{
  "prefix": {
    "pe": "https://proofengine.info/proofs/eml-k19-addition-tree/",
    "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 binary operator eml is defined by the expression \\(\\text{eml}(a, b) = \\exp(a) - \\ln(b)\\). There exists a finite binary tree consisting solely of eml operations, whose 10 leaves are drawn from \\(\\{1, x, y\\}\\), such that the tree evaluates exactly to \\(x + y\\). The tree has K = 19 tokens (9 eml operations and 10 leaves), and the identity holds for all real \\(x\\) and \\(y\\) (and formally for all complex \\(x, y\\) in the algebraic setting where \\(\\ln \\circ \\exp\\) is the identity).",
      "prov:value": "The binary operator eml is defined by the expression \\(\\text{eml}(a, b) = \\exp(a) - \\ln(b)\\). There exists a finite binary tree consisting solely of eml operations, whose 10 leaves are drawn from \\(\\{1, x, y\\}\\), such that the tree evaluates exactly to \\(x + y\\). The tree has K = 19 tokens (9 eml operations and 10 leaves), and the identity holds for all real \\(x\\) and \\(y\\) (and formally for all complex \\(x, y\\) in the algebraic setting where \\(\\ln \\circ \\exp\\) is the identity)."
    },
    "pe:evidence-A1": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Token count K = 19 (9 eml operations + 10 leaves)",
      "pe:factId": "A1",
      "pe:evidenceType": "computed",
      "pe:method": "Programmatic parsing of the expression string to count eml operation nodes and leaf nodes (1, x, y)",
      "pe:result": "Confirmed: 9 eml + 10 leaves = K = 19"
    },
    "pe:evidence-A2": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Step-by-step symbolic evaluation: E9 = x + y",
      "pe:factId": "A2",
      "pe:evidenceType": "computed",
      "pe:method": "SymPy symbolic evaluation through 9 layers: build each sub-expression E1..E9 with simplification at each step, verify residuals at 5 critical algebraic cancellation points (E1=exp(x), E2=e-x, E4=exp(e)/(e-x), E7=e-x-y, E9=x+y) all equal zero",
      "pe:result": "Confirmed: all 5 critical residuals = 0, E9 = x + y"
    },
    "pe:evidence-A3": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Full expression minus (x + y) = 0",
      "pe:factId": "A3",
      "pe:evidenceType": "computed",
      "pe:method": "SymPy simplify(E9 - (x + y)) for real symbols x, y; verify residual equals 0",
      "pe:result": "Confirmed: residual = 0"
    },
    "pe:evidence-A4": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Removable singularity at x = e: limit equals e + y",
      "pe:factId": "A4",
      "pe:evidenceType": "computed",
      "pe:method": "SymPy limit(E9, x, e) from left, right, and both sides; verify all three equal e + y",
      "pe:result": "Confirmed: limit from left = y + E, from right = y + E, both = e + y"
    },
    "pe:evidence-A5": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Numerical spot-check at 10 real-valued (x, y) pairs",
      "pe:factId": "A5",
      "pe:evidenceType": "computed",
      "pe:method": "Numerical evaluation of the full 9-layer chain at 10 real-valued (x, y) pairs spanning [-100, 100], including x = 0, x > e, x near the singularity at e (within 1e-12); verify |result - (x+y)| < 1e-6",
      "pe:result": "Confirmed: max |diff| = 1.26e-14"
    },
    "pe:evidence-A6": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Numerical spot-check at 5 complex-valued (x, y) pairs",
      "pe:factId": "A6",
      "pe:evidenceType": "computed",
      "pe:method": "Numerical evaluation of the full 9-layer chain at 5 complex-valued (x, y) pairs with |Im(x+y)| < pi (principal-branch domain); verify |result - (x+y)| < 1e-10",
      "pe:result": "Confirmed: max |diff| = 1.34e-15"
    },
    "pe:evidence-A7": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Exhaustive search: no K <= 15 eml tree computes x + y",
      "pe:factId": "A7",
      "pe:evidenceType": "computed",
      "pe:method": "Embedded exhaustive bottom-up search: enumerate all distinct eml binary trees with leaves {1, x, y} at each odd K from 1 to 15 using numerical fingerprinting at a generic complex test point; check if x+y appears",
      "pe:result": "Confirmed: 1,980,526 distinct values at K=15, x+y not found at any K <= 15"
    },
    "pe:verdict": {
      "prov:type": "pe:Verdict",
      "prov:label": "Verdict: PROVED",
      "prov:value": "PROVED",
      "pe:doi": "10.5281/zenodo.19635620"
    }
  },
  "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-A2-from-A1": {
      "prov:generatedEntity": "pe:evidence-A2",
      "prov:usedEntity": "pe:evidence-A1"
    },
    "pe:deriv-A3-from-A2": {
      "prov:generatedEntity": "pe:evidence-A3",
      "prov:usedEntity": "pe:evidence-A2"
    },
    "pe:deriv-A4-from-A2": {
      "prov:generatedEntity": "pe:evidence-A4",
      "prov:usedEntity": "pe:evidence-A2"
    }
  },
  "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"
    },
    "pe:used-A4": {
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      "prov:entity": "pe:evidence-A4"
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    "pe:used-A5": {
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      "prov:entity": "pe:evidence-A5"
    },
    "pe:used-A6": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A6"
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    "pe:used-A7": {
      "prov:activity": "pe:determine-verdict",
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