https://w3id.org/np/RAkyV1yQzZ3lP4Hg3ggMmg_AoPcv4Q_GEC8pJYRnB_uII
.trig | .trig.txt | .jelly | .jelly.txt | .jsonld | .jsonld.txt | .nq | .nq.txt | .xml | .xml.txt
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<https://w3id.org/delphi/resource/assessment/wolfram2002-ch05-07--claim-test> a <https://w3id.org/delphi/ontology/Assessment>;
dc:description "analysis.py grows every rule's pattern from a single cell of colour 1 (9 steps for 2x2 rules, 512 x 512; 6 steps for 3x3 rules, 729 x 729), checks the nesting identity P_n = sigma^j(P_{n-j}) with sigma^j given by the single-cell patterns P_j^(c) for every j = 1..n-1 by exact array comparison, and counts distinct 4 x 4 and 8 x 8 windows (exact 64-bit codes, or a 64-bit multiplicative hash for three-colour 8 x 8 windows) against the bound s^4 b^2 k^2. The same measures are applied to 2000 neighbour-dependent rules and a random image. Run time 3 min.";
<http://www.w3.org/2000/01/rdf-schema#comment> "Measured: [object Object] count of rules failing the nesting identity or the block-complexity bound (n = 6256, exact except that three-colour 8 x 8 window counts use a 64-bit hash (collision probability ~1e-8 per pattern, which could only lower a count)). Interpretation: Wolfram's 'completely regular nested structure' is formalized as exact hierarchical self-similarity: the pattern after n steps is the pattern after n - j steps with every cell replaced by the pattern grown in j steps from a cell of that colour, for every j. This holds for neighbour-independent rules by construction (the substitution acts cellwise and commutes with composition), so this test is a computational verification of a mathematical statement across an enumerated and a sampled rule space, plus a check that the formalization and the complexity bound genuinely discriminate (neighbour-dependent controls, as on NKS p. 191). Scope is limited to grid substitution systems with uniform b x b blocks (b = 2, 3; 2 or 3 colours); 'geometric replacement rules' of traditional fractal geometry with arbitrary affine maps, including overlapping ones, are covered by Hutchinson's theorem on self-similar sets in the literature but are not computed here.";
<http://www.w3.org/2000/01/rdf-schema#label> "Claim test of wolfram2002-ch05-07: supports, confidence 0.9";
<http://www.w3.org/2000/01/rdf-schema#seeAlso> <https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch05-07/result.json>;
prov:wasAttributedTo <https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-opus-5-5>;
<https://w3id.org/delphi/ontology/admissibleDatasets> "Patterns generated exactly by committed code from fully specified substitution rules (all two-colour 2 x 2 rules, seeded samples of larger rule spaces); controls generated by the same code.";
<https://w3id.org/delphi/ontology/assessmentType> "claim-test/0.1.0";
<https://w3id.org/delphi/ontology/dataKind> "computational";
<https://w3id.org/delphi/ontology/disconfirmingObservation> "A neighbour-independent rule whose pattern breaks the scale decomposition or exceeds the quadratic block-complexity bound.";
<https://w3id.org/delphi/ontology/evidenceAgainst> "All 4,256 neighbour-independent rules satisfy the exact nesting identity at every scale and stay at or below 18 % of the quadratic block-complexity bound (largest 748 distinct 8 x 8 blocks against a bound of 4,096), while 1,982 of 2,000 neighbour-dependent rules break the identity, 1,799 exceed the bound (up to 9.7 times), and a random image has 255,025 distinct 8 x 8 blocks.",
"Before the full run the functions were exercised on three hand-chosen or differently seeded rules and one random image to check the code; this did not involve the pre-registered rule samples.",
"For neighbour-independent rules the nesting identity is a theorem (the substitution acts cell by cell, so sigma^n = sigma^j composed with sigma^(n-j)); the computation verifies the formalization and its implementation on an exhaustive and a sampled rule space rather than discovering an empirical regularity. Its non-trivial content is that the formalization discriminates: the neighbour dependence Wolfram introduces on p. 191 destroys it in 99 % of sampled rules.",
"Scope: uniform b x b grid substitutions with b = 2, 3 and 2-3 colours, from a single seed cell. Geometric replacement rules with arbitrary affine maps (traditional IFS fractals), including overlapping ones where the drawn union can look solid, were not computed; their self-similarity is Hutchinson's theorem. 'Nested' here includes degenerate cases (18 of the 256 two-colour rules give all-white or all-black patterns), which are trivially nested.";
<https://w3id.org/delphi/ontology/expectedObservation> "Every enumerated or sampled neighbour-independent substitution rule yields a pattern that exactly decomposes into single-cell patterns at every scale and whose number of distinct 4 x 4 and 8 x 8 blocks stays below s^4 b^2 k^2, while some neighbour-dependent rules and a random image violate these.";
<https://w3id.org/delphi/ontology/failCriterion> "Pre-registered. 'refutes' if any neighbour-independent rule in F1-F3 fails (1) or (2). If the control condition fails (fewer than 20 neighbour-dependent rules flagged, or the random image not flagged), the measures are judged unable to discriminate and the verdict is 'inconclusive' (reason 'other').";
<https://w3id.org/delphi/ontology/hasConfidence> "0.9"^^xsd:double;
<https://w3id.org/delphi/ontology/hasEvidence> <https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-07--claim-test--dataset-1>,
<https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-07--claim-test--dataset-2>,
<https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-07--claim-test--dataset-3>;
<https://w3id.org/delphi/ontology/hasVerdict> "supports";
<https://w3id.org/delphi/ontology/hypothesis> "Every two-dimensional substitution system in which each cell is replaced by a fixed block depending only on its own colour produces, from a single cell, a pattern that is exactly nested (decomposes at every scale into the patterns grown from single cells) and has at most quadratic block complexity, whereas rules in which the replacement depends on neighbours can violate both.";
<https://w3id.org/delphi/ontology/inferenceMethod> "Exact array equality for the nesting identity at every scale j; exact (or 64-bit-hashed) counts of distinct k x k windows compared with the bound s^4 b^2 k^2; validity requires the neighbour-dependent controls and a random image to be flagged.";
<https://w3id.org/delphi/ontology/passCriterion> "Pre-registered. Neighbour-independent families: F1 = all 256 two-colour rules replacing each cell by a 2 x 2 block; F2 = 2000 random three-colour 2 x 2 rules; F3 = 2000 random two-colour 3 x 3 rules (numpy PCG64, seed 20261031). Each is iterated from a single cell of colour 1 for n steps (n = 9 for 2 x 2 rules, 512 x 512 cells; n = 6 for 3 x 3 rules, 729 x 729). Measures: (1) exact nesting: for every j = 1..n-1, P_n equals P_{n-j} with each cell of colour c replaced by P_j^(c), the j-step pattern grown from a single cell of colour c; (2) block complexity: the number N(k) of distinct k x k sub-blocks of P_n satisfies N(k) <= s^4 b^2 k^2 for k = 4 and 8 (s colours, b x b blocks; this bound follows from (1), since any k x k window lies in a 2 x 2 group of level-j supertiles with b^(j-1) < k <= b^j; a random s-colour image of this size has N(8) close to the number of positions, about 2.5 x 10^5). Control (validity): 2000 random two-colour neighbour-dependent 2 x 2 rules, where the block replacing a cell depends on its own colour and the colours of its right and lower neighbours (outside the pattern = colour 0), iterated and measured identically; at least 20 of them (1 %) must fail (1) or (2), and a Bernoulli(0.5) 512 x 512 image must fail (2), showing the measures can detect non-nested patterns. 'supports' if every rule in F1, F2 and F3 satisfies (1) and (2) and the control condition holds.";
<https://w3id.org/delphi/ontology/preregistered> "false"^^xsd:boolean .
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<https://w3id.org/delphi/resource/assessment/wolfram2002-ch05-07--claim-test> .
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a <https://w3id.org/delphi/ontology/EvidenceItem>;
dc:description "the rule class the claim is about";
dc:license "";
<http://www.w3.org/2000/01/rdf-schema#label> "Dataset: Neighbour-independent 2D substitution rules: all 256 two-colour 2x2 rules, 2000 random three-colour 2x2 rules and 2000 random two-colour 3x3 rules (numpy PCG64 seed 20261031), patterns grown from a single cell (generated by analysis.py)";
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<https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-07--claim-test--dataset-2>
a <https://w3id.org/delphi/ontology/EvidenceItem>;
dc:description "control showing the measures detect non-nested patterns";
dc:license "";
<http://www.w3.org/2000/01/rdf-schema#label> "Dataset: 2000 random two-colour 2x2 substitution rules whose block depends on the cell and its right and lower neighbours (generated by analysis.py, same seed stream)";
prov:wasDerivedFrom <https://www.wolframscience.com/nks/p191--substitution-systems-and-fractals/> .
<https://w3id.org/delphi/resource/evidence/wolfram2002-ch05-07--claim-test--dataset-3>
a <https://w3id.org/delphi/ontology/EvidenceItem>;
dc:description "random reference for block complexity";
dc:license "";
<http://www.w3.org/2000/01/rdf-schema#label> "Dataset: Bernoulli(0.5) 512 x 512 image (generated by analysis.py, same seed stream)" .
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<http://www.w3.org/2000/01/rdf-schema#label> "Claim test of wolfram2002-ch05-07: supports, confidence 0.9";
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