{
  "prefix": {
    "pe": "https://proofengine.info/proofs/2026-is-both-a-happy-number-and-mathematically-per/",
    "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": "2026 is both a \"happy number\" and mathematically \"perfect,\" proving the year is cosmically special.",
      "prov:value": "2026 is both a \"happy number\" and mathematically \"perfect,\" proving the year is cosmically special."
    },
    "pe:evidence-A1": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "SC1: Is 2026 a happy number? (iterative digit-square-sum algorithm)",
      "pe:factId": "A1",
      "pe:evidenceType": "computed",
      "pe:method": "is_happy_iterative(2026): Floyd cycle detection on digit-square-sum sequence",
      "pe:result": "True \u2014 sequence [2026, 44, 32, 13, 10, 1] terminates at 1"
    },
    "pe:evidence-A2": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "SC1 cross-check: Happy-number verification via cycle membership (OEIS A007770 structure)",
      "pe:factId": "A2",
      "pe:evidenceType": "computed",
      "pe:method": "is_happy_cycle_check(2026): halt on UNHAPPY_CYCLE membership",
      "pe:result": "True \u2014 agrees with A1"
    },
    "pe:evidence-A3": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "SC2: Is 2026 a perfect number? (compute sum of proper divisors)",
      "pe:factId": "A3",
      "pe:evidenceType": "computed",
      "pe:method": "sum_of_proper_divisors_direct(2026): O(sqrt(n)) enumeration",
      "pe:result": "1016 (need 2026 for perfect; deficit = 1010)"
    },
    "pe:evidence-A4": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "SC2 cross-check: Perfect-number verification via multiplicative sigma formula",
      "pe:factId": "A4",
      "pe:evidenceType": "computed",
      "pe:method": "sigma_multiplicative(2026): product formula \u03c3(n)=\u220f(p^(a+1)-1)/(p-1) minus n",
      "pe:result": "1016 \u2014 agrees with A3"
    },
    "pe:evidence-A5": {
      "prov:type": "pe:ComputedEvidence",
      "prov:label": "SC2 factorisation: prime factorisation of 2026",
      "pe:factId": "A5",
      "pe:evidenceType": "computed",
      "pe:method": "prime_factorisation(2026): trial division",
      "pe:result": "{2: 1, 1013: 1}"
    },
    "pe:verdict": {
      "prov:type": "pe:Verdict",
      "prov:label": "Verdict: PARTIALLY VERIFIED",
      "prov:value": "PARTIALLY VERIFIED"
    }
  },
  "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"
    },
    "pe:used-A4": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A4"
    },
    "pe:used-A5": {
      "prov:activity": "pe:determine-verdict",
      "prov:entity": "pe:evidence-A5"
    }
  }
}