{
  "prefix": {
    "pe": "https://proofengine.info/proofs/eml-k17-multiplication-tree/",
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    "xsd": "http://www.w3.org/2001/XMLSchema#",
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  "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 9 leaves are drawn from \\(\\{1, x, y\\}\\), such that the tree evaluates exactly to \\(x \\times y\\). The tree has K = 17 tokens (8 eml operations and 9 leaves), and the identity holds for all complex \\(x\\) and \\(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 9 leaves are drawn from \\(\\{1, x, y\\}\\), such that the tree evaluates exactly to \\(x \\times y\\). The tree has K = 17 tokens (8 eml operations and 9 leaves), and the identity holds for all complex \\(x\\) and \\(y\\) (in the algebraic setting where \\(\\ln \\circ \\exp\\) is the identity)."
    },
    "pe:evidence-A1": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Token count K = 17 (8 eml operations + 9 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: 8 eml + 9 leaves = K = 17"
    },
    "pe:evidence-A2": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Step-by-step symbolic evaluation: M8 = x * y",
      "pe:factId": "A2",
      "pe:evidenceType": "computed",
      "pe:method": "SymPy symbolic evaluation through 8 layers: build each sub-expression M1..M8 with simplification at each step, verify residuals at 5 critical algebraic cancellation points (M1=e-ln(x), M3=exp(e)/(e-ln(x)), M5=e-ln(xy), M7=ln(xy), M8=x*y) all equal zero",
      "pe:result": "Confirmed: all 5 critical residuals = 0, M8 = 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(M8 - x*y) for positive 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^e: limit equals e^e * y",
      "pe:factId": "A4",
      "pe:evidenceType": "computed",
      "pe:method": "SymPy limit(M8, x, exp(e)) from left, right, and both sides; verify all three equal exp(e) * y",
      "pe:result": "Confirmed: limit from left = y*exp(E), from right = y*exp(E), both = e^e * y"
    },
    "pe:evidence-A5": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Numerical spot-check at 12 real-valued (x, y) pairs",
      "pe:factId": "A5",
      "pe:evidenceType": "computed",
      "pe:method": "Numerical evaluation of the full 8-layer chain at 12 real-valued (x, y) pairs including positive, negative, near-zero, near-singularity (x near e^e within 1e-12); verify |result - x*y| < 1e-6",
      "pe:result": "Confirmed: max |diff| = 9.38e-15"
    },
    "pe:evidence-A6": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "Numerical spot-check at 6 complex-valued (x, y) pairs",
      "pe:factId": "A6",
      "pe:evidenceType": "computed",
      "pe:method": "Numerical evaluation of the full 8-layer chain at 6 complex-valued (x, y) pairs including purely imaginary and negative real inputs; verify |result - x*y| < 1e-10",
      "pe:result": "Confirmed: max |diff| = 6.47e-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.19635621"
    }
  },
  "activity": {
    "pe:determine-verdict": {
      "prov:type": "pe:VerdictDetermination",
      "prov:label": "Determine verdict from evidence",
      "prov:endTime": "2026-04-16"
    }
  },
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    "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"
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  "wasGeneratedBy": {
    "pe:gen-verdict": {
      "prov:entity": "pe:verdict",
      "prov:activity": "pe:determine-verdict"
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  },
  "wasDerivedFrom": {
    "pe:deriv-A2-from-A1": {
      "prov:generatedEntity": "pe:evidence-A2",
      "prov:usedEntity": "pe:evidence-A1"
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    "pe:deriv-A3-from-A2": {
      "prov:generatedEntity": "pe:evidence-A3",
      "prov:usedEntity": "pe:evidence-A2"
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    "pe:deriv-A4-from-A2": {
      "prov:generatedEntity": "pe:evidence-A4",
      "prov:usedEntity": "pe:evidence-A2"
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  "wasAttributedTo": {
    "pe:attr-verdict": {
      "prov:entity": "pe:verdict",
      "prov:agent": "pe:proof-engine"
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  "used": {
    "pe:used-A1": {
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    "pe:used-A2": {
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