{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "lindemann-pi-transcendence-1882",
    "title": "Lindemann: π is transcendental and the classical circle cannot be squared",
    "summary": "Lindemann adapted Hermite's exponential-integral arithmetic to prove that π is not a root of any nonzero polynomial with rational coefficients. The proof uses symmetry among algebraic conjugates, integer or algebraic-integer relations that become too small, and a nonzero determinant; the application combines the exponential relation e^(iπ)=−1 with the algebraic hypothesis. Transcendence excludes exact squaring of a circle by a finite construction using only a compass and unmarked straightedge.",
    "boundary": "The admitted result is transcendence of π and its classical Euclidean constructibility consequence, not simply irrationality, a finite numerical test or an approximation algorithm. Rational coefficients mean the usual rational-number algebraic notion; historical real-or-complex rational wording is not arbitrary complex coefficients. Hermite's e theorem is antecedent, not this new result. Section4states broader exponential-transcendence/linear-independence results with proofs expressly reserved for fuller later treatment; those are not promoted here to a fully proved Lindemann–Weierstrass theorem or a claim of algebraic independence of e and π. Other construction tools and approximate quadratures are not ruled out.",
    "area": "Transcendental number theory",
    "topic": "Mathematics",
    "year": 1882,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "lindemann-pi-transcendence-1882",
    "target": {
      "record_id": "urn:uuid:bfc59d30-5412-4dde-abc9-4cb9845fe1a8",
      "version": 1,
      "content_hash": "sha256:8f39f922e1e5b3845e8b4f7732e14acec39085c9514f27d1da8ebf0aea8d9260"
    },
    "humans": [
      "F. Lindemann"
    ],
    "authors_complete": true,
    "human_role_note": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.",
    "agents": [],
    "related": [
      "hermite-e-transcendence-1873"
    ],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "cur20261020-lindemann-zahl-pi-1882",
        "title": "Ueber die Zahl π (1882 original MathAnn20,213–225)",
        "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0020/LOG_0031.pdf",
        "date": "1882"
      }
    ],
    "points": "320000",
    "published_grade": {
      "id": "lindemann-pi-transcendence-1882",
      "entry_id": "lindemann-pi-transcendence-1882",
      "band": "S",
      "modifier": "-",
      "target": {
        "record_id": "urn:uuid:bfc59d30-5412-4dde-abc9-4cb9845fe1a8",
        "version": 1,
        "content_hash": "sha256:8f39f922e1e5b3845e8b4f7732e14acec39085c9514f27d1da8ebf0aea8d9260"
      },
      "what_changed": "Lindemann adapted Hermite's exponential-integral arithmetic to prove that π is not a root of any nonzero polynomial with rational coefficients. The proof uses symmetry among algebraic conjugates, integer or algebraic-integer relations that become too small, and a nonzero determinant; the application combines the exponential relation e^(iπ)=−1 with the algebraic hypothesis. Transcendence excludes exact squaring of a circle by a finite construction using only a compass and unmarked straightedge.",
      "scope": "Lindemann adapted Hermite's exponential-integral arithmetic to prove that π is not a root of any nonzero polynomial with rational coefficients. The proof uses symmetry among algebraic conjugates, integer or algebraic-integer relations that become too small, and a nonzero determinant; the application combines the exponential relation e^(iπ)=−1 with the algebraic hypothesis. Transcendence excludes exact squaring of a circle by a finite construction using only a compass and unmarked straightedge. The admitted result is transcendence of π and its classical Euclidean constructibility consequence, not simply irrationality, a finite numerical test or an approximation algorithm. Rational coefficients mean the usual rational-number algebraic notion; historical real-or-complex rational wording is not arbitrary complex coefficients. Hermite's e theorem is antecedent, not this new result. Section4states broader exponential-transcendence/linear-independence results with proofs expressly reserved for fuller later treatment; those are not promoted here to a fully proved Lindemann–Weierstrass theorem or a claim of algebraic independence of e and π. Other construction tools and approximate quadratures are not ruled out.",
      "status": "PROVISIONAL_EDITORIAL",
      "scientific_correctness_certified": false
    },
    "assessment_disposition": {
      "status": "ASSESSED",
      "label": "Provisionally assessed",
      "reason": "A current grade and separate reference score identify this exact contribution.",
      "target": {
        "record_id": "urn:uuid:bfc59d30-5412-4dde-abc9-4cb9845fe1a8",
        "version": 1,
        "content_hash": "sha256:8f39f922e1e5b3845e8b4f7732e14acec39085c9514f27d1da8ebf0aea8d9260"
      },
      "edition": "ratings-e039106f30dec177"
    }
  },
  "version": {
    "kind": "PUBLIC_RECORD_VIEW",
    "previous": null,
    "published_at": "2026-10-23T00:00:00Z",
    "publisher": {
      "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33",
      "role": "WRS_EDITORIAL_PUBLICATION"
    },
    "record": {
      "contributors": [
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "F. Lindemann",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Ueber die Zahl π (1882 original MathAnn20,213–225)",
            "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0020/LOG_0031.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Ueber die Zahl π (1882 original MathAnn20,213–225)",
            "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0020/LOG_0031.pdf"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "The admitted result is transcendence of π and its classical Euclidean constructibility consequence, not simply irrationality, a finite numerical test or an approximation algorithm. Rational coefficients mean the usual rational-number algebraic notion; historical real-or-complex rational wording is not arbitrary complex coefficients. Hermite's e theorem is antecedent, not this new result. Section4states broader exponential-transcendence/linear-independence results with proofs expressly reserved for fuller later treatment; those are not promoted here to a fully proved Lindemann–Weierstrass theorem or a claim of algebraic independence of e and π. Other construction tools and approximate quadratures are not ruled out.",
        "domain": "Transcendental number theory"
      },
      "sources": [
        {
          "citation": "Ueber die Zahl π (1882 original MathAnn20,213–225)",
          "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0020/LOG_0031.pdf"
        }
      ],
      "statement": "# Lindemann: π is transcendental and the classical circle cannot be squared\n\n## Why better rulers cannot solve the old puzzle\n\nSquaring the circle asks for an exact construction of a square with the same area as a given circle, using only a compass and an unmarked straightedge in finitely many steps. It is not a challenge to draw a very accurate approximation. Lindemann's proof that pi is transcendental settles the classical exact-construction problem. The permitted tools are part of the question; changing them can change what is possible.\n\n## Follow the area calculation\n\nTake a circle of radius one. Its area is pi, so a square of equal area would need side length the square root of pi. Lengths obtainable through the classical construction rules are algebraic: they arise through the allowed arithmetic and quadratic constructions. If the square root of pi were algebraic, its square would be algebraic too. Pi is not, so the required side cannot be constructed by those rules.\n\n## Stronger than an irrational decimal\n\nIrrational means not expressible as a ratio of integers. Transcendental means not a root of any nonzero integer-coefficient polynomial. The square root of two demonstrates the difference: it is irrational but algebraic. Thus the proof cannot come from observing that pi has a long, nonrepeating decimal expansion. Numerical approximations, however accurate, leave the universal polynomial question unresolved. The contribution is an exact arithmetic obstruction, not a new digit-computation method.\n\n## What is inherited and what is new\n\nHermite's earlier transcendence proof for e supplies a major antecedent. Lindemann adapts exponential-integral arithmetic and combines it with the relation between the complex exponential and pi. That relation alone is not a proof: the assumed algebraic relations, integer structure and nonvanishing arguments must do real work. The source's last section also states broader exponential claims while reserving fuller proofs, so those broader results are not silently counted as fully established by this paper.\n\n## A practical explanation task\n\nDraw a diagram of the logical chain: an allowed exact square would give an algebraic side, which would make its area algebraic, contradicting pi's transcendence. Label the theorem and the construction-rule input separately. Then contrast an approximate square with an exact one. This makes a useful beginner explanation without claiming to reconstruct the whole transcendence proof or to rule out constructions using other tools, infinite procedures or different geometric operations.\n\n## Scientific scope\n\nThe admitted result is transcendence of π and its classical Euclidean constructibility consequence, not simply irrationality, a finite numerical test or an approximation algorithm. Rational coefficients mean the usual rational-number algebraic notion; historical real-or-complex rational wording is not arbitrary complex coefficients. Hermite's e theorem is antecedent, not this new result. Section4states broader exponential-transcendence/linear-independence results with proofs expressly reserved for fuller later treatment; those are not promoted here to a fully proved Lindemann–Weierstrass theorem or a claim of algebraic independence of e and π. Other construction tools and approximate quadratures are not ruled out.\n\n## Sources for this explanation\n\n- [Ueber die Zahl π (1882 original MathAnn20,213–225)](https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0020/LOG_0031.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "Lindemann adapted Hermite's exponential-integral arithmetic to prove that π is not a root of any nonzero polynomial with rational coefficients. The proof uses symmetry among algebraic conjugates, integer or algebraic-integer relations that become too small, and a nonzero determinant; the application combines the exponential relation e^(iπ)=−1 with the algebraic hypothesis. Transcendence excludes exact squaring of a circle by a finite construction using only a compass and unmarked straightedge.",
      "title": "Lindemann: π is transcendental and the classical circle cannot be squared"
    },
    "record_id": "urn:uuid:bfc59d30-5412-4dde-abc9-4cb9845fe1a8",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:bfc59d30-5412-4dde-abc9-4cb9845fe1a8",
      "version": 1,
      "content_hash": "sha256:8f39f922e1e5b3845e8b4f7732e14acec39085c9514f27d1da8ebf0aea8d9260"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:9b3334bbb4548b631e2a5c336d5a64c1148e98c04b7397944f273e629ab1527e"
  },
  "target": {
    "record_id": "urn:uuid:bfc59d30-5412-4dde-abc9-4cb9845fe1a8",
    "version": 1,
    "content_hash": "sha256:8f39f922e1e5b3845e8b4f7732e14acec39085c9514f27d1da8ebf0aea8d9260"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "lindemann-pi-transcendence-1882",
    "entry_id": "lindemann-pi-transcendence-1882",
    "revision": 1,
    "title": "Lindemann: π is transcendental and the classical circle cannot be squared",
    "context": "Transcendental number theory",
    "track": "Exact theorem and reusable mathematical method",
    "band": "S",
    "modifier": "-",
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-23",
    "evidence_cutoff": "2026-10-23",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "Lindemann adapted Hermite's exponential-integral arithmetic to prove that π is not a root of any nonzero polynomial with rational coefficients. The proof uses symmetry among algebraic conjugates, integer or algebraic-integer relations that become too small, and a nonzero determinant; the application combines the exponential relation e^(iπ)=−1 with the algebraic hypothesis. Transcendence excludes exact squaring of a circle by a finite construction using only a compass and unmarked straightedge.",
    "scope": "Lindemann adapted Hermite's exponential-integral arithmetic to prove that π is not a root of any nonzero polynomial with rational coefficients. The proof uses symmetry among algebraic conjugates, integer or algebraic-integer relations that become too small, and a nonzero determinant; the application combines the exponential relation e^(iπ)=−1 with the algebraic hypothesis. Transcendence excludes exact squaring of a circle by a finite construction using only a compass and unmarked straightedge. The admitted result is transcendence of π and its classical Euclidean constructibility consequence, not simply irrationality, a finite numerical test or an approximation algorithm. Rational coefficients mean the usual rational-number algebraic notion; historical real-or-complex rational wording is not arbitrary complex coefficients. Hermite's e theorem is antecedent, not this new result. Section4states broader exponential-transcendence/linear-independence results with proofs expressly reserved for fuller later treatment; those are not promoted here to a fully proved Lindemann–Weierstrass theorem or a claim of algebraic independence of e and π. Other construction tools and approximate quadratures are not ruled out.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The admitted result is transcendence of π and its classical Euclidean constructibility consequence, not simply irrationality, a finite numerical test or an approximation algorithm. Rational coefficients mean the usual rational-number algebraic notion; historical real-or-complex rational wording is not arbitrary complex coefficients. Hermite's e theorem is antecedent, not this new result. Section4states broader exponential-transcendence/linear-independence results with proofs expressly reserved for fuller later treatment; those are not promoted here to a fully proved Lindemann–Weierstrass theorem or a claim of algebraic independence of e and π. Other construction tools and approximate quadratures are not ruled out.",
    "references": [
      {
        "title": "Ueber die Zahl π (1882 original MathAnn20,213–225)",
        "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0020/LOG_0031.pdf",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      }
    ],
    "roles": [
      {
        "who": "The named authors in the research record",
        "credit": "SCIENCE",
        "did": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred."
      }
    ],
    "target": {
      "record_id": "urn:uuid:bfc59d30-5412-4dde-abc9-4cb9845fe1a8",
      "version": 1,
      "content_hash": "sha256:8f39f922e1e5b3845e8b4f7732e14acec39085c9514f27d1da8ebf0aea8d9260"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "320000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": {
    "id": "lindemann-pi-transcendence-1882",
    "target": {
      "record_id": "urn:uuid:bfc59d30-5412-4dde-abc9-4cb9845fe1a8",
      "version": 1,
      "content_hash": "sha256:8f39f922e1e5b3845e8b4f7732e14acec39085c9514f27d1da8ebf0aea8d9260"
    },
    "sections": [
      {
        "title": "Why better rulers cannot solve the old puzzle",
        "text": "Squaring the circle asks for an exact construction of a square with the same area as a given circle, using only a compass and an unmarked straightedge in finitely many steps. It is not a challenge to draw a very accurate approximation. Lindemann's proof that pi is transcendental settles the classical exact-construction problem. The permitted tools are part of the question; changing them can change what is possible."
      },
      {
        "title": "Follow the area calculation",
        "text": "Take a circle of radius one. Its area is pi, so a square of equal area would need side length the square root of pi. Lengths obtainable through the classical construction rules are algebraic: they arise through the allowed arithmetic and quadratic constructions. If the square root of pi were algebraic, its square would be algebraic too. Pi is not, so the required side cannot be constructed by those rules."
      },
      {
        "title": "Stronger than an irrational decimal",
        "text": "Irrational means not expressible as a ratio of integers. Transcendental means not a root of any nonzero integer-coefficient polynomial. The square root of two demonstrates the difference: it is irrational but algebraic. Thus the proof cannot come from observing that pi has a long, nonrepeating decimal expansion. Numerical approximations, however accurate, leave the universal polynomial question unresolved. The contribution is an exact arithmetic obstruction, not a new digit-computation method."
      },
      {
        "title": "What is inherited and what is new",
        "text": "Hermite's earlier transcendence proof for e supplies a major antecedent. Lindemann adapts exponential-integral arithmetic and combines it with the relation between the complex exponential and pi. That relation alone is not a proof: the assumed algebraic relations, integer structure and nonvanishing arguments must do real work. The source's last section also states broader exponential claims while reserving fuller proofs, so those broader results are not silently counted as fully established by this paper."
      },
      {
        "title": "A practical explanation task",
        "text": "Draw a diagram of the logical chain: an allowed exact square would give an algebraic side, which would make its area algebraic, contradicting pi's transcendence. Label the theorem and the construction-rule input separately. Then contrast an approximate square with an exact one. This makes a useful beginner explanation without claiming to reconstruct the whole transcendence proof or to rule out constructions using other tools, infinite procedures or different geometric operations."
      }
    ],
    "references": [
      {
        "title": "Ueber die Zahl π (1882 original MathAnn20,213–225)",
        "url": "https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0020/LOG_0031.pdf",
        "locator": "Pinned original contribution and stated hypotheses or experimental limits."
      }
    ]
  },
  "reading_status": "EXPLANATORY_GUIDE"
}