{
  "study": "Independent learned-function sampling on the fixed cube benchmark",
  "date": "2026-09-10",
  "successCriterion": "Match every supplied cube example within 40 generations",
  "matchedControls": {
    "seeds": [
      0,
      1,
      2,
      3,
      4
    ],
    "population": 2000,
    "generationLimit": 40,
    "inheritedFunctions": [
      "multiply",
      "sum_to_n"
    ],
    "primitives": [
      "sum"
    ],
    "initialSubtreeDepth": 1,
    "terminalProbability": 0.01,
    "survivorFraction": 0.95,
    "calculationLimitMs": 100,
    "cubeTemplateSupplied": false
  },
  "arms": [
    {
      "name": "previous_sampler",
      "successfulSeeds": 0,
      "seedCount": 5,
      "firstMatchingGenerations": [
        null,
        null,
        null,
        null,
        null
      ],
      "recordedEvaluations": 400001,
      "evaluationsContainingSavedCall": 851,
      "calculationTimeouts": 0,
      "runs": [
        {
          "seed": 0,
          "accuracy": 0.3333333333333333,
          "first_matching_generation": null,
          "expression": "n"
        },
        {
          "seed": 1,
          "accuracy": 0.3333333333333333,
          "first_matching_generation": null,
          "expression": "n"
        },
        {
          "seed": 2,
          "accuracy": 0.5,
          "first_matching_generation": null,
          "expression": "\u03a3(k=\u03a3(k=n..1)[8]..8.87)[n]"
        },
        {
          "seed": 3,
          "accuracy": 0.5,
          "first_matching_generation": null,
          "expression": "\u03a3(i=0.28..\u03a3(i=2..n)[6])[n]"
        },
        {
          "seed": 4,
          "accuracy": 0.3333333333333333,
          "first_matching_generation": null,
          "expression": "n"
        }
      ]
    },
    {
      "name": "independent_helper_choice",
      "successfulSeeds": 5,
      "seedCount": 5,
      "firstMatchingGenerations": [
        14,
        18,
        11,
        15,
        9
      ],
      "recordedEvaluations": 400000,
      "evaluationsContainingSavedCall": 198158,
      "calculationTimeouts": 0,
      "runs": [
        {
          "seed": 0,
          "accuracy": 1,
          "first_matching_generation": 14,
          "expression": "multiply(multiply(n, n), n)"
        },
        {
          "seed": 1,
          "accuracy": 1,
          "first_matching_generation": 18,
          "expression": "multiply(multiply(n, n), n)"
        },
        {
          "seed": 2,
          "accuracy": 1,
          "first_matching_generation": 11,
          "expression": "multiply(n, multiply(n, n))"
        },
        {
          "seed": 3,
          "accuracy": 1,
          "first_matching_generation": 15,
          "expression": "multiply(multiply(n, n), n)"
        },
        {
          "seed": 4,
          "accuracy": 1,
          "first_matching_generation": 9,
          "expression": "multiply(n, multiply(n, n))"
        }
      ]
    }
  ],
  "interpretation": "Independent helper sampling improved construction on this fixed benchmark. The successes compose retained multiplication; they are not a proof of unrestricted arithmetic. Multiplication retains integer repetition-count semantics and the original execution cap."
}