{
  "prefix": {
    "pe": "https://proofengine.info/proofs/eml-pi-and-i-from-1/",
    "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": "Using only the eml operator applied to the constant 1 (and allowing complex intermediates), there exist finite expressions that evaluate exactly to the mathematical constant \\(\\pi\\) and to the imaginary unit \\(i\\). These expressions can be verified by symbolic simplification or by numerical evaluation that matches the known values \\(\\pi \\approx 3.1415926535\\ldots\\) and \\(i^2 = -1\\) to machine precision.",
      "prov:value": "Using only the eml operator applied to the constant 1 (and allowing complex intermediates), there exist finite expressions that evaluate exactly to the mathematical constant \\(\\pi\\) and to the imaginary unit \\(i\\). These expressions can be verified by symbolic simplification or by numerical evaluation that matches the known values \\(\\pi \\approx 3.1415926535\\ldots\\) and \\(i^2 = -1\\) to machine precision."
    },
    "pe:evidence-A1": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Token count of the pi-expression: K = 137",
      "pe:factId": "A1",
      "pe:evidenceType": "computed",
      "pe:method": "Recursive tree-walk K(t) = 1 + K(left) + K(right) with K('1') = 1, applied to the constructed pi-expression",
      "pe:result": "Confirmed: K_pi = 137"
    },
    "pe:evidence-A2": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Token count of the i-expression: K = 91",
      "pe:factId": "A2",
      "pe:evidenceType": "computed",
      "pe:method": "Recursive tree-walk K(t) = 1 + K(left) + K(right) with K('1') = 1, applied to the constructed i-expression",
      "pe:result": "Confirmed: K_i = 91"
    },
    "pe:evidence-A3": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Symbolic verification: Z = eml(1, eml(1, eml(eml(eml(1,1),1),1))) has Im(Z) = -pi",
      "pe:factId": "A3",
      "pe:evidenceType": "computed",
      "pe:method": "SymPy construction of Z as eml_sym(1, eml_sym(1, eml_sym(eml_sym(eml_sym(1,1),1),1))); apply .as_real_imag() to extract imaginary component; verify simplify(Im(Z) - (-pi)) == 0",
      "pe:result": "Confirmed: Im(Z) = -pi exactly; residual = 0"
    },
    "pe:evidence-A4": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Numerical evaluation of pi-expression matches math.pi to machine precision",
      "pe:factId": "A4",
      "pe:evidenceType": "computed",
      "pe:method": "Recursive numerical evaluation of the pi-expression tree via cmath.exp and cmath.log; compare to math.pi",
      "pe:result": "Confirmed: |pi-expr - math.pi| = 3.52e-15"
    },
    "pe:evidence-A5": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Numerical evaluation of i-expression matches 1j to machine precision",
      "pe:factId": "A5",
      "pe:evidenceType": "computed",
      "pe:method": "Recursive numerical evaluation of the i-expression tree via cmath.exp and cmath.log; compare to 1j",
      "pe:result": "Confirmed: |i-expr - 1j| = 1.30e-15"
    },
    "pe:evidence-A6": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Numerical cross-check: (i-expression)^2 equals -1 to machine precision",
      "pe:factId": "A6",
      "pe:evidenceType": "computed",
      "pe:method": "Numerical evaluation of (i-expression)^2 and comparison to -1",
      "pe:result": "Confirmed: |(i-expr)^2 + 1| = 2.59e-15"
    },
    "pe:evidence-A7": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Every leaf in both expressions is the constant 1 (no variables)",
      "pe:factId": "A7",
      "pe:evidenceType": "computed",
      "pe:method": "Recursive tree-walk confirming every leaf node equals the constant 1 (no x, y, or other variables)",
      "pe:result": "Confirmed: pi-expression all-ones = True, i-expression all-ones = True"
    },
    "pe:verdict": {
      "prov:type": "pe:Verdict",
      "prov:label": "Verdict: PROVED",
      "prov:value": "PROVED",
      "pe:doi": "10.5281/zenodo.19635622"
    }
  },
  "activity": {
    "pe:determine-verdict": {
      "prov:type": "pe:VerdictDetermination",
      "prov:label": "Determine verdict from evidence",
      "prov:endTime": "2026-04-16"
    }
  },
  "agent": {
    "pe:proof-engine": {
      "prov:type": "prov:SoftwareAgent",
      "prov:label": "Proof Engine",
      "schema:version": "1.18.0",
      "schema:url": "https://github.com/yaniv-golan/proof-engine"
    }
  },
  "wasGeneratedBy": {
    "pe:gen-verdict": {
      "prov:entity": "pe:verdict",
      "prov:activity": "pe:determine-verdict"
    }
  },
  "wasDerivedFrom": {
    "pe:deriv-A3-from-A1": {
      "prov:generatedEntity": "pe:evidence-A3",
      "prov:usedEntity": "pe:evidence-A1"
    },
    "pe:deriv-A3-from-A2": {
      "prov:generatedEntity": "pe:evidence-A3",
      "prov:usedEntity": "pe:evidence-A2"
    },
    "pe:deriv-A4-from-A3": {
      "prov:generatedEntity": "pe:evidence-A4",
      "prov:usedEntity": "pe:evidence-A3"
    },
    "pe:deriv-A5-from-A3": {
      "prov:generatedEntity": "pe:evidence-A5",
      "prov:usedEntity": "pe:evidence-A3"
    },
    "pe:deriv-A6-from-A5": {
      "prov:generatedEntity": "pe:evidence-A6",
      "prov:usedEntity": "pe:evidence-A5"
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  "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"
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    "pe:used-A2": {
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    "pe:used-A3": {
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    "pe:used-A4": {
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