{
  "prefix": {
    "pe": "https://proofengine.info/proofs/the-positive-root-of-x-x-1-0-is-exactly-1-5-2/",
    "prov": "http://www.w3.org/ns/prov#",
    "xsd": "http://www.w3.org/2001/XMLSchema#",
    "schema": "http://schema.org/"
  },
  "entity": {
    "pe:claim": {
      "prov:type": "pe:Claim",
      "prov:label": "The positive root of x\u00b2 - x - 1 = 0 is exactly (1 + \u221a5)/2",
      "prov:value": "The positive root of x\u00b2 - x - 1 = 0 is exactly (1 + \u221a5)/2"
    },
    "pe:evidence-A1": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Direct substitution: (1+\u221a5)/2 satisfies x\u00b2 - x - 1 = 0",
      "pe:factId": "A1",
      "pe:evidenceType": "computed",
      "pe:method": "Direct substitution with exact integer arithmetic over Q(\u221a5)",
      "pe:result": "\u03c6\u00b2 - \u03c6 - 1 = 0 (exact)"
    },
    "pe:evidence-A2": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Quadratic formula yields (1+\u221a5)/2 as the positive root",
      "pe:factId": "A2",
      "pe:evidenceType": "computed",
      "pe:method": "Quadratic formula: x = (-b \u00b1 \u221a(b\u00b2-4ac)) / 2a with a=1, b=-1, c=-1",
      "pe:result": "Positive root = (1+\u221a5)/2 = 1.618033988749895"
    },
    "pe:evidence-A3": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "(1+\u221a5)/2 is positive",
      "pe:factId": "A3",
      "pe:evidenceType": "computed",
      "pe:method": "Direct evaluation: (1+\u221a5)/2 > 0",
      "pe:result": "\u03c6 = 1.618033988749895 > 0"
    },
    "pe:verdict": {
      "prov:type": "pe:Verdict",
      "prov:label": "Verdict: PROVED",
      "prov:value": "PROVED"
    }
  },
  "activity": {
    "pe:determine-verdict": {
      "prov:type": "pe:VerdictDetermination",
      "prov:label": "Determine verdict from evidence",
      "prov:endTime": "2026-03-28"
    }
  },
  "agent": {
    "pe:proof-engine": {
      "prov:type": "prov:SoftwareAgent",
      "prov:label": "Proof Engine",
      "schema:version": "0.10.0",
      "schema:url": "https://github.com/yaniv-golan/proof-engine"
    }
  },
  "wasGeneratedBy": {
    "pe:gen-verdict": {
      "prov:entity": "pe:verdict",
      "prov:activity": "pe:determine-verdict"
    }
  },
  "wasDerivedFrom": {},
  "wasAttributedTo": {
    "pe:attr-verdict": {
      "prov:entity": "pe:verdict",
      "prov:agent": "pe:proof-engine"
    }
  },
  "used": {
    "pe:used-A1": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A1"
    },
    "pe:used-A2": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A2"
    },
    "pe:used-A3": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A3"
    }
  }
}