[
  {
    "local_id": "C1",
    "type": "empirical",
    "core": true,
    "statement": "The least x at which the primes up to x congruent to 1 modulo 4 outnumber those congruent to 3 modulo 4 is 26861.",
    "evidence": [{"result": "R1.mod4.first_loser_lead", "produced_by": "code/races.py"}],
    "depends_on": [],
    "falsified_if": "Some x below 26861 has more primes up to x congruent to 1 than to 3 modulo 4, or 26861 doesn't.",
    "confidence": 0.99
  },
  {
    "local_id": "C2",
    "type": "empirical",
    "core": true,
    "statement": "Of the integers x from 1 to 100000000, the primes congruent to 1 modulo 4 lead at exactly 30624, in 128 separate stretches, the last ending at x = 12382326, and the two classes are tied at exactly 3866 integers, the last being x = 12424002.",
    "evidence": [
      {"result": "R1.mod4.integers_loser_leads", "produced_by": "code/races.py"},
      {"result": "R1.mod4.loser_stretches", "produced_by": "code/races.py"},
      {"result": "R1.mod4.last_loser_lead", "produced_by": "code/races.py"},
      {"result": "R1.mod4.integers_tied", "produced_by": "code/races.py"},
      {"result": "R1.mod4.last_tie", "produced_by": "code/races.py"}
    ],
    "depends_on": ["C1"],
    "falsified_if": "An exact count of the race modulo 4 up to 100000000 gives any of these five numbers differently.",
    "confidence": 0.98
  },
  {
    "local_id": "C3",
    "type": "empirical",
    "core": true,
    "statement": "Weighting each integer x from 1 to 100000000 by 1/x, the primes congruent to 1 modulo 4 lead on a share 0.00040587 of the race, about a tenth of the limiting share of about 0.0041 implied by Rubinstein and Sarnak's logarithmic density of 0.9959 for the other side.",
    "evidence": [{"result": "R1.mod4.log_weighted_share_loser_leads", "produced_by": "code/races.py", "tolerance": 0.000000002}],
    "depends_on": ["C2"],
    "falsified_if": "The sum of 1/x over the integers up to 100000000 at which primes congruent to 1 modulo 4 lead, divided by the sum of 1/x over all of them, lies outside 0.000405868 to 0.000405872.",
    "confidence": 0.97
  },
  {
    "local_id": "C4",
    "type": "empirical",
    "core": true,
    "statement": "For every integer x from 2 to 100000000, the primes up to x congruent to 2 modulo 3 strictly outnumber those congruent to 1 modulo 3, the two classes being tied only at x = 1.",
    "evidence": [
      {"result": "R1.mod3.integers_loser_leads", "produced_by": "code/races.py"},
      {"result": "R1.mod3.integers_tied", "produced_by": "code/races.py"},
      {"result": "R1.mod3.last_tie", "produced_by": "code/races.py"}
    ],
    "depends_on": [],
    "falsified_if": "Some x from 2 to 100000000 has at least as many primes up to x congruent to 1 as to 2 modulo 3.",
    "confidence": 0.98
  },
  {
    "local_id": "C5",
    "type": "empirical",
    "core": false,
    "statement": "Up to 100000000 there are 2880950 primes congruent to 3 modulo 4 and 2880504 congruent to 1, and 2880937 primes congruent to 2 modulo 3 and 2880517 congruent to 1.",
    "evidence": [
      {"result": "R1.mod4.primes_in_winner_class", "produced_by": "code/races.py"},
      {"result": "R1.mod4.primes_in_loser_class", "produced_by": "code/races.py"},
      {"result": "R1.mod3.primes_in_winner_class", "produced_by": "code/races.py"},
      {"result": "R1.mod3.primes_in_loser_class", "produced_by": "code/races.py"}
    ],
    "depends_on": [],
    "falsified_if": "A count of the primes up to 100000000 in these four residue classes gives any count differently.",
    "confidence": 0.99
  }
]