{
 "what": "AlphaProof Nexus's nine Erdős proofs read against the formal statements they prove: the proved theorem equal to formal-conjectures' at the commit the repository builds against; the fixed definitions verbatim outside the agent's EVOLVE blocks; no EVOLVE declaration shadowing a statement identifier; no kernel-bypassing construct; and what formal-conjectures did to each statement afterwards. Textual and exact; the elaborated statements are Comparator's (the kernel lane).",
 "decider": "54c76d9a42edecc2e7678ed279415744930ee3271ddb0924e9f152c12eb307cf",
 "pins": "corpus/alphaproof-nexus/PINS.json",
 "apnCommit": "0647711a71183c1ea492ad60860776617ce1ea88",
 "fcPin": "7862bf6a6976c26acb086b6b2cec12ad1d92ce5d",
 "fcHead": "e30c4c3defa569bb11dd45abfd71bfc2ead85669",
 "rows": [
  {
   "theorem": "erdos_12.parts.i",
   "answer": "True",
   "S1": true,
   "S2": true,
   "fixedDeclarations": 1,
   "S3": true,
   "S4": true,
   "S5": "unchanged",
   "problem": 12,
   "kind": "part"
  },
  {
   "theorem": "erdos_12.parts.ii",
   "answer": "False",
   "S1": true,
   "S2": true,
   "fixedDeclarations": 1,
   "S3": true,
   "S4": true,
   "S5": "unchanged",
   "problem": 12,
   "kind": "part"
  },
  {
   "theorem": "erdos_125.variants.positive_lower_density",
   "answer": "False",
   "S1": true,
   "S2": true,
   "fixedDeclarations": 2,
   "S3": true,
   "S4": true,
   "S5": "unchanged",
   "problem": 125,
   "kind": "variant"
  },
  {
   "theorem": "erdos_138.variants.difference",
   "answer": "True",
   "S1": true,
   "S2": true,
   "fixedDeclarations": 3,
   "S3": true,
   "S4": true,
   "S5": "unchanged",
   "problem": 138,
   "kind": "variant"
  },
  {
   "theorem": "erdos_152",
   "answer": "True",
   "S1": true,
   "S2": true,
   "fixedDeclarations": 1,
   "S3": true,
   "S4": true,
   "S5": "unchanged",
   "problem": 152,
   "kind": "main"
  },
  {
   "theorem": "erdos_26.variants.tenenbaum",
   "answer": "False",
   "S1": true,
   "S2": true,
   "fixedDeclarations": 4,
   "S3": true,
   "S4": true,
   "S5": "unchanged",
   "problem": 26,
   "kind": "variant"
  },
  {
   "theorem": "erdos_741.parts.i",
   "answer": "False",
   "S1": true,
   "S2": true,
   "fixedDeclarations": 0,
   "S3": true,
   "S4": true,
   "S5": "changed",
   "headStatement": ": ∀ A : Set ℕ, 0 < upperDensity (A + A) → ∃ A₁ A₂, A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ 0 < upperDensity (A₁ + A₁) ∧ 0 < upperDensity (A₂ + A₂)",
   "pinStatement": ": answer(False) ↔ ∀ A : Set ℕ, HasPosDensity (A + A) → ∃ A₁ A₂, A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ HasPosDensity (A₁ + A₁) ∧ HasPosDensity (A₂ + A₂)",
   "problem": 741,
   "kind": "part"
  },
  {
   "theorem": "erdos_741.parts.ii",
   "answer": "True",
   "S1": true,
   "S2": true,
   "fixedDeclarations": 0,
   "S3": true,
   "S4": true,
   "S5": "unchanged",
   "problem": 741,
   "kind": "part"
  },
  {
   "theorem": "erdos_846",
   "answer": "False",
   "S1": true,
   "S2": true,
   "fixedDeclarations": 2,
   "S3": true,
   "S4": true,
   "S5": "unchanged",
   "problem": 846,
   "kind": "main"
  }
 ],
 "counts": {
  "statements": 9,
  "problemNumbers": 7,
  "problems": [
   12,
   26,
   125,
   138,
   152,
   741,
   846
  ],
  "parts": 4,
  "variants": 3,
  "S1": 9,
  "S2": 9,
  "S3": 9,
  "S4": 9,
  "changedAtHead": [
   "erdos_741.parts.i"
  ]
 },
 "redControls": [
  {
   "id": "statement-weakened",
   "what": "ε > 0 replaced by ε ≥ 0 in the proved statement",
   "caughtBy": "S1",
   "fired": true
  },
  {
   "id": "answer-flipped",
   "what": "answer(False) replaced by answer(True)",
   "caughtBy": "S1",
   "fired": true
  },
  {
   "id": "definition-changed",
   "what": "the definition of NonTrilinearFor weakened (≤ to <)",
   "caughtBy": "S2",
   "fired": true
  },
  {
   "id": "shadowing-def",
   "what": "a definition named WeaklyNonTrilinear declared inside an EVOLVE block",
   "caughtBy": "S3",
   "fired": true
  },
  {
   "id": "axiom-planted",
   "what": "an axiom declared inside an EVOLVE block",
   "caughtBy": "S4",
   "fired": true
  }
 ]
}