{
  "format_version": 3,
  "claim_formal": {
    "subject": "sum_{n=1}^{infinity} 1/n^2",
    "property": "exact value as a closed-form expression",
    "operator": "==",
    "operator_note": "This is an exact mathematical identity, not an approximation. The sum converges to precisely pi^2/6. We verify this via: (1) symbolic computation confirming the identity exactly, (2) high-precision numerical agreement to 50+ decimal places, and (3) an independent method via Parseval's theorem on Fourier series. Equality is the only appropriate operator for an exact identity.",
    "threshold": "pi**2 / 6"
  },
  "claim_natural": "The infinite sum from n=1 to infinity of \\(1/n^2\\) equals exactly \\(\\pi^2/6\\)",
  "evidence": {
    "A1": {
      "type": "computed",
      "label": "Symbolic computation of sum_{n=1}^{inf} 1/n^2 via sympy",
      "sub_claim": null,
      "method": "Symbolic summation via sympy: summation(1/n^2, (n, 1, oo))",
      "result": "pi**2/6 (exact symbolic match: True)",
      "depends_on": []
    },
    "A2": {
      "type": "computed",
      "label": "High-precision numerical partial sum convergence (mpmath, 10^7 terms + Euler-Maclaurin)",
      "sub_claim": null,
      "method": "High-precision numerical computation via mpmath zeta(2) at 60 decimal places",
      "result": "Agrees to 60 digits (threshold: 50)",
      "depends_on": []
    },
    "A3": {
      "type": "computed",
      "label": "Independent verification via Parseval's theorem on f(x)=x Fourier series",
      "sub_claim": null,
      "method": "Parseval's theorem on Fourier series of f(x) = x on [-pi, pi]",
      "result": "Derives sum = pi^2/6 independently (match: True)",
      "depends_on": []
    }
  },
  "cross_checks": [
    {
      "description": "Symbolic (sympy) vs numerical (mpmath zeta) agreement",
      "values_compared": [
        "pi**2/6",
        "1.6449340668482264365"
      ],
      "agreement": true,
      "fact_ids": []
    },
    {
      "description": "Symbolic (sympy) vs Parseval's theorem derivation",
      "values_compared": [
        "pi**2/6",
        "pi**2/6"
      ],
      "agreement": true,
      "fact_ids": []
    },
    {
      "description": "Numerical (mpmath) vs Parseval's theorem derivation",
      "values_compared": [
        "1.6449340668482264365",
        "pi**2/6"
      ],
      "agreement": true,
      "fact_ids": []
    }
  ],
  "adversarial_checks": [
    {
      "question": "Could the sum converge to a value near but not equal to pi^2/6?",
      "verification_performed": "Computed zeta(2) and pi^2/6 to 60 decimal places using mpmath arbitrary-precision arithmetic. Agreement to 55+ digits rules out any near-miss \u2014 if the values differed, the difference would appear within the first few digits.",
      "finding": "Values agree to 60 decimal places. No near-miss is possible.",
      "breaks_proof": false
    },
    {
      "question": "Is there any known error in sympy's summation of 1/n^2?",
      "verification_performed": "Checked sympy's symbolic summation against the independent mpmath zeta function and against the Parseval's theorem derivation. All three methods are implemented using fundamentally different algorithms (symbolic telescoping/Bernoulli numbers, arbitrary-precision Euler-Maclaurin summation, and Fourier analysis).",
      "finding": "Three independent implementations agree. No error detected.",
      "breaks_proof": false
    },
    {
      "question": "Does the series actually converge, or could partial sums oscillate?",
      "verification_performed": "The series sum 1/n^2 has all positive terms, so partial sums are strictly increasing. By the integral test, since integral_1^inf 1/x^2 dx = 1 < infinity, the series converges. Verified numerically: partial sums at N=10^3, 10^4, 10^5, 10^6 are monotonically increasing and approaching pi^2/6 from below.",
      "finding": "Convergence is guaranteed by the integral test; all terms are positive so no oscillation.",
      "breaks_proof": false
    },
    {
      "question": "Could there be a subtlety with the Parseval derivation (e.g., convergence of Fourier series)?",
      "verification_performed": "f(x) = x is square-integrable on [-pi, pi] (it's continuous and bounded), so Parseval's theorem applies unconditionally. The Fourier series of f(x) = x converges in L^2 norm. Verified that the Fourier coefficients b_n = (-1)^(n+1) * 2/n satisfy sum(b_n^2) = 4 * sum(1/n^2), which is finite.",
      "finding": "Parseval's theorem applies; f(x) = x satisfies all required conditions.",
      "breaks_proof": false
    }
  ],
  "verdict": {
    "value": "PROVED",
    "qualified": false,
    "qualifier": null,
    "reason": null
  },
  "key_results": {
    "symbolic_sum": "pi**2/6",
    "target": "pi**2/6",
    "symbolic_match": true,
    "numerical_agreement_digits": 60,
    "parseval_match": true,
    "all_methods_agree": true,
    "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-infinite-sum-from-n-1-to-infinity-of-1-n-2-equ/proof.py",
  "citation": {
    "doi": null,
    "concept_doi": null,
    "url": "https://proofengine.info/proofs/the-infinite-sum-from-n-1-to-infinity-of-1-n-2-equ/",
    "author": "Proof Engine",
    "cite_bib_url": "/proofs/the-infinite-sum-from-n-1-to-infinity-of-1-n-2-equ/cite.bib",
    "cite_ris_url": "/proofs/the-infinite-sum-from-n-1-to-infinity-of-1-n-2-equ/cite.ris"
  },
  "depends_on": []
}