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It then runs each of the 256 rules for 4096 steps from a single black cell (array width 2(T+128)+1 so that zero-boundary defects of background-flipping rules cannot reach the strip), computes rho, tests the strip for time-periodicity with period <= 64 over rows 2048-4095, measures the black-cell extent at rows 2047 and 4095, and classifies: random-looking if rho >= tau; nested if not random, not periodic and growing; regular otherwise. Random-looking rules are grouped into orbits under reflection and complementation. Run time about 50 s." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: [object Object] rule counts (of 256) (n = 256, deterministic given the calibrated threshold tau = 0.190; the counts are sensitive to the classification proxies (see caveats)). Interpretation: 'Seem in many respects random' is read as 'the central region of the pattern is essentially incompressible, as judged against i.i.d. bits and classical nested patterns by a threshold fixed before the run'; 'nested' is read as 'grows, is not time-periodic, and is highly compressible'. Both are proxies for Wolfram's visual classification, so counts within a few of his 10 and 24 are treated as agreement. 'Basic forms' for the random rules is read as symmetry classes (the natural reading given that 10 = 4 + 4 + 2 under the 88-class symmetry reduction); for the nested rules Wolfram's three forms are visual (rule 22/60 type, rule 150 type, rule 225 type) and are only approximated by the dimension/width clustering. All patterns are generated by analysis.py; there is no external download." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch03-03: supports, confidence 0.6" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch03-03/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-ch03-03-eca-single-seed-generator" } ], "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. 'refutes' if the random-looking count is <= 5 or >= 16, or the nested count is <= 14 or >= 36, under the calibrated criterion. Otherwise 'mixed' (e.g. counts near but outside the pass windows, or random-looking rules forming a number of symmetry classes other than 3). 'inconclusive' with reason 'other' if the calibration fails to separate random from nested references." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "A bug was fixed between the first and the final run: with array width 2T+1, zero-boundary defects of rules with f(000) = 1 (background flips every step) reached the last 64 rows of the central strip and made 54 regular rules look non-periodic (first-run nested count 78 -> 'refutes'). The array was widened by 128 cells on each side so the strip is causally isolated from the boundary; nothing else in the pre-registered procedure was changed." }, { "@value": "Both pre-registered count windows were met: 9 random-looking rules (Wolfram: 10) in exactly 3 symmetry classes, represented by rules 30, 45 and 73 (the classes {30, 86, 135, 149}, {45, 75, 89, 101} and {73, 109}), and 24 nested rules (Wolfram: 24). The one missing random-looking rule is 109, the complement-reflection partner of 73: under this criterion its central strip is time-periodic (period 3), i.e. the randomness of rule 109's single-seed pattern lies outside the central 128 columns or is of a kind the strip statistic does not capture. The random/non-random gap is huge: the nine rules have rho 0.57-1.11, the next rule 0.085." }, { "@value": "The nested count of 24 agrees with Wolfram's number but not entirely with his membership: this census includes the rule 110 class (110, 124, 137, 193), which is compressible, growing and non-periodic but which Wolfram would not call nested, and excludes rules such as 60/195 (growth is one-sided, so the seed-centred strip sits on the pattern's periodic edge), 151/183 and 225 (background-flipping rules whose central strip is periodic at this scale). The agreement in count is therefore partly coincidental and the membership differences are reported for transparency." }, { "@value": "The secondary 'three forms' check for nested rules was not reproduced: the black-cell exponent D separates the rule 90/22 form (D = 1.585, 13 rules), the rule 150 form (1.694) and the rule 110/182 group (about 2.0), but is meaningless for background-flipping rules (D 0.73-0.99), so 7 bins were found. Wolfram's forms are visual and his third form (rule 225, width ~ sqrt(t)) is not in this nested list at all." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered. Each rule is run for T = 4096 steps from a single black cell on a white background (zero boundary, width 2T+1). Statistics: (i) rho = 8 * lzma_bytes / cells of the central strip (columns -64..63 relative to the seed, rows 1024..4095, one byte per cell, preset 9); (ii) periodic: the strip rows are time-periodic with some period p <= 64 over rows 2048..4095; (iii) growing: the black-cell extent of row 4095 is at least 1.2 times that of row 2047, or the extent of row 4095 exceeds 64 cells while the strip is not periodic. Calibration (performed before the rules are run): random references are i.i.d. fair bits (numpy seed 20260303) in a 128 x 3072 strip; nested references are the same-size central strips of the Sierpinski gasket (Pascal's triangle mod 2), Pascal's triangle mod 3 thresholded to nonzero, and the outer products t_i XOR t_j of the Thue-Morse and of the period-doubling sequences; periodic references are random patterns of period 1..64 in time and 1..16 in space. tau_random = geometric mean of the smallest random-reference rho and the largest nested-reference rho. Classification: 'random-looking' iff rho >= tau_random; 'nested' iff not random-looking, not periodic, and growing; 'regular' otherwise. Symmetry classes: orbits under reflection (swap left and right inputs) and complementation (conjugation by colour swap). Secondary, reported but not decisive: for nested rules, the black-cell count exponent D = log2(N(4096)/N(2048)) with N(t) = black cells in rows 0..t-1, and the width exponent log2(width(4095)/width(2047)); the number of clusters when D is binned to 0.1 and the width exponent to {about 1, about 0.5} is compared with Wolfram's 'three fundamentally different forms'. 'supports' if the random-looking count is in [8, 12] and forms exactly 3 symmetry classes, and the nested count is in [20, 28]." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered. 'refutes' if the random-looking count is <= 5 or >= 16, or the nested count is <= 14 or >= 36, under the calibrated criterion. Otherwise 'mixed' (e.g. counts near but outside the pass windows, or random-looking rules forming a number of symmetry classes other than 3). 'inconclusive' with reason 'other' if the calibration fails to separate random from nested references." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.6", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch03-03--claim-test--dataset-1" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "When the 256 elementary cellular automata are run from a single black cell and classified by a fixed, calibrated compressibility criterion, about 10 produce random-looking patterns and these fall into three classes under left-right reflection and black-white complementation, while about 24 produce growing, non-periodic but highly compressible (nested) patterns." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py first calibrates the strip-compressibility statistic rho (bits per cell after lzma, 128 x 3072 central strip) on i.i.d. random bits (rho = 1.110-1.116), four classical nested patterns (0.009-0.033) and six periodic tilings (0.004-0.013), giving tau = 0.190. It then runs each of the 256 rules for 4096 steps from a single black cell (array width 2(T+128)+1 so that zero-boundary defects of background-flipping rules cannot reach the strip), computes rho, tests the strip for time-periodicity with period <= 64 over rows 2048-4095, measures the black-cell extent at rows 2047 and 4095, and classifies: random-looking if rho >= tau; nested if not random, not periodic and growing; regular otherwise. Random-looking rules are grouped into orbits under reflection and complementation. Run time about 50 s." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered. Each rule is run for T = 4096 steps from a single black cell on a white background (zero boundary, width 2T+1). Statistics: (i) rho = 8 * lzma_bytes / cells of the central strip (columns -64..63 relative to the seed, rows 1024..4095, one byte per cell, preset 9); (ii) periodic: the strip rows are time-periodic with some period p <= 64 over rows 2048..4095; (iii) growing: the black-cell extent of row 4095 is at least 1.2 times that of row 2047, or the extent of row 4095 exceeds 64 cells while the strip is not periodic. Calibration (performed before the rules are run): random references are i.i.d. fair bits (numpy seed 20260303) in a 128 x 3072 strip; nested references are the same-size central strips of the Sierpinski gasket (Pascal's triangle mod 2), Pascal's triangle mod 3 thresholded to nonzero, and the outer products t_i XOR t_j of the Thue-Morse and of the period-doubling sequences; periodic references are random patterns of period 1..64 in time and 1..16 in space. tau_random = geometric mean of the smallest random-reference rho and the largest nested-reference rho. Classification: 'random-looking' iff rho >= tau_random; 'nested' iff not random-looking, not periodic, and growing; 'regular' otherwise. Symmetry classes: orbits under reflection (swap left and right inputs) and complementation (conjugation by colour swap). Secondary, reported but not decisive: for nested rules, the black-cell count exponent D = log2(N(4096)/N(2048)) with N(t) = black cells in rows 0..t-1, and the width exponent log2(width(4095)/width(2047)); the number of clusters when D is binned to 0.1 and the width exponent to {about 1, about 0.5} is compared with Wolfram's 'three fundamentally different forms'. 'supports' if the random-looking count is in [8, 12] and forms exactly 3 symmetry classes, and the nested count is in [20, 28]." } ], "https://w3id.org/delphi/ontology/preregistered": [ { "@value": "false", "@type": "http://www.w3.org/2001/XMLSchema#boolean" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch03-03--claim-test--dataset-1", "http://purl.org/dc/terms/description": [ { "@value": "the finite rule space whose census the claim reports" } ], "http://purl.org/dc/terms/license": [ { "@value": "generated" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: All 256 elementary cellular automata from a single black cell, 4096 steps, generated by analysis.py together with the calibration reference patterns described in pass_criterion" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://www.wolframscience.com/nks/p58--more-cellular-automata/" } ] }, { "@id": "https://w3id.org/delphi/resource/claim/wolfram2002-ch03-03", "https://w3id.org/delphi/ontology/hasAssessment": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch03-03--claim-test" } ] } ] } ]