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The same function is applied to a checkerboard of length 12288 and to 12288 bits from random.Random(20261102). Thresholds in the script are the pre-registered ones. The run writes derived/measured.json. A second run matched that file byte for byte." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: 0.9983424040186749 normalized 4-bit block entropy h4 of bits j=32..95 of 3^n for n=64..255 (n = 3072, plugin Shannon entropy on this finite window; no sampling of the powers). Interpretation: Wolfram's pattern is the base-2 digits of 3^n stacked by n. 'Small-scale structure' is read as the lowest 4 bits, which repeat because the multiplicative order of 3 modulo 16 is 4; that fact fixes the column window and is not a measurement of the bulk. 'Appears random for all practical purposes' on a larger scale is read as normalized 4-bit block entropy of interior bits j = 32..95, compared with a random bit string of equal length rather than with a proof of normality. The repetitive control is a checkerboard, not another arithmetic sequence. The computation shows only how these finite digit strings behave. It says nothing about organisms." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch04-03: supports, confidence 0.78" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch04-03/result.json" } ], "http://www.w3.org/ns/prov#wasAttributedTo": [ { "@id": "https://w3id.org/delphi/resource/agent/delphisci-claim-tester-grok-4-7" } ], "https://w3id.org/delphi/ontology/admissibleDatasets": [ { "@value": "The dataset is the deterministic sequence 3^n written in base 2, generated by the checksummed script, for the locked range n = 64..255 and the locked columns. A same-length checkerboard and a same-length seeded pseudorandom string are the controls. No organism data are admissible for this computational claim." } ], "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": "With the checkerboard still at block entropy at most 0.25, the interior digits have block entropy at least 0.20 below the random reference. If the checkerboard itself exceeds 0.25, the statistic is not discriminating and the test is inconclusive." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "Small-scale structure is demonstrated only for the lowest 4 bits. Columns between bit 4 and bit 31 were not part of the pass rule." }, { "@value": "The check is finite: n stops at 255 and the blocks are 4 bits. A longer run, or blocks long enough to see multiplication by 3, could show structure that this statistic treats as random." }, { "@value": "The random reference is a seeded Python generator of the same length, not a cryptographic source, and h4 near 1 is a local frequency statement, not a proof that the digits are normal." }, { "@value": "The result is a computational observation about these powers. It says nothing about organisms." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Interior base-2 digits of 3^n for n = 64..255 have 4-bit block entropy within 0.05 of a seeded random string of the same length, the lowest 4 bits have block entropy at least 0.15 lower than that interior, and a checkerboard control has block entropy at most 0.25." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered. Refutes if h4(checkerboard) <= 0.25 and h4(bulk) <= h4(random) - 0.20 (the control is repetitive and the bulk is clearly less random than the reference). Inconclusive if h4(checkerboard) > 0.25 (the repetitive control was not repetitive under this statistic). Mixed in every other case." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.78", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-03--claim-test--dataset-1" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-03--claim-test--dataset-2" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-03--claim-test--dataset-3" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "The base-2 digits of successive powers of 3, away from the least significant bits, have 4-bit block entropy close to a same-length random reference, while the lowest 4 bits are substantially less random, and a checkerboard of the same length is far less random than either." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "Normalized Shannon entropy of non-overlapping 4-bit blocks. Supports only when the bulk is within 0.05 of the random reference, the small-scale window is at least 0.15 below the bulk, and the checkerboard is at most 0.25. Refutes when the control works and the bulk is at least 0.20 below the reference. Otherwise mixed, or inconclusive if the control fails." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered, all three required. Rows n = 64 through 255 (192 powers). Bit 0 is the least significant bit. Small-scale stream: bits j = 0,1,2,3 of each power, concatenated in that order (192 blocks of 4 bits). Bulk stream: bits j = 32 through 95 of each power, concatenated (12288 bits, 3072 blocks). Statistic h4 is the Shannon entropy, in bits, of non-overlapping 4-bit blocks, divided by 4. Repetitive control: the checkerboard bit k = k mod 2 of length 12288. Random reference: 12288 bits from random.Random(20261102).randrange(2), in order. 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