": [
        {
          "stay": 1,
          "theta": [
            "1/4",
            "1/4",
            "1/2"
          ],
          "qstar": [
            "11/20",
            "9/20"
          ],
          "incumbent_mean": [
            "11/20",
            "9/20"
          ],
          "returned_mean": [
            "13/20",
            "7/20"
          ],
          "error": [
            "1/10",
            "-1/10"
          ]
        },
        {
          "stay": 2,
          "theta": [
            "3/8",
            "3/8",
            "1/4"
          ],
          "qstar": [
            "11/20",
            "9/20"
          ],
          "incumbent_mean": [
            "11/20",
            "9/20"
          ],
          "returned_mean": [
            "3/5",
            "2/5"
          ],
          "error": [
            "1/20",
            "-1/20"
          ]
        },
        {
          "stay": 3,
          "theta": [
            "5/16",
            "5/16",
            "3/8"
          ],
          "qstar": [
            "11/20",
            "9/20"
          ],
          "incumbent_mean": [
            "11/20",
            "9/20"
          ],
          "returned_mean": [
            "5/8",
            "3/8"
          ],
          "error": [
            "3/40",
            "-3/40"
          ]
        }
      ],
      "conclusion": "With a fixed symmetric mixing matrix, endpoint error need not increase with stay length."
    },
    "inadmissible_restorer": {
      "result": {
        "theta": [
          "9/10",
          "0",
          "1/10"
        ],
        "qstar": [
          "-31/10",
          "41/10"
        ],
        "incumbent_mean": [
          "1/2",
          "1/2"
        ],
        "returned_mean": [
          "43/50",
          "7/50"
        ],
        "error": [
          "9/25",
          "-9/25"
        ]
      },
      "conclusion": "Required q* may lie outside the strictly positive simplex; restoration is not guaranteed."
    }
  },
  "not_proved": [
    "The general theorem by finite examples",
    "Auditor influence equals capture with opposite sign",
    "Founder influence or tenure ordering",
    "Succession is mathematically endpoint restoration"
  ]
}