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(2) It draws 20,000 random eventually periodic patterns (base in {2,3,10}, preperiod 0-20 digits, block 1-30 digits, all-(b-1) blocks rejected), reconstructs the exact rational with Python Fractions and re-expands it, counting patterns not reproduced digit for digit. (3) It downloads the first 10,000 decimal digits of sqrt(2), e and pi from OEIS and, for every p <= N/3, finds the smallest index after which the digit sequence agrees with its shift by p, flagging any p for which two full periods already agree. Run time about 9 s including downloads." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: 0 rational expansions violating period <= q-1 or preperiod + period <= q (out of 194,550) (n = 194550, exact). Interpretation: Wolfram's 'repetitive digit sequences' are eventually periodic positional expansions in an integer base; the clause 'period at most q - 1' in the claim text is the standard long-division bound (there are only q - 1 non-zero remainders). Terminating expansions are treated as eventually periodic with repeating 0, as the assessors note; period blocks consisting only of the digit b - 1 are excluded from the random patterns because they denote the alternative expansion of a terminating rational and would re-expand in the terminating form. The converse ('rationals are the only numbers with repetitive digit sequences') is tested constructively (every eventually periodic pattern reconstructs to a rational that reproduces it) and, as a sanity check only, on the published digits of three irrational constants; a finite digit string can never prove non-periodicity, so part (3) is a consistency check rather than a proof." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch04-14: supports, confidence 0.97" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch04-14/result.json" } ], "http://www.w3.org/ns/prov#wasAttributedTo": [ { "@id": "https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1" } ], "https://w3id.org/delphi/ontology/admissibleDatasets": [ { "@value": "Planned datasets: rational-expansions-bases-2-3-10-q-le-20000, oeis-a007732-bfile, random-eventually-periodic-patterns, oeis-a002193-bfile, oeis-a001113-bfile, oeis-a000796-bfile" } ], "https://w3id.org/delphi/ontology/assessmentType": [ { "@value": "claim-test/0.1.0" } ], "https://w3id.org/delphi/ontology/dataKind": [ { "@value": "computational" } ], "https://w3id.org/delphi/ontology/disconfirmingObservation": [ { "@value": "Pre-registered: any rational whose expansion has minimal period > q - 1 or does not become periodic within q digits; any disagreement with A007732; any eventually periodic pattern whose reconstructed rational re-expands differently; or an irrational constant's published digits exhibiting at least two full agreeing periods for some p <= N/3." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "A finite digit string cannot establish that a number is not eventually periodic; part (3) only checks that 10,000 published digits of sqrt(2), e and pi show no two agreeing repeats for any period up to N/3 (the maximal agreement found was 1.33 periods, consistent with chance), which is a consistency check, not a proof of the 'only rationals' clause." }, { "@value": "Both directions are classical theorems; the computation confirms the forward direction exhaustively for all denominators up to 20,000 in three bases (the bound period = q - 1 is attained, e.g. for full-reptend primes), agrees with OEIS A007732 on all 10,000 tabulated decimal periods, and confirms the converse constructively on 20,000 random eventually periodic patterns." }, { "@value": "Terminating expansions are counted as eventually periodic with repeating 0 (period 1), following OEIS A007732; under the alternative convention (repeating b-1) they are still eventually periodic, so the verdict does not depend on this choice." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered: (1) for bases 2, 3 and 10 and all denominators q = 1..20000, the digit expansion of 1/q (and of p/q for all 1 <= p < q when q <= 300) has period <= q - 1 and preperiod + period <= q, with zero violations, and the base-10 periods of 1/q agree with OEIS A007732 for every n covered by its b-file; (2) 100% of 20000 random eventually periodic digit patterns (random preperiod of length 0..20, random non-trivial period block of length 1..30, bases 2, 3, 10) are reproduced exactly when the rational number reconstructed from them is re-expanded by long division; (3) none of the first 10000 decimal digits of sqrt(2), e and pi (OEIS b-files) is eventually periodic in the sense that, for some period p <= N/3, the last 2p digits already repeat with period p." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered: any rational whose expansion has minimal period > q - 1 or does not become periodic within q digits; any disagreement with A007732; any eventually periodic pattern whose reconstructed rational re-expands differently; or an irrational constant's published digits exhibiting at least two full agreeing periods for some p <= N/3." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.97", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-1" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-2" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-3" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-4" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-5" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-6" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "In any integer base, every rational p/q has an eventually periodic digit expansion with period at most q - 1 (and preperiod + period at most q), every eventually periodic digit sequence is the expansion of a rational number, and the published digit expansions of irrational constants show no eventual periodicity." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py (1) runs long division for 1/q, q = 1..20000, and for all p/q with q <= 300, in bases 2, 3 and 10 (64,850 expansions per base), recording preperiod and period from the first repeated remainder and counting violations of period <= q-1 or preperiod + period <= q; base-10 periods of 1/n are compared with the 10,000-term OEIS A007732 b-file. (2) It draws 20,000 random eventually periodic patterns (base in {2,3,10}, preperiod 0-20 digits, block 1-30 digits, all-(b-1) blocks rejected), reconstructs the exact rational with Python Fractions and re-expands it, counting patterns not reproduced digit for digit. (3) It downloads the first 10,000 decimal digits of sqrt(2), e and pi from OEIS and, for every p <= N/3, finds the smallest index after which the digit sequence agrees with its shift by p, flagging any p for which two full periods already agree. Run time about 9 s including downloads." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered: (1) for bases 2, 3 and 10 and all denominators q = 1..20000, the digit expansion of 1/q (and of p/q for all 1 <= p < q when q <= 300) has period <= q - 1 and preperiod + period <= q, with zero violations, and the base-10 periods of 1/q agree with OEIS A007732 for every n covered by its b-file; (2) 100% of 20000 random eventually periodic digit patterns (random preperiod of length 0..20, random non-trivial period block of length 1..30, bases 2, 3, 10) are reproduced exactly when the rational number reconstructed from them is re-expanded by long division; (3) none of the first 10000 decimal digits of sqrt(2), e and pi (OEIS b-files) is eventually periodic in the sense that, for some period p <= N/3, the last 2p digits already repeat with period p." } ], "https://w3id.org/delphi/ontology/preregistered": [ { "@value": "false", "@type": "http://www.w3.org/2001/XMLSchema#boolean" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-1", "http://purl.org/dc/terms/description": [ { "@value": "direction rational => eventually periodic with period <= q - 1" } ], "http://purl.org/dc/terms/license": [ { "@value": "" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: Preperiod and period of 1/q (q <= 20000) and p/q (q <= 300) in bases 2, 3, 10 by long division (generated by analysis.py)" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-2", "http://purl.org/dc/terms/description": [ { "@value": "independent tabulation of decimal periods for cross-checking" } ], "http://purl.org/dc/terms/license": [ { "@value": "CC BY-SA 4.0 (OEIS)" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: OEIS A007732: period of decimal representation of 1/n, b-file" } ], "http://www.w3.org/ns/prov#atLocation": [ { "@id": "https://oeis.org/A007732/b007732.txt" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://oeis.org/A007732" } ], "https://w3id.org/delphi/ontology/hasChecksum": [ { "@value": "sha256:eb0e8d499b46a8c12bdb8d9f747ca6d09f322a2de1ddd26971b95d4c89dca730" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-3", "http://purl.org/dc/terms/description": [ { "@value": "direction eventually periodic => rational (constructive)" } ], "http://purl.org/dc/terms/license": [ { "@value": "" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: 20000 random eventually periodic digit patterns (numpy default_rng seed 20261031; generated by analysis.py)" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-4", "http://purl.org/dc/terms/description": [ { "@value": "irrational sanity check" } ], "http://purl.org/dc/terms/license": [ { "@value": "CC BY-SA 4.0 (OEIS)" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: OEIS A002193: decimal expansion of sqrt(2), b-file" } ], "http://www.w3.org/ns/prov#atLocation": [ { "@id": "https://oeis.org/A002193/b002193.txt" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://oeis.org/A002193" } ], "https://w3id.org/delphi/ontology/hasChecksum": [ { "@value": "sha256:ca6c74f68ed066a0d703ce5bf2eb0f721fa07a7d233596ec9f86355d22a796e8" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-5", "http://purl.org/dc/terms/description": [ { "@value": "irrational sanity check" } ], "http://purl.org/dc/terms/license": [ { "@value": "CC BY-SA 4.0 (OEIS)" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: OEIS A001113: decimal expansion of e, b-file" } ], "http://www.w3.org/ns/prov#atLocation": [ { "@id": "https://oeis.org/A001113/b001113.txt" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://oeis.org/A001113" } ], "https://w3id.org/delphi/ontology/hasChecksum": [ { "@value": "sha256:9eef3a3fe766065b330d8f1f398efc62f19cd3af299af8ca252ef98818a3dc9d" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-14--claim-test--dataset-6", "http://purl.org/dc/terms/description": [ { "@value": "irrational sanity check" } ], "http://purl.org/dc/terms/license": [ { "@value": "CC BY-SA 4.0 (OEIS)" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: OEIS A000796: decimal expansion of pi, b-file" } ], "http://www.w3.org/ns/prov#atLocation": [ { "@id": "https://oeis.org/A000796/b000796.txt" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://oeis.org/A000796" } ], "https://w3id.org/delphi/ontology/hasChecksum": [ { "@value": "sha256:f8772370d624d403f056e3e3d4aadb15fec135d5075b5506053eb86460f4e20b" } ] }, { "@id": "https://w3id.org/delphi/resource/claim/wolfram2002-ch04-14", "https://w3id.org/delphi/ontology/hasAssessment": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch04-14--claim-test" } ] } ] }, { "@id": "https://w3id.org/np/RAHoH2699Qk2gXT9TKw-nttCfxqgM5R4o5QEy6ZchQ2kA/pubinfo", "@graph": [ { "@id": "https://w3id.org/np/RAHoH2699Qk2gXT9TKw-nttCfxqgM5R4o5QEy6ZchQ2kA", "http://purl.org/dc/terms/created": [ { "@value": "2026-11-08T16:24:31.923Z", "@type": "http://www.w3.org/2001/XMLSchema#dateTime" } ], "http://purl.org/dc/terms/creator": [ { "@id": "https://orcid.org/0009-0007-4948-9192" } ], "http://purl.org/dc/terms/license": [ { "@id": "https://creativecommons.org/licenses/by/4.0/" } ], "http://purl.org/nanopub/x/hasNanopubType": [ { "@id": "https://w3id.org/delphi/ontology/Assessment" } ], "http://purl.org/nanopub/x/introduces": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch04-14--claim-test" } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch04-14: supports, confidence 0.97" } ], "https://w3id.org/delphi/ontology/domain": [ { "@id": "https://w3id.org/delphi/resource/domain/wolfram-nks-2002" } ] }, { "@id": "https://w3id.org/np/RAHoH2699Qk2gXT9TKw-nttCfxqgM5R4o5QEy6ZchQ2kA/sig", "http://purl.org/nanopub/x/hasAlgorithm": [ { "@value": "RSA" } ], "http://purl.org/nanopub/x/hasPublicKey": [ { "@value": "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB" } ], "http://purl.org/nanopub/x/hasSignature": [ { "@value": "AK3GRGXQQi0JkTIIEyTeS0bxaQhVyX+urF7D68y+4LsBlULvhVaJskXdE760BRF2yuQBXa1nawRxTDCXTYzYrO9nDMZwIG+ZjfQkGhESZzSfXBRYUpzxo/WCIVe2fc3rax5QI3VV1K/z6MI5e4uUeUZoFd6/r1I3aWgYQyYgf7X22zRbKkqi6yjVSgUy+jODNnNwNu0cJo6Kaw4f86XSeRknW0qrackhqJWll1ZBvAvFnjv9E5/wyHkzU9Tzqa3mZ/aRdJie7cbhhyttyR2cO+CeoT+au19rbG3L3i0I6w8olZaqgJzmH3hMw/ePWK+C9KzsgrqvmeO08BZJG4qgog==" } ], "http://purl.org/nanopub/x/hasSignatureTarget": [ { "@id": "https://w3id.org/np/RAHoH2699Qk2gXT9TKw-nttCfxqgM5R4o5QEy6ZchQ2kA" } ], "http://purl.org/nanopub/x/signedBy": [ { "@id": "https://orcid.org/0009-0007-4948-9192" } ] } ] } ]