{
  "status": "passed",
  "scope": "Finite evidence: exhaustive RM words for k=1..4; two full greedy-cover examples; n=8 quotient/decoder checked for all 65536 pairs u,v; exact integer covering conditions at n=16,32,64,128,256. No formal or asymptotic verification; no large Cohen witness synthesized.",
  "RM_checks": [
    {
      "k": 1,
      "length": 2,
      "all_words": 4,
      "radius": 0,
      "bound": 0
    },
    {
      "k": 2,
      "length": 4,
      "all_words": 16,
      "radius": 1,
      "bound": 1
    },
    {
      "k": 3,
      "length": 8,
      "all_words": 256,
      "radius": 2,
      "bound": 2
    },
    {
      "k": 4,
      "length": 16,
      "all_words": 65536,
      "radius": 6,
      "bound": 6
    }
  ],
  "greedy_examples": [
    {
      "S": 8,
      "dimension_budget": 5,
      "R": 3,
      "uncovered_sizes": [
        163,
        70,
        0
      ]
    },
    {
      "S": 10,
      "dimension_budget": 6,
      "R": 3,
      "uncovered_sizes": [
        848,
        672,
        360,
        96,
        0
      ]
    }
  ],
  "constructed_encoding": {
    "n": 8,
    "S": 72,
    "query_pairs": 65536,
    "max_observed_cost": 28,
    "proved_bound": 32
  },
  "parameter_checks": [
    {
      "n": 16,
      "S": 272,
      "cohen_integer_sufficient_R": 114,
      "RM_bound": 128,
      "product_optimum": 128,
      "B": 1,
      "L": 256,
      "tail": 16
    },
    {
      "n": 32,
      "S": 1056,
      "cohen_integer_sufficient_R": 446,
      "RM_bound": 504,
      "product_optimum": 512,
      "B": 2,
      "L": 512,
      "tail": 32
    },
    {
      "n": 64,
      "S": 4160,
      "cohen_integer_sufficient_R": 1818,
      "RM_bound": 2016,
      "product_optimum": 2048,
      "B": 4,
      "L": 1024,
      "tail": 64
    },
    {
      "n": 128,
      "S": 16512,
      "cohen_integer_sufficient_R": 7467,
      "RM_bound": 8072,
      "product_optimum": 8192,
      "B": 8,
      "L": 2048,
      "tail": 128
    },
    {
      "n": 256,
      "S": 65792,
      "cohen_integer_sufficient_R": 30585,
      "RM_bound": 32416,
      "product_optimum": 32768,
      "B": 15,
      "L": 4096,
      "tail": 4352
    }
  ]
}