{
  "format_version": 3,
  "claim_formal": {
    "subject": "the equation x\u00b2 - x - 1 = 0",
    "property": "positive root equals (1 + \u221a5)/2",
    "operator": "==",
    "operator_note": "Exact algebraic equality. We verify that substituting \u03c6 = (1 + \u221a5)/2 into x\u00b2 - x - 1 yields exactly 0 (symbolically), and independently derive the positive root via the quadratic formula to confirm it equals (1 + \u221a5)/2. Numerical verification uses tolerance for floating-point, but the symbolic argument is exact.",
    "threshold": true
  },
  "claim_natural": "The positive root of x\u00b2 - x - 1 = 0 is exactly (1 + \u221a5)/2",
  "evidence": {
    "A1": {
      "type": "computed",
      "label": "Direct substitution: (1+\u221a5)/2 satisfies x\u00b2 - x - 1 = 0",
      "sub_claim": null,
      "method": "Direct substitution with exact integer arithmetic over Q(\u221a5)",
      "result": "\u03c6\u00b2 - \u03c6 - 1 = 0 (exact)",
      "depends_on": []
    },
    "A2": {
      "type": "computed",
      "label": "Quadratic formula yields (1+\u221a5)/2 as the positive root",
      "sub_claim": null,
      "method": "Quadratic formula: x = (-b \u00b1 \u221a(b\u00b2-4ac)) / 2a with a=1, b=-1, c=-1",
      "result": "Positive root = (1+\u221a5)/2 = 1.618033988749895",
      "depends_on": []
    },
    "A3": {
      "type": "computed",
      "label": "(1+\u221a5)/2 is positive",
      "sub_claim": null,
      "method": "Direct evaluation: (1+\u221a5)/2 > 0",
      "result": "\u03c6 = 1.618033988749895 > 0",
      "depends_on": []
    }
  },
  "cross_checks": [
    {
      "description": "Direct substitution (A1) vs quadratic formula derivation (A2)",
      "values_compared": [
        "substitution yields 0",
        "quadratic formula yields 1.618033988749895"
      ],
      "agreement": true,
      "fact_ids": []
    }
  ],
  "adversarial_checks": [
    {
      "question": "Could there be another positive root we're missing?",
      "verification_performed": "A degree-2 polynomial has at most 2 roots (Fundamental Theorem of Algebra). The quadratic formula gives both roots: (1+\u221a5)/2 \u2248 1.618 and (1-\u221a5)/2 \u2248 -0.618. Only one root is positive. Verified computationally: root_negative < 0.",
      "finding": "The other root is (1-\u221a5)/2 \u2248 -0.618034, which is negative. Only one positive root exists.",
      "breaks_proof": false
    },
    {
      "question": "Could floating-point error make the substitution appear to be zero when it isn't?",
      "verification_performed": "The proof uses exact integer arithmetic for the symbolic verification: representing \u03c6 as (1 + 1\u00b7\u221a5)/2 and computing \u03c6\u00b2 - \u03c6 - 1 using rational coefficients of {1, \u221a5}. The irrational parts cancel exactly (coefficient = 0) and the rational remainder is exactly 0. No floating-point arithmetic is involved in the symbolic path.",
      "finding": "Symbolic verification uses only integer/rational arithmetic. Result is exactly 0, not approximately 0.",
      "breaks_proof": false
    },
    {
      "question": "Is the equation x\u00b2 - x - 1 = 0 the correct equation (not x\u00b2 + x - 1 or x\u00b2 - x + 1)?",
      "verification_performed": "Checked: x\u00b2 - x - 1 evaluated at \u03c6 = (1+\u221a5)/2 gives 0. If the equation were x\u00b2 + x - 1, the roots would be (-1 \u00b1 \u221a5)/2, and the positive root would be (-1+\u221a5)/2 \u2248 0.618, not (1+\u221a5)/2. If the equation were x\u00b2 - x + 1, discriminant = 1-4 = -3 < 0, no real roots.",
      "finding": "The equation x\u00b2 - x - 1 = 0 is the unique quadratic with (1+\u221a5)/2 as a root and integer coefficients (up to scaling).",
      "breaks_proof": false
    }
  ],
  "verdict": {
    "value": "PROVED",
    "qualified": false,
    "qualifier": null,
    "reason": null
  },
  "key_results": {
    "phi": 1.618033988749895,
    "substitution_exact_zero": true,
    "quadratic_positive_root": 1.618033988749895,
    "positive": true,
    "operator": "==",
    "claim_holds": true
  },
  "generator": {
    "name": "proof-engine",
    "version": "0.10.0",
    "repo": "https://github.com/yaniv-golan/proof-engine",
    "generated_at": "2026-03-28"
  },
  "proof_py_url": "/proofs/the-positive-root-of-x-x-1-0-is-exactly-1-5-2/proof.py",
  "citation": {
    "doi": null,
    "concept_doi": null,
    "url": "https://proofengine.info/proofs/the-positive-root-of-x-x-1-0-is-exactly-1-5-2/",
    "author": "Proof Engine",
    "cite_bib_url": "/proofs/the-positive-root-of-x-x-1-0-is-exactly-1-5-2/cite.bib",
    "cite_ris_url": "/proofs/the-positive-root-of-x-x-1-0-is-exactly-1-5-2/cite.ris"
  },
  "depends_on": []
}