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Nesting of the non-trivial additive rules was checked as exact scale-2 self-similarity p[2t, 2i] = p[t, i] of the single-black-cell pattern over 256 steps, and the sublattice relation p90[t, 2j - t] = p60[t, j] was verified over 256 steps before being used to merge rules 60/102 with 90. Equivalence classes of non-trivial additive rules were formed by union-find under reflection and the verified sublattice relation." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: [object Object] rule numbers; fraction of the 248 non-additive rules with at least one block self-emulation (b <= 4, t <= 4) (n = 256, exhaustive finite check; exact within the searched block sizes and step counts). Interpretation: 'Additive' is read as linear over GF(2) (Wolfram's own definition on p. 264: patterns from two initial conditions can be superimposed by XOR). 'Emulate itself' is read as block emulation with 2 to 4 cells per block and 1 to 4 steps per emulated step. 'Fundamentally different' is read as 'not related by left-right reflection and not related by the sublattice/shear relation that maps the rule 60 pattern onto the even sublattice of the rule 90 pattern'; without the second identification rule 60 (and its mirror 102) would be a third additive rule, which the test records explicitly. 'Rather rare' is given the threshold 25 % of the non-additive rules. 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-ch06-20: supports, confidence 0.85" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch06-20/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-ch06-20-eca-enumerator" } ], "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 clause A or clause B fails (an additive rule that does not self-emulate with b = t = 2, a non-trivial additive rule that is not nested, or a third inequivalent non-trivial additive class). 'mixed' if A and B pass but C fails (rule 184 does not self-emulate with 3-cell blocks within the search, or self-emulation is found in more than 25 % of non-additive rules, so 'rather rare' would be an overstatement)." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "'Fundamentally different' required an interpretive choice: rule 60 (and its mirror 102) is additive and nested but is not related to 90 by reflection or colour complement; it is the rule 90 pattern restricted to the sublattice i + t even. Without that identification the claim's 'precisely rules 90 and 150' would be 'rules 60, 90 and 150'. The test records this explicitly." }, { "@value": "All three pre-registered clauses passed: exactly 8 additive rules exist, each self-emulates with 2-cell blocks and 2 steps (the Frobenius identity f^2 = f at stride 2 over GF(2)) and the four non-trivial ones are exactly self-similar at scale 2; after identifying 60/102 with 90 through the verified sublattice relation the additive rules fall into the two classes {60, 90, 102} and {150}; rule 184 self-emulates with the 3-cell blocks 010/101 and 3 steps, as Wolfram says, and only 15 of the 248 non-additive rules (6 %) have any block self-emulation in the searched range." }, { "@value": "The 15 non-additive self-emulating rules are the affine rules 15, 51, 85, 153, 165, 195 (complements/XOR with 1), the AND/OR-type class-1 rules 128, 136, 192, 238, 252, 254, 255, the shift-like rule 226 and rule 184. Larger blocks or more steps (b > 4, t > 4) might reveal further self-emulating rules; 'rather rare' is confirmed only within the searched range." }, { "@value": "The claim's sub-clause that additivity implies self-emulation in general (for any number of colours and neighbours) was tested only for the elementary rules; the argument via the Frobenius endomorphism is standard but was not checked beyond them." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered. Definitions: a rule f is additive iff f(x XOR y) = f(x) XOR f(y) for all pairs of 3-cell neighbourhoods x, y (which forces f(000) = 0). Rule f self-emulates with block size b >= 2 and t >= 1 steps iff there are two distinct b-cell blocks B0, B1 such that for every arrangement of 2k+1 blocks (k = ceil(t/b)) drawn from {B0, B1}, running f for t steps leaves the centre block equal to B_{f(c-1, c0, c1)}, where c-1, c0, c1 are the bits represented by the three central blocks; this is the block-emulation notion of NKS pp. 269-271 and note p. 1118, which allows t != b. The search covers b in {2, 3, 4}, t in {1, 2, 3, 4} and all block pairs. Trivial rules: 0, 204 (identity), 170 and 240 (shifts). Equivalences for 'fundamentally different': reflection (rule 60 <-> 102) and the sublattice relation p90[t, 2j - t] = p60[t, j], verified computationally on the first 256 steps from a single black cell. Nested: the single-cell pattern p satisfies p[2t, 2i] = p[t, i] for all 0 <= t < 256 and |i| <= t. Clause A passes if every additive rule self-emulates with b = t = 2 and every non-trivial additive rule is nested in this sense. Clause B passes if the non-trivial additive rules fall into exactly two equivalence classes, one containing 90 and one containing 150. Clause C passes if rule 184 self-emulates with b = 3 for some t <= 4 and the fraction of the 248 non-additive rules with at least one self-emulation in the searched range is <= 0.25. 'supports' if A, B and C all pass." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered. 'refutes' if clause A or clause B fails (an additive rule that does not self-emulate with b = t = 2, a non-trivial additive rule that is not nested, or a third inequivalent non-trivial additive class). 'mixed' if A and B pass but C fails (rule 184 does not self-emulate with 3-cell blocks within the search, or self-emulation is found in more than 25 % of non-additive rules, so 'rather rare' would be an overstatement)." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.85", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch06-20--claim-test--dataset-1" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "Among the 256 elementary cellular automata, every rule that is additive (linear over GF(2)) can emulate itself with blocks of cells and yields an exactly self-similar (nested) pattern from a single black cell; after removing trivial rules and identifying rules related by reflection or by the sublattice (shear) relation between rules 60 and 90, exactly two additive classes remain, represented by rules 90 and 150; self-emulation among the remaining 248 rules is uncommon, but rule 184 has it with blocks of three cells." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py tests all 256 elementary rules for GF(2)-linearity over all 64 neighbourhood pairs, then searches exhaustively for block self-emulations with b = 2, 3, 4 cells per block, t = 1..4 steps and all ordered pairs of distinct blocks, checking every arrangement of 2k+1 blocks (k = ceil(t/b)) so that the centre block's light cone is covered. Nesting of the non-trivial additive rules was checked as exact scale-2 self-similarity p[2t, 2i] = p[t, i] of the single-black-cell pattern over 256 steps, and the sublattice relation p90[t, 2j - t] = p60[t, j] was verified over 256 steps before being used to merge rules 60/102 with 90. Equivalence classes of non-trivial additive rules were formed by union-find under reflection and the verified sublattice relation." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered. Definitions: a rule f is additive iff f(x XOR y) = f(x) XOR f(y) for all pairs of 3-cell neighbourhoods x, y (which forces f(000) = 0). Rule f self-emulates with block size b >= 2 and t >= 1 steps iff there are two distinct b-cell blocks B0, B1 such that for every arrangement of 2k+1 blocks (k = ceil(t/b)) drawn from {B0, B1}, running f for t steps leaves the centre block equal to B_{f(c-1, c0, c1)}, where c-1, c0, c1 are the bits represented by the three central blocks; this is the block-emulation notion of NKS pp. 269-271 and note p. 1118, which allows t != b. The search covers b in {2, 3, 4}, t in {1, 2, 3, 4} and all block pairs. Trivial rules: 0, 204 (identity), 170 and 240 (shifts). Equivalences for 'fundamentally different': reflection (rule 60 <-> 102) and the sublattice relation p90[t, 2j - t] = p60[t, j], verified computationally on the first 256 steps from a single black cell. Nested: the single-cell pattern p satisfies p[2t, 2i] = p[t, i] for all 0 <= t < 256 and |i| <= t. Clause A passes if every additive rule self-emulates with b = t = 2 and every non-trivial additive rule is nested in this sense. Clause B passes if the non-trivial additive rules fall into exactly two equivalence classes, one containing 90 and one containing 150. Clause C passes if rule 184 self-emulates with b = 3 for some t <= 4 and the fraction of the 248 non-additive rules with at least one self-emulation in the searched range is <= 0.25. 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