{
  "status": "Self-reviewed parameter proof, pending later independent review",
  "theorem": "For integers D=3L^2-K^2>0 and K>=max(100,10D^2), both checkerboard phases are globally optimal at their cardinalities.",
  "uniformConstantMargins": {
    "D_over_K_below_one_thirtieth": "1/9000",
    "local_epsilon_cap": "48829/9000000",
    "U_squared_below_seven_sixths_squared": "83/3000",
    "U_below_uniform_local_cap": "28300114529/2250000000000",
    "L_over_K_below_three_fifths": "239/9000",
    "n_below_K_squared_over_three": "757/30000",
    "packing_sqrt_coefficient": "33/700",
    "UD_below_three_over_K": "23/150",
    "covering_delta_above_eight_over_five_K": "472/1925",
    "full_K_step_angle_below_point_one": "587/12000",
    "projection_barrier": "74/125",
    "two_ended_angle_coefficient": "7/400",
    "zigzag_error_coefficient": "15967/192000",
    "endpoint_height_coefficient": "5/12",
    "universal_competitor_exclusion": "89/1000",
    "refined_corner_direction": "1044864217863264412820881/2000000000000000000000000"
  },
  "sequences": [
    {
      "defect": 2,
      "terms": [
        {
          "K": 19,
          "L": 11,
          "ns": [
            120
          ],
          "M2": "482/361",
          "theoremApplies": false
        },
        {
          "K": 71,
          "L": 41,
          "ns": [
            1512
          ],
          "M2": "6722/5041",
          "theoremApplies": false
        },
        {
          "K": 265,
          "L": 153,
          "ns": [
            20482
          ],
          "M2": "93634/70225",
          "theoremApplies": true
        },
        {
          "K": 989,
          "L": 571,
          "ns": [
            283140
          ],
          "M2": "1304162/978121",
          "theoremApplies": true
        },
        {
          "K": 3691,
          "L": 2131,
          "ns": [
            3935672
          ],
          "M2": "18164642/13623481",
          "theoremApplies": true
        },
        {
          "K": 13775,
          "L": 7953,
          "ns": [
            54787152
          ],
          "M2": "253000834/189750625",
          "theoremApplies": true
        }
      ],
      "asymptotic_n_times_gap": "1/12"
    },
    {
      "defect": 3,
      "terms": [
        {
          "K": 12,
          "L": 7,
          "ns": [
            52
          ],
          "M2": "193/144",
          "theoremApplies": false
        },
        {
          "K": 45,
          "L": 26,
          "ns": [
            621
          ],
          "M2": "2701/2025",
          "theoremApplies": false
        },
        {
          "K": 168,
          "L": 97,
          "ns": [
            8281
          ],
          "M2": "37633/28224",
          "theoremApplies": true
        },
        {
          "K": 627,
          "L": 362,
          "ns": [
            113982
          ],
          "M2": "524173/393129",
          "theoremApplies": true
        },
        {
          "K": 2340,
          "L": 1351,
          "ns": [
            1582516
          ],
          "M2": "7300801/5475600",
          "theoremApplies": true
        },
        {
          "K": 8733,
          "L": 5042,
          "ns": [
            22022781
          ],
          "M2": "101687053/76265289",
          "theoremApplies": true
        }
      ],
      "asymptotic_n_times_gap": "1/8"
    },
    {
      "defect": 11,
      "terms": [
        {
          "K": 17,
          "L": 10,
          "ns": [
            99
          ],
          "M2": "389/289",
          "theoremApplies": false
        },
        {
          "K": 64,
          "L": 37,
          "ns": [
            1235
          ],
          "M2": "5465/4096",
          "theoremApplies": false
        },
        {
          "K": 239,
          "L": 138,
          "ns": [
            16680
          ],
          "M2": "76165/57121",
          "theoremApplies": false
        },
        {
          "K": 892,
          "L": 515,
          "ns": [
            230394
          ],
          "M2": "1060889/795664",
          "theoremApplies": false
        },
        {
          "K": 3329,
          "L": 1922,
          "ns": [
            3201795
          ],
          "M2": "14776325/11082241",
          "theoremApplies": true
        },
        {
          "K": 12424,
          "L": 7173,
          "ns": [
            44568475
          ],
          "M2": "205807705/154355776",
          "theoremApplies": true
        }
      ],
      "asymptotic_n_times_gap": "11/24"
    },
    {
      "defect": 12,
      "terms": [
        {
          "K": 24,
          "L": 14,
          "ns": [
            187,
            188
          ],
          "M2": "193/144",
          "theoremApplies": false
        },
        {
          "K": 90,
          "L": 52,
          "ns": [
            2411,
            2412
          ],
          "M2": "2701/2025",
          "theoremApplies": false
        },
        {
          "K": 336,
          "L": 194,
          "ns": [
            32857,
            32858
          ],
          "M2": "37633/28224",
          "theoremApplies": false
        },
        {
          "K": 1254,
          "L": 724,
          "ns": [
            454937,
            454938
          ],
          "M2": "524173/393129",
          "theoremApplies": false
        },
        {
          "K": 4680,
          "L": 2702,
          "ns": [
            6326371,
            6326372
          ],
          "M2": "7300801/5475600",
          "theoremApplies": true
        },
        {
          "K": 17466,
          "L": 10084,
          "ns": [
            88077347,
            88077348
          ],
          "M2": "101687053/76265289",
          "theoremApplies": true
        }
      ],
      "asymptotic_n_times_gap": "1/2"
    }
  ],
  "finitePrefixChecks": [
    {
      "n": 621,
      "candidateSquared": "2701/2025",
      "candidateApprox": 1.1549143520165586,
      "candidate": {
        "K": 45,
        "L": 26,
        "phase": 0,
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      "epsilonCap": "106907/500000000",
      "deltaLower": "10000000000000000000000000000000000000000000000/255058921571048445034083035587495477860430497489",
      "deltaUpper": "125000000000000000000000/2635582647597399071436417",
      "wallSitesMinimum": 19,
      "constants": {
        "A": "814263053397448309615661/7500000000000000000000000",
        "beta": "320721/1000000000",
        "B": "400222229330990590370836794754222587454258300449/400000000000000000000000000000000000000000000000",
        "S": "26/45"
      },
      "localChecks": {
        "wall_cone": "3899246911258517295428453/324000000000000000000000000",
        "five_angles": "1565985511794896619231319/1500000000000000000000000",
        "seven_angles": "3134857512589793238462637/3000000000000000000000000",
        "acute": "3137744001589793238462641/6000000000000000000000000",
        "wall_fan_one": "12556748984359172953850567/15000000000000000000000000",
        "wall_fan_five": "12542316539359172953850551/15000000000000000000000000",
        "corner_fan_three": "1565985511794896619231317/5000000000000000000000000",
        "mixed_modes": "290283297598358996639940615280211212344824841142802556840783672943631838293/1265625000000000000000000000000000000000000000000000000000000000000000000000",
        "same_sign_large": "6244698787179586476925283/7500000000000000000000000",
        "adjacent_straight": "661935302396435172502002791248648645016749683577857756264308172943631838293/3456000000000000000000000000000000000000000000000000000000000000000000000000",
        "adjacent_zigzag": "3077448453589793238462643/10000000000000000000000000",
        "corner_color": "3103106133589793238462643/10000000000000000000000000",
        "corner_direction": "1163664800096172464423491/3750000000000000000000000",
        "endpoint_monotonicity": "112115421311333905319930719565917724301765338002846379/113906250000000000000000000000000000000000000000000000",
        "epsilon_cap": "125640557076223926921963012545741699551/346410161513775458705489400000000000000000000000",
        "h_lt_quarter": "3129138746124345934657611/14056440787186128380994224",
        "wall_sites_at_least_four": "16"
      },
      "rowBarrier": {
        "forbiddenSteps": 26,
        "angleUpper": "1175074178397448309615661/7500000000000000000000000",
        "projectionLower": "1651078057351616066230034856837446539347253189090056045724988376213625775009805001256004018950634641909559594669770088765634135970850806251991494684076507/1634220297292625827996698043446833818595980951196610351562500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "projectionMarginApprox": 0.010315475879793007
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      "representations": [
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          "family": "A",
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          ],
          "areaBoundSquared": "3293/2809",
          "result": "excluded by row projection barrier"
        },
        {
          "family": "A",
          "rows": [
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          ],
          "areaBoundSquared": "2701/2025",
          "result": "area bound reaches or exceeds candidate"
        }
      ],
      "status": "criterion passes (self-reviewed proof, not independent review)"
    },
    {
      "n": 1512,
      "candidateSquared": "6722/5041",
      "candidateApprox": 1.1547578024095568,
      "candidate": {
        "K": 71,
        "L": 41,
        "phase": 0,
        "rows": [
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      "epsilonCap": "11453/200000000",
      "deltaLower": "500000000000000000000000000000000000000000000000/19896731803750610674667235700212066988196958160903",
      "deltaUpper": "1000000000000000000000000/33460441422839345923428063",
      "wallSitesMinimum": 29,
      "constants": {
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        "beta": "34359/400000000",
        "B": "1000148782913299859608523782744807334605404488589/1000000000000000000000000000000000000000000000000",
        "S": "41/71"
      },
      "localChecks": {
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        "five_angles": "1569507864294896619231319/1500000000000000000000000",
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        "corner_direction": "1174231857596172464423491/3750000000000000000000000",
        "endpoint_monotonicity": "139663876143064239661866399623925382995210403934678519/141778125000000000000000000000000000000000000000000000",
        "epsilon_cap": "839780331169669159167085165394595511411/866025403784438646763723500000000000000000000000",
        "h_lt_quarter": "31150925818020232466435315/133841765691357383693712252",
        "wall_sites_at_least_four": "26"
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      "rowBarrier": {
        "forbiddenSteps": 41,
        "angleUpper": "947744438397448309615661/7500000000000000000000000",
        "projectionLower": "127280110780092924410509513324873001680432678136724656155834599124501956341388393789855267324961591181313847321859846509614172430108225392076268821852039/124373530263345804722267276513997114624199926156330585937500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "projectionMarginApprox": 0.023369767751970933
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      "representations": [
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          ],
          "areaBoundSquared": "8114/6889",
          "result": "excluded by row projection barrier"
        },
        {
          "family": "A",
          "rows": [
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          "areaBoundSquared": "6722/5041",
          "result": "area bound reaches or exceeds candidate"
        },
        {
          "family": "B0",
          "rows": [
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          ],
          "areaBoundSquared": "6053/3844",
          "result": "area bound reaches or exceeds candidate"
        },
        {
          "family": "B1",
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          "areaBoundSquared": "6053/3844",
          "result": "area bound reaches or exceeds candidate"
        }
      ],
      "status": "criterion passes (self-reviewed proof, not independent review)"
    }
  ],
  "constantRatioCounterexample": {
    "t": 545,
    "n": 12480228,
    "originalWindow": [
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    ],
    "improvedWindow": [
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    ],
    "originalM2": "193/144",
    "improvedM2": "57595250/43178041",
    "exactSquaredImprovement": "39645913/6217637904",
    "scope": "Analytic construction certificate; no millions-of-points array generated."
  }
}