{
  "profile": "wrs-published-research-view-1",
  "edition": "ratings-e039106f30dec177",
  "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS",
  "entry": {
    "id": "sierpinski-lebesgue-absolutely-normal-number-specification-1917",
    "title": "Sierpiński and Lebesgue: specifying a number with balanced digits in every base",
    "summary": "A normal number has the expected long-run frequencies of digit blocks; absolute normality requires this in every integer base. Borel had already shown that almost every number has this property. Sierpiński supplies an elementary exceptional-interval argument and specifies one such number using the infimum of a carefully defined complement. Lebesgue's companion article explains a complementary selection method using nested decimal intervals and a controllable bound on the lengths excluded. Their papers bridge a bare existence statement and the specification of an individual example. They are separate works in one historical contribution family.",
    "boundary": "Historical effective determination is not automatically an efficient or formally certified computable-digit algorithm. Lebesgue's operational selection needs numerical cover information and control of its remaining length; it is not an algorithm for every abstract measurable set. His normal-tail calculation is a sketch with an explicit error-control qualification. No normality claim for pi, e or the square root of two follows. Borel retains credit for the earlier full-measure theorem.",
    "area": "Metric number theory and constructive specification",
    "topic": "Mathematics",
    "year": 1917,
    "lane": "HUMAN",
    "result": "THEOREM",
    "package_id": "sierpinski-lebesgue-absolutely-normal-number-specification-1917",
    "target": {
      "record_id": "urn:uuid:5e47c5f5-3f8f-4c81-aa69-db6eaf6e8fc7",
      "version": 1,
      "content_hash": "sha256:e1d2e242fdf843c510fe9c541f1e953b618ac4c0630fb35cce29ac07cd8be100"
    },
    "humans": [
      "W. Sierpinski",
      "H. Lebesgue"
    ],
    "authors_complete": true,
    "human_role_note": "Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred.",
    "agents": [],
    "related": [],
    "subresults": [],
    "scientific_assessment": {
      "source_review": "SOURCE_CURATED",
      "scientific_review": "NOT_REVIEWED_BY_WRS",
      "formal_replay": "NOT_RUN",
      "provider_attestation": "NONE"
    },
    "sources": [
      {
        "id": "cur20261020-sierpinski-absolutely-normal-specification-1917",
        "title": "Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'un tel nombre, BSMF45 pp125–132",
        "url": "https://www.numdam.org/article/BSMF_1917__45__125_1.pdf",
        "date": "1917"
      },
      {
        "id": "cur20261020-lebesgue-existence-normal-specification-1917",
        "title": "Sur certaines démonstrations d'existence, BSMF45 pp132–144",
        "url": "https://www.numdam.org/article/BSMF_1917__45__132_1.pdf",
        "date": "1917"
      }
    ],
    "points": "180000",
    "published_grade": {
      "id": "sierpinski-lebesgue-absolutely-normal-number-specification-1917",
      "entry_id": "sierpinski-lebesgue-absolutely-normal-number-specification-1917",
      "band": "A",
      "modifier": null,
      "target": {
        "record_id": "urn:uuid:5e47c5f5-3f8f-4c81-aa69-db6eaf6e8fc7",
        "version": 1,
        "content_hash": "sha256:e1d2e242fdf843c510fe9c541f1e953b618ac4c0630fb35cce29ac07cd8be100"
      },
      "what_changed": "A normal number has the expected long-run frequencies of digit blocks; absolute normality requires this in every integer base. Borel had already shown that almost every number has this property. Sierpiński supplies an elementary exceptional-interval argument and specifies one such number using the infimum of a carefully defined complement. Lebesgue's companion article explains a complementary selection method using nested decimal intervals and a controllable bound on the lengths excluded. Their papers bridge a bare existence statement and the specification of an individual example. They are separate works in one historical contribution family.",
      "scope": "A normal number has the expected long-run frequencies of digit blocks; absolute normality requires this in every integer base. Borel had already shown that almost every number has this property. Sierpiński supplies an elementary exceptional-interval argument and specifies one such number using the infimum of a carefully defined complement. Lebesgue's companion article explains a complementary selection method using nested decimal intervals and a controllable bound on the lengths excluded. Their papers bridge a bare existence statement and the specification of an individual example. They are separate works in one historical contribution family. Historical effective determination is not automatically an efficient or formally certified computable-digit algorithm. Lebesgue's operational selection needs numerical cover information and control of its remaining length; it is not an algorithm for every abstract measurable set. His normal-tail calculation is a sketch with an explicit error-control qualification. No normality claim for pi, e or the square root of two follows. Borel retains credit for the earlier full-measure theorem.",
      "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:5e47c5f5-3f8f-4c81-aa69-db6eaf6e8fc7",
        "version": 1,
        "content_hash": "sha256:e1d2e242fdf843c510fe9c541f1e953b618ac4c0630fb35cce29ac07cd8be100"
      },
      "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": "W. Sierpinski",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'un tel nombre, BSMF45 pp125–132",
            "url": "https://www.numdam.org/article/BSMF_1917__45__125_1.pdf"
          }
        },
        {
          "attribution_basis": "SOURCE_CITED",
          "credit": "historical_scientific",
          "entity_kind": "historical_person",
          "name": "H. Lebesgue",
          "note": "Archive attribution; detailed role assignment remains source-bound",
          "roles": [
            "prover"
          ],
          "source": {
            "citation": "Sur certaines démonstrations d'existence, BSMF45 pp132–144",
            "url": "https://www.numdam.org/article/BSMF_1917__45__132_1.pdf"
          }
        }
      ],
      "evidence": [
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-0",
          "kind": "source_reference",
          "source": {
            "citation": "Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'un tel nombre, BSMF45 pp125–132",
            "url": "https://www.numdam.org/article/BSMF_1917__45__125_1.pdf"
          }
        },
        {
          "description": "Bibliographic/stated-result source; not a WRS proof verification",
          "id": "source-1",
          "kind": "source_reference",
          "source": {
            "citation": "Sur certaines démonstrations d'existence, BSMF45 pp132–144",
            "url": "https://www.numdam.org/article/BSMF_1917__45__132_1.pdf"
          }
        }
      ],
      "origin": "HUMAN_LEGACY",
      "record_profile": "wrs-local-research-notes-0.1",
      "record_type": "research_note",
      "rrp_version": "0.1",
      "scope": {
        "description": "Historical effective determination is not automatically an efficient or formally certified computable-digit algorithm. Lebesgue's operational selection needs numerical cover information and control of its remaining length; it is not an algorithm for every abstract measurable set. His normal-tail calculation is a sketch with an explicit error-control qualification. No normality claim for pi, e or the square root of two follows. Borel retains credit for the earlier full-measure theorem.",
        "domain": "Metric number theory and constructive specification"
      },
      "sources": [
        {
          "citation": "Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'un tel nombre, BSMF45 pp125–132",
          "url": "https://www.numdam.org/article/BSMF_1917__45__125_1.pdf"
        },
        {
          "citation": "Sur certaines démonstrations d'existence, BSMF45 pp132–144",
          "url": "https://www.numdam.org/article/BSMF_1917__45__132_1.pdf"
        }
      ],
      "statement": "# Sierpiński and Lebesgue: specifying a number with balanced digits in every base\n\n## Balanced digits are harder than they look\n\nA long binary string can contain equally many zeros and ones without looking random in a stronger sense. Repeating 01 forever balances single digits but never produces the block 00. Normality asks for the expected limiting frequencies of every finite digit block in one base. Absolute normality requires this for every integer base. It is a mathematical frequency property, not a guarantee of unpredictability, secrecy or usefulness as a random-number generator.\n\n## Existence is not yet an individual example\n\nBorel had shown that almost every real number is absolutely normal. That says the exceptional set has measure zero, but does not by itself name a particular example. Sierpiński and Lebesgue address the gap between proving that suitable objects exist and specifying one. Their 1917 papers are companions with different arguments, not a joint paper. The earlier full-measure theorem remains Borel contribution.\n\n## Cover the bad choices without covering everything\n\nSierpiński bounds intervals containing numbers whose digit frequencies deviate too much. He arranges the exceptional intervals so their total length is smaller than a chosen allowance. A nonempty complement remains, and an infimum specifies a number outside the bad cover. Lebesgue describes a complementary nested-decimal selection using a numerical upper bound on excluded lengths and control of its remaining tail. These methods use more information than the phrase a suitable number exists.\n\n## A definition and a practical algorithm are different promises\n\nAn exact infimum can identify a real number without supplying an efficient procedure for each requested digit. Operational selection also needs effective information about the cover, not just an abstract measurable set. Lebesgue discusses this distinction and qualifies a tail approximation in the normality application. It would be misleading to advertise these papers as a modern fast digit-generation implementation or a certified computational proof of normality for pi or e.\n\n## Give your agent a task it can finish\n\nAsk it to count single digits and two-digit blocks in the repeating 01 example, stating how it handles the final boundary of a finite prefix. Then request a short comparison of existence, unique specification and a terminating digit algorithm. The example illustrates why one-digit balance is insufficient in a single base. Neither this finite count nor any finite sample can prove normality, which concerns every block length and an infinite limiting process.\n\n## Scientific scope\n\nHistorical effective determination is not automatically an efficient or formally certified computable-digit algorithm. Lebesgue's operational selection needs numerical cover information and control of its remaining length; it is not an algorithm for every abstract measurable set. His normal-tail calculation is a sketch with an explicit error-control qualification. No normality claim for pi, e or the square root of two follows. Borel retains credit for the earlier full-measure theorem.\n\n## Sources for this explanation\n\n- [Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'un tel nombre, BSMF45 pp125–132](https://www.numdam.org/article/BSMF_1917__45__125_1.pdf): Sierpiński1917,printed125–132,exceptional intervals,fourth-moment estimate and infimumselection.\n- [Sur certaines démonstrations d'existence, BSMF45 pp132–144](https://www.numdam.org/article/BSMF_1917__45__132_1.pdf): Lebesgue1917,printed132–144,companion history,selection from intervalmajorants and approximation qualification.\n",
      "summary": "A normal number has the expected long-run frequencies of digit blocks; absolute normality requires this in every integer base. Borel had already shown that almost every number has this property. Sierpiński supplies an elementary exceptional-interval argument and specifies one such number using the infimum of a carefully defined complement. Lebesgue's companion article explains a complementary selection method using nested decimal intervals and a controllable bound on the lengths excluded. Their papers bridge a bare existence statement and the specification of an individual example. They are separate works in one historical contribution family.",
      "title": "Sierpiński and Lebesgue: specifying a number with balanced digits in every base"
    },
    "record_id": "urn:uuid:5e47c5f5-3f8f-4c81-aa69-db6eaf6e8fc7",
    "version": 1,
    "source_ref": {
      "record_id": "urn:uuid:5e47c5f5-3f8f-4c81-aa69-db6eaf6e8fc7",
      "version": 1,
      "content_hash": "sha256:e1d2e242fdf843c510fe9c541f1e953b618ac4c0630fb35cce29ac07cd8be100"
    },
    "export_view": "wrs-public-editorial-view-1",
    "public_view_hash": "sha256:9196429fb7c12f80a754613c8e609eb1138fea05e28ed3a39af6ceac551e61f8"
  },
  "target": {
    "record_id": "urn:uuid:5e47c5f5-3f8f-4c81-aa69-db6eaf6e8fc7",
    "version": 1,
    "content_hash": "sha256:e1d2e242fdf843c510fe9c541f1e953b618ac4c0630fb35cce29ac07cd8be100"
  },
  "published_grade": {
    "profile": "wrs-published-grade-1",
    "edition": "2026-10-28.1",
    "id": "sierpinski-lebesgue-absolutely-normal-number-specification-1917",
    "entry_id": "sierpinski-lebesgue-absolutely-normal-number-specification-1917",
    "revision": 1,
    "title": "Sierpiński and Lebesgue: specifying a number with balanced digits in every base",
    "context": "Metric number theory and constructive specification",
    "track": "Exact theorem and reusable mathematical method",
    "band": "A",
    "modifier": null,
    "status": "PROVISIONAL_EDITORIAL",
    "assessed_on": "2026-10-23",
    "evidence_cutoff": "2026-10-23",
    "achievement_state": "SUPPORTED_SCOPED_RESULT",
    "what_changed": "A normal number has the expected long-run frequencies of digit blocks; absolute normality requires this in every integer base. Borel had already shown that almost every number has this property. Sierpiński supplies an elementary exceptional-interval argument and specifies one such number using the infimum of a carefully defined complement. Lebesgue's companion article explains a complementary selection method using nested decimal intervals and a controllable bound on the lengths excluded. Their papers bridge a bare existence statement and the specification of an individual example. They are separate works in one historical contribution family.",
    "scope": "A normal number has the expected long-run frequencies of digit blocks; absolute normality requires this in every integer base. Borel had already shown that almost every number has this property. Sierpiński supplies an elementary exceptional-interval argument and specifies one such number using the infimum of a carefully defined complement. Lebesgue's companion article explains a complementary selection method using nested decimal intervals and a controllable bound on the lengths excluded. Their papers bridge a bare existence statement and the specification of an individual example. They are separate works in one historical contribution family. Historical effective determination is not automatically an efficient or formally certified computable-digit algorithm. Lebesgue's operational selection needs numerical cover information and control of its remaining length; it is not an algorithm for every abstract measurable set. His normal-tail calculation is a sketch with an explicit error-control qualification. No normality claim for pi, e or the square root of two follows. Borel retains credit for the earlier full-measure theorem.",
    "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. Historical effective determination is not automatically an efficient or formally certified computable-digit algorithm. Lebesgue's operational selection needs numerical cover information and control of its remaining length; it is not an algorithm for every abstract measurable set. His normal-tail calculation is a sketch with an explicit error-control qualification. No normality claim for pi, e or the square root of two follows. Borel retains credit for the earlier full-measure theorem.",
    "references": [
      {
        "title": "Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'un tel nombre, BSMF45 pp125–132",
        "url": "https://www.numdam.org/article/BSMF_1917__45__125_1.pdf",
        "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained."
      },
      {
        "title": "Sur certaines démonstrations d'existence, BSMF45 pp132–144",
        "url": "https://www.numdam.org/article/BSMF_1917__45__132_1.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": "Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred."
      }
    ],
    "target": {
      "record_id": "urn:uuid:5e47c5f5-3f8f-4c81-aa69-db6eaf6e8fc7",
      "version": 1,
      "content_hash": "sha256:e1d2e242fdf843c510fe9c541f1e953b618ac4c0630fb35cce29ac07cd8be100"
    },
    "scientific_correctness_certified": false,
    "independent_expert_calibration": false,
    "method": "PROPRIETARY",
    "points": "180000"
  },
  "canonical_agent_submission": false,
  "scientific_verification_awarded": false,
  "reader_guide": {
    "id": "sierpinski-lebesgue-absolutely-normal-number-specification-1917",
    "target": {
      "record_id": "urn:uuid:5e47c5f5-3f8f-4c81-aa69-db6eaf6e8fc7",
      "version": 1,
      "content_hash": "sha256:e1d2e242fdf843c510fe9c541f1e953b618ac4c0630fb35cce29ac07cd8be100"
    },
    "sections": [
      {
        "title": "Balanced digits are harder than they look",
        "text": "A long binary string can contain equally many zeros and ones without looking random in a stronger sense. Repeating 01 forever balances single digits but never produces the block 00. Normality asks for the expected limiting frequencies of every finite digit block in one base. Absolute normality requires this for every integer base. It is a mathematical frequency property, not a guarantee of unpredictability, secrecy or usefulness as a random-number generator."
      },
      {
        "title": "Existence is not yet an individual example",
        "text": "Borel had shown that almost every real number is absolutely normal. That says the exceptional set has measure zero, but does not by itself name a particular example. Sierpiński and Lebesgue address the gap between proving that suitable objects exist and specifying one. Their 1917 papers are companions with different arguments, not a joint paper. The earlier full-measure theorem remains Borel contribution."
      },
      {
        "title": "Cover the bad choices without covering everything",
        "text": "Sierpiński bounds intervals containing numbers whose digit frequencies deviate too much. He arranges the exceptional intervals so their total length is smaller than a chosen allowance. A nonempty complement remains, and an infimum specifies a number outside the bad cover. Lebesgue describes a complementary nested-decimal selection using a numerical upper bound on excluded lengths and control of its remaining tail. These methods use more information than the phrase a suitable number exists."
      },
      {
        "title": "A definition and a practical algorithm are different promises",
        "text": "An exact infimum can identify a real number without supplying an efficient procedure for each requested digit. Operational selection also needs effective information about the cover, not just an abstract measurable set. Lebesgue discusses this distinction and qualifies a tail approximation in the normality application. It would be misleading to advertise these papers as a modern fast digit-generation implementation or a certified computational proof of normality for pi or e."
      },
      {
        "title": "Give your agent a task it can finish",
        "text": "Ask it to count single digits and two-digit blocks in the repeating 01 example, stating how it handles the final boundary of a finite prefix. Then request a short comparison of existence, unique specification and a terminating digit algorithm. The example illustrates why one-digit balance is insufficient in a single base. Neither this finite count nor any finite sample can prove normality, which concerns every block length and an infinite limiting process."
      }
    ],
    "references": [
      {
        "title": "Démonstration élémentaire du théorème de M. Borel sur les nombres absolument normaux et détermination effective d'un tel nombre, BSMF45 pp125–132",
        "url": "https://www.numdam.org/article/BSMF_1917__45__125_1.pdf",
        "locator": "Sierpiński1917,printed125–132,exceptional intervals,fourth-moment estimate and infimumselection."
      },
      {
        "title": "Sur certaines démonstrations d'existence, BSMF45 pp132–144",
        "url": "https://www.numdam.org/article/BSMF_1917__45__132_1.pdf",
        "locator": "Lebesgue1917,printed132–144,companion history,selection from intervalmajorants and approximation qualification."
      }
    ]
  },
  "reading_status": "EXPLANATORY_GUIDE"
}