{
  "campaign": "CSM_RH_Campaign_07",
  "identity": {
    "name": "Vaughan identity",
    "formula": "Lambda = Lambda_<=V + mu_<=U*L - mu_<=U*Lambda_<=V*1 + mu_>U*Lambda_>V*1",
    "source": "Terence Tao, 254A Notes 3, Lemma 18, equation (32)",
    "source_url": "https://terrytao.wordpress.com/2015/01/10/254a-notes-3-the-large-sieve-and-the-bombieri-vinogradov-theorem/"
  },
  "cutoffs": {
    "X": "2N",
    "U": "floor(X^(1/3))",
    "V": "floor(X^(1/3))"
  },
  "blocks": [
    {
      "block_id": "V-SHORT",
      "formula": "sum_{n<=V} w_N(n)(Lambda(n)-1)A(n-1)",
      "trivial_exponent": "5/3+o(1)",
      "proved_exponent": "5/3+o(1)",
      "required_exponent": "<=3-kappa",
      "saving_source": "short support",
      "status": "CLOSED"
    },
    {
      "block_id": "V-I1",
      "formula": "sum_{d<=U} mu(d) sum_{V<dm<X} log(m) w_N(dm) A(dm-1)",
      "trivial_exponent": "3+o(1)",
      "proved_exponent": "3-o(1) only from known PNT subpower input; no fixed exponent",
      "required_exponent": "3-kappa if estimated separately",
      "saving_source": "OPEN",
      "unit_divisor_atom": true,
      "status": "CRITICAL"
    },
    {
      "block_id": "V-I2",
      "formula": "-sum_{r<=UV} alpha_UV(r) sum_{V<rm<X} w_N(rm)A(rm-1)",
      "trivial_exponent": "3+o(1)",
      "proved_exponent": "no fixed-power block saving certified",
      "required_exponent": "3-kappa if estimated separately",
      "status": "CRITICAL"
    },
    {
      "block_id": "V-II",
      "formula": "sum_{r>U} b_U(r) sum_{l>V,r*l<X} Lambda(l)w_N(r*l)A(r*l-1)",
      "trivial_exponent": "3+o(1)",
      "proved_exponent": "no fixed-power zero-frequency Type-II saving certified",
      "required_exponent": "3-kappa if estimated separately",
      "status": "CRITICAL"
    },
    {
      "block_id": "V-C",
      "formula": "-sum_{V<n<X} w_N(n)A(n-1)",
      "trivial_exponent": "3",
      "proved_exponent": "subpower only from known PNT error",
      "required_exponent": "3-kappa if estimated separately",
      "status": "CRITICAL"
    },
    {
      "block_id": "V-LONG",
      "formula": "V-I1 + V-I2 + V-II + V-C",
      "trivial_exponent": "3+o(1)",
      "proved_exponent": "exactly original long CSSA frontier",
      "required_exponent": "3-kappa",
      "status": "OPEN_SAME_STRENGTH"
    }
  ]
}