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Per-pair results are in derived/pairs.csv. Runtime about 4 minutes." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: 43 pairs (m,b) with m not a power of b whose interior digit array is 'complex' by the pre-registered criteria (n = 53). Interpretation: Wolfram's 'even rather simple systems based on numbers can lead to highly complex behavior' (p. 115) is illustrated by the digit sequences of successive powers (p. 116-120: 2^t in base 2 is trivial, 3^t in base 2 looks random). 'Highly complex behavior' is read as digit arrays that a general-purpose compressor cannot reduce below 80% of their entropy bound and whose digit frequencies are uniform to within 2%, in explicit contrast to the trivial members of the same family that compress to below 10%. The claim is existential; the family is the smallest natural one containing the book's opening examples. The window excludes the leading digits (biased by Benford-type effects) and trailing digits (eventually periodic in t at fixed position, and all zero when b divides m)." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch04-02: supports, confidence 0.9" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch04-02/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: nks-powers-digit-arrays" } ], "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. The claim FAILS if no pair with m not a power of b is 'complex' (every such digit array compresses below 0.8 log2(b) bits per digit or has digit frequencies deviating from uniform by more than 0.02). If any trivial pair (m a power of b) is not 'simple', the measure lacks discriminating power and the verdict is inconclusive (not-operationalizable)." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "Incompressibility by lzma over about 3.6 million interior digits (for 3^t in base 2: 1.086 bits/digit, digit frequencies within 0.0004 of 1/2) is evidence against periodic or nested structure at these scales, not a proof of normality; the digits of 3^t in base 2 are not proven to be normal." }, { "@value": "The 10 non-complex non-trivial pairs split into two explicable groups: (2,4), (2,8), (4,8), (8,4), (3,9) where m and b are powers of a common integer (the pre-registered 'm is a power of b' control missed this symmetric case; they compress to < 0.001 of the bound, i.e. are trivially regular), and the five m = 6 pairs in bases 2, 3, 4, 8, 9 where the trailing zeros of 6^t = 2^t 3^t occupy 1/log_b(6) of each row and intrude into the fixed 25%-75% window, lowering compression ratios to 0.30-0.83 and biasing digit frequencies." }, { "@value": "The test covers the book's opening family (powers in positional notation) only; the claim's other illustrations in Chapter 4 (e.g. the 3n/2 system, iterated maps) were not tested." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered. Family: all pairs (m, b) with multiplier m in 2..9 and base b in 2..9, iterating x_{t+1} = m x_t from x_0 = 1 for t = 1..T = 3000 (i.e. the digits of m^t). For each pair build the digit array of rows m^t written in base b; take the interior window of each row consisting of digit positions between 25% and 75% of that row's length (least significant digit = position 0), concatenate all interior digits in row order into one byte string (one digit per byte). Measure (i) bits per digit = 8 * lzma(preset 9|EXTREME) compressed size / number of digits; (ii) maximum absolute deviation of any digit-value frequency from 1/b. A pair is 'complex' if (i) >= 0.8 * log2(b) and (ii) <= 0.02; a pair is 'simple' if (i) < 0.1 * log2(b). The claim PASSES if at least one pair with m not a power of b is 'complex' AND every pair with m a power of b (m = b^j, j >= 1, e.g. (2,2), (4,2), (8,2), (9,3), (3,3), ...) is 'simple' (validity check that the measure separates trivially regular digit arrays). Reported, not decisive: how many of the 64 pairs are complex, and whether the pairs with m and b sharing all prime factors (e.g. (6,2), (4,8), (2,4)) behave differently." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered. The claim FAILS if no pair with m not a power of b is 'complex' (every such digit array compresses below 0.8 log2(b) bits per digit or has digit frequencies deviating from uniform by more than 0.02). If any trivial pair (m a power of b) is not 'simple', the measure lacks discriminating power and the verdict is inconclusive (not-operationalizable)." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.9", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-02--claim-test--dataset-1" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "Within the family of the simplest arithmetic iterations, repeated multiplication by a fixed small integer m viewed as a base-b digit array (Wolfram's opening example, pp. 116-120), some members produce digit arrays whose interior is incompressible with digit frequencies indistinguishable from uniform, while the trivial members (m a power of b) are detected as highly compressible." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py iterates x -> m x from 1 for 3000 steps for each of the 64 (m, b) pairs using Python integers, converts each power to base-b digits, keeps the interior 25%-75% of digit positions per row, concatenates (one byte per digit) and lzma-compresses; it also computes digit-value frequencies in the window. Per-pair results are in derived/pairs.csv. Runtime about 4 minutes." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered. Family: all pairs (m, b) with multiplier m in 2..9 and base b in 2..9, iterating x_{t+1} = m x_t from x_0 = 1 for t = 1..T = 3000 (i.e. the digits of m^t). For each pair build the digit array of rows m^t written in base b; take the interior window of each row consisting of digit positions between 25% and 75% of that row's length (least significant digit = position 0), concatenate all interior digits in row order into one byte string (one digit per byte). Measure (i) bits per digit = 8 * lzma(preset 9|EXTREME) compressed size / number of digits; (ii) maximum absolute deviation of any digit-value frequency from 1/b. A pair is 'complex' if (i) >= 0.8 * log2(b) and (ii) <= 0.02; a pair is 'simple' if (i) < 0.1 * log2(b). The claim PASSES if at least one pair with m not a power of b is 'complex' AND every pair with m a power of b (m = b^j, j >= 1, e.g. (2,2), (4,2), (8,2), (9,3), (3,3), ...) is 'simple' (validity check that the measure separates trivially regular digit arrays). Reported, not decisive: how many of the 64 pairs are complex, and whether the pairs with m and b sharing all prime factors (e.g. (6,2), (4,8), (2,4)) behave differently." } ], "https://w3id.org/delphi/ontology/preregistered": [ { "@value": "false", "@type": "http://www.w3.org/2001/XMLSchema#boolean" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-02--claim-test--dataset-1", "http://purl.org/dc/terms/description": [ { "@value": "program family under test" } ], "http://purl.org/dc/terms/license": [ { "@value": "generated data" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: Digit arrays of m^t in base b for m, b in 2..9, t = 1..3000 (generated by analysis.py)" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://www.wolframscience.com/nks/p116--elementary-arithmetic/" } ] }, { "@id": "https://w3id.org/delphi/resource/claim/wolfram2002-ch04-02", "https://w3id.org/delphi/ontology/hasAssessment": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch04-02--claim-test" } ] } ] } ]