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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." } g_iri { prefix_id: 2 name_id: 4 } } } rows { prefix { value: "https://w3id.org/delphi/ontology/" } } rows { name { value: "Assessment" } } rows { quad { p_iri { prefix_id: 4 name_id: 10 } o_iri { prefix_id: 10 name_id: 14 } } } rows { prefix { value: "http://www.w3.org/2000/01/rdf-schema#" } } rows { name { value: "comment" } } rows { quad { p_iri { prefix_id: 11 } o_literal { lex: "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." } } } rows { name { value: "label" } } rows { quad { p_iri { } o_literal { lex: "Claim test of wolfram2002-ch04-14: supports, confidence 0.97" } } } rows { name { value: "seeAlso" } } rows { prefix { value: "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch04-14/" } } rows { name { value: "result.json" } } rows { quad { p_iri { } o_iri { prefix_id: 12 } } } rows { name { value: "wasAttributedTo" } } rows { prefix { value: "https://w3id.org/delphi/resource/agent/" } } rows { name { value: "delphisci-claim-tester-claude-fable-5-1" } } rows { quad { p_iri { prefix_id: 5 } o_iri { prefix_id: 13 } } } rows { name { value: "admissibleDatasets" } } rows { quad { p_iri { prefix_id: 10 } o_literal { lex: "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" } } } rows { name { value: "assessmentType" } } rows { quad { p_iri { } o_literal { lex: "claim-test/0.1.0" } } } rows { name { value: "dataKind" } } rows { quad { p_iri { } o_literal { lex: "computational" } } } rows { name { value: "disconfirmingObservation" } } rows { quad { p_iri { } o_literal { lex: "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." } } } rows { name { value: "evidenceAgainst" } } rows { quad { p_iri { } o_literal { lex: "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." } } } rows { quad { o_literal { lex: "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." } } } rows { quad { o_literal { lex: "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." } } } rows { name { value: "expectedObservation" } } rows { quad { p_iri { } o_literal { lex: "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." } } } rows { name { value: "failCriterion" } } rows { quad { p_iri { } o_literal { lex: "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." } } } rows { name { value: "hasConfidence" } } rows { datatype { value: "http://www.w3.org/2001/XMLSchema#double" } } rows { quad { p_iri { } o_literal { lex: "0.97" datatype: 1 } } } rows { name { value: "hasEvidence" } } rows { prefix { value: "https://w3id.org/delphi/resource/evidence/" } } rows { name { value: "wolfram2002-ch04-14--claim-test--dataset-1" } } rows { quad { p_iri { } o_iri { prefix_id: 14 } } } rows { name { value: "wolfram2002-ch04-14--claim-test--dataset-2" } } rows { quad { o_iri { } } } rows { name { value: "wolfram2002-ch04-14--claim-test--dataset-3" } } rows { quad { o_iri { } } } rows { name { value: "wolfram2002-ch04-14--claim-test--dataset-4" } } rows { quad { o_iri { } } } rows { name { value: "wolfram2002-ch04-14--claim-test--dataset-5" } } rows { quad { o_iri { } } } rows { name { value: "wolfram2002-ch04-14--claim-test--dataset-6" } } rows { quad { o_iri { } } } rows { name { value: "hasVerdict" } } rows { quad { p_iri { prefix_id: 10 } o_literal { lex: "supports" } } } rows { name { value: "hypothesis" } } rows { quad { p_iri { } o_literal { lex: "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." } } } rows { name { value: "inferenceMethod" } } rows { quad { p_iri { } o_literal { lex: "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." } } } rows { name { value: "passCriterion" } } rows { quad { p_iri { } o_literal { lex: "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." } } } rows { name { value: "preregistered" } } rows { datatype { value: "http://www.w3.org/2001/XMLSchema#boolean" } } rows { quad { p_iri { } o_literal { lex: "false" datatype: 2 } } } rows { prefix { value: "https://w3id.org/delphi/resource/claim/" } } rows { name { value: "wolfram2002-ch04-14" } } rows { name { value: "hasAssessment" } } rows { quad { s_iri { prefix_id: 15 } p_iri { prefix_id: 10 } o_iri { prefix_id: 9 name_id: 12 } } } rows { quad { s_iri { prefix_id: 14 name_id: 30 } p_iri { prefix_id: 7 name_id: 13 } o_literal { lex: "direction rational => eventually periodic with period <= q - 1" } } } rows { name { value: "license" } } rows { quad { p_iri { name_id: 43 } o_literal { } } } rows { name { value: "EvidenceItem" } } rows { quad { p_iri { prefix_id: 4 name_id: 10 } o_iri { prefix_id: 10 name_id: 44 } } } rows { quad { p_iri { prefix_id: 11 name_id: 16 } o_literal { lex: "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)" } } } rows { quad { s_iri { prefix_id: 14 name_id: 31 } p_iri { prefix_id: 7 name_id: 13 } o_literal { lex: "independent tabulation of decimal periods for cross-checking" } } } rows { quad { p_iri { name_id: 43 } o_literal { lex: "CC BY-SA 4.0 (OEIS)" } } } rows { quad { p_iri { prefix_id: 4 name_id: 10 } o_iri { prefix_id: 10 name_id: 44 } } } rows { quad { p_iri { prefix_id: 11 name_id: 16 } o_literal { lex: "Dataset: OEIS A007732: period of decimal representation of 1/n, b-file" } } } rows { name { value: "atLocation" } } rows { prefix { value: "https://oeis.org/A007732/" } } rows { name { value: "b007732.txt" } } rows { quad { p_iri { prefix_id: 5 name_id: 45 } o_iri { prefix_id: 16 } } } rows { name { value: "wasDerivedFrom" } } rows { prefix { id: 6 value: "https://oeis.org/" } } rows { name { value: "A007732" } } rows { quad { p_iri { prefix_id: 5 } o_iri { prefix_id: 6 } } } rows { name { value: "hasChecksum" } } rows { quad { p_iri { prefix_id: 10 } o_literal { lex: "sha256:eb0e8d499b46a8c12bdb8d9f747ca6d09f322a2de1ddd26971b95d4c89dca730" } } } rows { quad { s_iri { prefix_id: 14 name_id: 32 } p_iri { prefix_id: 7 name_id: 13 } o_literal { lex: "direction eventually periodic => rational (constructive)" } } } rows { quad { p_iri { name_id: 43 } o_literal { } } } rows { quad { p_iri { prefix_id: 4 name_id: 10 } o_iri { prefix_id: 10 name_id: 44 } } } rows { quad { p_iri { prefix_id: 11 name_id: 16 } o_literal { lex: "Dataset: 20000 random eventually periodic digit patterns (numpy default_rng seed 20261031; 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