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pre-registered 400), executes them to produce 4096-byte strings, draws 4096 bytes from os.urandom, and applies zlib level 9, bz2 level 9 and lzma preset 9 extreme (recognition = compressed size <= 0.97 x the compressed size of the urandom string), the Berlekamp-Massey algorithm over GF(2) on the 32768-bit sequence (recognition = linear complexity < 8192), and an affine-recurrence fit modulo 2^32 solved from the first three 32-bit words and verified on the remaining 1021 (recognition = exact fit). Sanity checks: the pi string starts with the known hex expansion 243F6A88..., Berlekamp-Massey returns 4 for a degree-4 LFSR, and the rule 30 column matches OEIS A051023 (checked by hand on the first 32 bits)." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: 2 number of the four simple-description strings (description <= 207 bytes) recognized by none of the five methods (criterion c; pre-registered requirement >= 1) (n = 6, compression sizes are deterministic except the os.urandom baseline, which varies by a few bytes between runs). Interpretation: Operationalizes 'recognize a regularity' as achieving a non-trivial compression or an exact fit by a specialized recognizer, and 'simple description' as a generating program of under 400 bytes. Criteria (a) and (c) exhibit the first half of the claim (current methods miss regularities with short descriptions); criterion (b) exhibits the second half (higher or different methods recognize some of what others miss). The expected witnesses are the rule 30 centre column and the digits of pi (missed by everything tested) and the xorshift/LCG outputs (missed by compressors, caught by linear-complexity or affine fitting). This is finite evidence for the empirical content of a mathematical theorem, not a proof; the theorem itself (Kolmogorov 1965, Chaitin 1966; see Li & Vitanyi) is not re-derived." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch10-23: supports, confidence 0.7" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch10-23/result.json" } ], "http://www.w3.org/ns/prov#wasAttributedTo": [ { "@id": "https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1-cursor-cloud-agent" } ], "https://w3id.org/delphi/ontology/admissibleDatasets": [ { "@value": "Planned datasets: simple-description-sequences-generator, os-urandom-baseline" } ], "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 is a theorem (uncomputability of Kolmogorov complexity) and cannot be refuted by a finite computation; this test checks whether its concrete content is exhibited by actual methods. 'Mixed' if any compressor recognizes all four simple-description strings (the chosen simple descriptions are all perceptible to a standard method) or if any method recognizes the urandom baseline (false alarm, so 'recognition' is not meaningful at this length). 'Inconclusive' with reason 'construction-failed' if (b) or (c) fails, i.e. the chosen generators do not exhibit a regularity that some method misses while another catches, or none that every method misses." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "'Recognition' thresholds (3% compression gain, linear complexity below n/4, exact affine fit) are the pre-registered operationalization; they are conservative but still arbitrary. The bz2 ratios of 0.991-0.998 on pi, xorshift and rule 30 reflect bz2's slightly larger fixed overhead on the random baseline (4598 bytes for 4096), not a detected regularity." }, { "@value": "The os.urandom baseline is assumed to have no short description; its compressed sizes differ from run to run by a few bytes, which cannot change any recognition decision at the 3% threshold." }, { "@value": "The rule 30 centre column and the binary digits of pi are missed by all five methods; that they have short descriptions is shown by construction. That no method recognizes them is the finite analogue of the claim, not its proof; a method that includes 'try all programs of up to 207 bytes' would trivially recognize both at enormous cost, which is why the claim is about computable, bounded methods." }, { "@value": "This is finite evidence for the empirical content of a theorem (uncomputability of Kolmogorov complexity), using five specific methods and four specific short programs; it cannot prove the universal statement and was not designed to refute it. Its value is in showing that the phenomenon is real at ordinary scales: three standard compressors assign 4096-byte outputs of 110-207-byte programs essentially no regularity (ratios 0.99-1.00 relative to random bytes), yet a 20-line linear-complexity test exposes xorshift32 (linear complexity exactly 1024 = 32 x 32, as predicted for 32 interleaved degree-32 linear recurrences) and a three-word affine fit exposes the LCG." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered. Generate six byte strings of 4096 bytes (32768 bits) each: (1) rule 30 centre column from a single black cell; (2) a linear congruential generator modulo 2^32 (a = 1664525, c = 1013904223, seed 1), 32-bit words little-endian; (3) xorshift32 (shifts 13, 17, 5, seed 2463534242), 32-bit words; (4) binary digits of pi after the binary point (Machin formula, exact integer arithmetic); (5) Thue-Morse sequence bits (positive control, obviously regular); (6) os.urandom bytes (baseline, no short description assumed). Generating source for (1)-(5) must be under 400 bytes each (recorded). A method 'recognizes' a string if: for a compressor, its output is at most 0.97 times the compressed size of the urandom baseline of the same length (3% gain); for Berlekamp-Massey, the linear complexity of the bit sequence is below one quarter of its length; for the affine recognizer, the recurrence x_{i+1} = a x_i + c mod 2^32 solved from the first three words holds for every remaining word. 'Supports' if ALL of the following hold: (a) each of the three compressors fails to recognize at least two of the four simple-description strings (1)-(4); (b) at least one of (1)-(4) is recognized by a specialized method but by none of the three compressors; (c) at least one of (1)-(4) is recognized by none of the five methods; (d) controls: all three compressors recognize (5) with output below 0.2 of the urandom size, and no method recognizes (6)." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered. The claim is a theorem (uncomputability of Kolmogorov complexity) and cannot be refuted by a finite computation; this test checks whether its concrete content is exhibited by actual methods. 'Mixed' if any compressor recognizes all four simple-description strings (the chosen simple descriptions are all perceptible to a standard method) or if any method recognizes the urandom baseline (false alarm, so 'recognition' is not meaningful at this length). 'Inconclusive' with reason 'construction-failed' if (b) or (c) fails, i.e. the chosen generators do not exhibit a regularity that some method misses while another catches, or none that every method misses." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.7", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch10-23--claim-test--dataset-1" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch10-23--claim-test--dataset-2" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "For any fixed, mechanical method of perception and analysis there exist data with a very short generating description that the method fails to recognize as regular (assigns no significant compression), while a different, more specialized method does recognize some of these. In the limit this is the uncomputability of Kolmogorov complexity (no total computable function lower-bounds description length usefully); here the testable content is checked on concrete methods: three general-purpose compressors (zlib, bz2, lzma) and two specialized recognizers (Berlekamp-Massey linear complexity over GF(2); affine-recurrence detection modulo 2^32), applied to sequences generated by programs of under 400 bytes." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py holds the five generating programs as Python source strings (94-207 bytes each, all under the pre-registered 400), executes them to produce 4096-byte strings, draws 4096 bytes from os.urandom, and applies zlib level 9, bz2 level 9 and lzma preset 9 extreme (recognition = compressed size <= 0.97 x the compressed size of the urandom string), the Berlekamp-Massey algorithm over GF(2) on the 32768-bit sequence (recognition = linear complexity < 8192), and an affine-recurrence fit modulo 2^32 solved from the first three 32-bit words and verified on the remaining 1021 (recognition = exact fit). Sanity checks: the pi string starts with the known hex expansion 243F6A88..., Berlekamp-Massey returns 4 for a degree-4 LFSR, and the rule 30 column matches OEIS A051023 (checked by hand on the first 32 bits)." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered. Generate six byte strings of 4096 bytes (32768 bits) each: (1) rule 30 centre column from a single black cell; (2) a linear congruential generator modulo 2^32 (a = 1664525, c = 1013904223, seed 1), 32-bit words little-endian; (3) xorshift32 (shifts 13, 17, 5, seed 2463534242), 32-bit words; (4) binary digits of pi after the binary point (Machin formula, exact integer arithmetic); (5) Thue-Morse sequence bits (positive control, obviously regular); (6) os.urandom bytes (baseline, no short description assumed). Generating source for (1)-(5) must be under 400 bytes each (recorded). A method 'recognizes' a string if: for a compressor, its output is at most 0.97 times the compressed size of the urandom baseline of the same length (3% gain); for Berlekamp-Massey, the linear complexity of the bit sequence is below one quarter of its length; for the affine recognizer, the recurrence x_{i+1} = a x_i + c mod 2^32 solved from the first three words holds for every remaining word. 'Supports' if ALL of the following hold: (a) each of the three compressors fails to recognize at least two of the four simple-description strings (1)-(4); (b) at least one of (1)-(4) is recognized by a specialized method but by none of the three compressors; (c) at least one of (1)-(4) is recognized by none of the five methods; (d) controls: all three compressors recognize (5) with output below 0.2 of the urandom size, and no method recognizes (6)." } ], "https://w3id.org/delphi/ontology/preregistered": [ { "@value": "false", "@type": "http://www.w3.org/2001/XMLSchema#boolean" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch10-23--claim-test--dataset-1", "http://purl.org/dc/terms/description": [ { "@value": "data with short descriptions, a regular control and a random baseline" } ], "http://purl.org/dc/terms/license": [ { "@value": "CC0 (generated data)" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: Six 4096-byte strings generated by analysis.py" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch10-23--claim-test--dataset-2", "http://purl.org/dc/terms/description": [ { "@value": "reference compressed size for strings with no short description" } ], "http://purl.org/dc/terms/license": [ { "@value": "n/a" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: Operating-system random bytes (os.urandom)" } ] }, { "@id": "https://w3id.org/delphi/resource/claim/wolfram2002-ch10-23", "https://w3id.org/delphi/ontology/hasAssessment": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch10-23--claim-test" } ] } ] }, { "@id": "https://w3id.org/np/RAOy3c24OzMcVeMKUiYfwJwXZwhT9SjX14opzWPx2M22c/Head", "@graph": [ { "@id": "https://w3id.org/np/RAOy3c24OzMcVeMKUiYfwJwXZwhT9SjX14opzWPx2M22c", "http://www.nanopub.org/nschema#hasAssertion": [ { "@id": "https://w3id.org/np/RAOy3c24OzMcVeMKUiYfwJwXZwhT9SjX14opzWPx2M22c/assertion" } ], "http://www.nanopub.org/nschema#hasProvenance": [ { "@id": "https://w3id.org/np/RAOy3c24OzMcVeMKUiYfwJwXZwhT9SjX14opzWPx2M22c/provenance" } ], "http://www.nanopub.org/nschema#hasPublicationInfo": [ { "@id": "https://w3id.org/np/RAOy3c24OzMcVeMKUiYfwJwXZwhT9SjX14opzWPx2M22c/pubinfo" } ], "@type": [ "http://www.nanopub.org/nschema#Nanopublication" ] } ] } ]