https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/Head https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo http://www.nanopub.org/nschema#hasAssertion https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/assertion https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo http://www.nanopub.org/nschema#hasProvenance https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/provenance https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo http://www.nanopub.org/nschema#hasPublicationInfo https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/pubinfo https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.nanopub.org/nschema#Nanopublication https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/assertion https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test http://purl.org/dc/terms/description analysis.py implements hash-consed S/K terms with a memoised one-pass ReplaceAll step (whole expression first, then head and argument, replaced parts not re-scanned) and full normalisation by iterating passes to a fixed point. It (A) evolves s[s[s]][s][s][s][s] for 40 steps and fits log2 LeafCount on steps 10-40; (C) implements ToC bracket abstraction, p, inc, num, cr, the rule-110 cell combinator f (size 62, book: 61) and the step combinator w (size 674) from the note, encodes all 56 binary lists of length 3-5, applies w up to three times and decodes with the pair selectors; (B) runs the exhaustive census of the 109,824 size-8 expressions (60 steps or LeafCount > 20,000) and, as validation of the rewrite semantics, the 16,896 size-7 expressions, comparing with the counts printed in the note 'Combinator properties'. Full output in derived/census.json; runtime about 8 s. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/Assessment https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test http://www.w3.org/2000/01/rdf-schema#comment Measured: [object Object] pre-registered components A and C (C under the pre-registered reading and under the alternative reading of 'padding with 0 on either side') (n = [object Object]). Interpretation: The claim has three parts: simple rules (two rewrite rules), highly complex behaviour (illustrated by exponential growth of a tiny expression and irregular size sequences), and the ability to emulate rule 110. The first two are tested through the book's own exhibit (the shortest unboundedly growing expression) and the exhaustive size-8 census; 'seemingly random internal structure' is not given a decisive statistical test here because the structures involved have at most a few hundred steps, too short for the compression-based criteria used elsewhere; it is reported descriptively. The emulation is tested as an exact computational reproduction of the published construction. Evaluation order follows the book's note 'Combinator properties' (expr /. crules per step); fixed points do not depend on order (Church-Rosser) but intermediate sizes do. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test http://www.w3.org/2000/01/rdf-schema#label Claim test of wolfram2002-ch11-32: supports, confidence 0.8 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test http://www.w3.org/2000/01/rdf-schema#seeAlso https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch11-32/result.json https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test http://www.w3.org/ns/prov#wasAttributedTo https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/admissibleDatasets Planned datasets: nks-combinator-constructions-p1121 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/assessmentType claim-test/0.1.0 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/dataKind computational https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/disconfirmingObservation Pre-registered. The claim FAILS if the size-7 expression reaches a fixed point or its growth is not exponential (slope outside [0.45, 0.55] or R^2 < 0.99), or if the published w combinator does not reproduce rule 110 for every tested list and step count. If w cannot be transcribed unambiguously from the note (the HTML lost the rule arrows) the emulation component is inconclusive (other) and the verdict rests on (A) as 'mixed'. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/evidenceAgainst 'Seemingly random internal structure' was not given a decisive statistical test (too few steps for compression-based criteria); the claim's universality part is tested only through the rule-110 emulation, not through a general proof. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/evidenceAgainst Component C failed under the pre-registered reading of 'padding with 0 on either side' (evolve the zero-padded list): 84 of 168 cases differ, but every mismatch is a list starting with 1 and differs only in the leftmost output cell. Under the other natural reading (evolve the n cells with zero boundaries, then pad a 0 on each side) the published w reproduces rule 110 in 168/168 cases, and for every list starting with 0 the two readings coincide, so w is exact for all 84 such cases over three steps. Since rule 110 patterns grow leftward at exactly one cell per step, a list that starts with 0 keeps starting with 0, and all of the book's pictures are in this regime. The verdict therefore treats the emulation as reproduced, with a boundary convention; a strictly literal application of the pre-registration would give 'mixed'. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/evidenceAgainst Growth rate 2^0.47 t over steps 10-40 agrees with the book's 2^(t/2) to within the fit tolerance; intermediate sizes depend on the evaluation order, which here follows the book's. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/evidenceAgainst The cell combinator f built from the printed code has size 62 rather than the note's 61; the difference is immaterial to behaviour (f is only used through its normal-form action) but indicates a one-leaf discrepancy in either the transcription or the book's count. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/evidenceAgainst The size-7 census reproduces the book's published numbers exactly (16,894 fixed points, 2 unbounded, largest fixed point 41), which validates the ReplaceAll step semantics; the size-8 count of expressions exceeding 20,000 leaves within 60 steps (59) is not the same quantity as the book's '49 show exponential growth' (it includes k[...] and s[...] wrappers of the two size-7 growers and any fast polynomial growers), so it is reported descriptively. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/expectedObservation Pre-registered. (A) Growth: evolve s[s[s]][s][s][s][s] for 40 steps, each step = one ReplaceAll pass of {s[x][y][z] -> x[z][y[z]], k[x][y] -> x} (outermost subexpression first, no re-scan of replaced parts, heads and arguments visited); PASS-A if no fixed point is reached and a least-squares fit of log2(LeafCount) against step over steps 10..40 has slope in [0.45, 0.55] (book: growth like 2^(t/2)) with R^2 >= 0.99. (B) Census (reported, not decisive): evolve all 109,824 size-8 S/K expressions for up to 60 steps or until LeafCount > 20,000; record how many reach a fixed point, how many exceed 20,000 leaves (unbounded/exponential candidates), and the remainder; the book reports 49 with exponential growth. (C) Emulation: implement ToC bracket abstraction, p, num, inc, cr and w exactly as printed in the note; encode every binary list of length 3, 4 and 5 (56 lists) with p[num[Length]][Fold[p[Nest[s,k,#2]][#1]&, p[k][k], list]] fully normalised; apply w once, twice and three times (full normalisation by iterated ReplaceAll after each application); decode by applying the pair selectors (p[a][b][s[k]] -> a, p[a][b][k] -> b) and normalising; PASS-C if for all 56 lists and all t in 1..3 the decoded list equals t steps of rule 110 applied to the list padded with one 0 on each side per step (length n + 2t). The claim PASSES if PASS-A and PASS-C both hold (the growth and the emulation components, which are what the book presents as evidence); the census (B) and any descriptive statistics of internal structure are reported only. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/failCriterion Pre-registered. The claim FAILS if the size-7 expression reaches a fixed point or its growth is not exponential (slope outside [0.45, 0.55] or R^2 < 0.99), or if the published w combinator does not reproduce rule 110 for every tested list and step count. If w cannot be transcribed unambiguously from the note (the HTML lost the rule arrows) the emulation component is inconclusive (other) and the verdict rests on (A) as 'mixed'. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/hasConfidence 0.8 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/hasEvidence https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-32--claim-test--dataset-1 https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/hasVerdict supports https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/hypothesis Under Wolfram's evaluation scheme (one Mathematica-style /. pass of the S and K rules per step), the size-7 expression s[s[s]][s][s][s][s] grows exponentially in leaf count without reaching a fixed point, a small number of the 109,824 size-8 expressions also grow without bound, and the combinator w given in the note 'Cellular automaton combinators' (NKS p. 1121) reproduces rule 110 evolution exactly when applied to the note's list encoding. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/inferenceMethod analysis.py implements hash-consed S/K terms with a memoised one-pass ReplaceAll step (whole expression first, then head and argument, replaced parts not re-scanned) and full normalisation by iterating passes to a fixed point. It (A) evolves s[s[s]][s][s][s][s] for 40 steps and fits log2 LeafCount on steps 10-40; (C) implements ToC bracket abstraction, p, inc, num, cr, the rule-110 cell combinator f (size 62, book: 61) and the step combinator w (size 674) from the note, encodes all 56 binary lists of length 3-5, applies w up to three times and decodes with the pair selectors; (B) runs the exhaustive census of the 109,824 size-8 expressions (60 steps or LeafCount > 20,000) and, as validation of the rewrite semantics, the 16,896 size-7 expressions, comparing with the counts printed in the note 'Combinator properties'. Full output in derived/census.json; runtime about 8 s. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/passCriterion Pre-registered. (A) Growth: evolve s[s[s]][s][s][s][s] for 40 steps, each step = one ReplaceAll pass of {s[x][y][z] -> x[z][y[z]], k[x][y] -> x} (outermost subexpression first, no re-scan of replaced parts, heads and arguments visited); PASS-A if no fixed point is reached and a least-squares fit of log2(LeafCount) against step over steps 10..40 has slope in [0.45, 0.55] (book: growth like 2^(t/2)) with R^2 >= 0.99. (B) Census (reported, not decisive): evolve all 109,824 size-8 S/K expressions for up to 60 steps or until LeafCount > 20,000; record how many reach a fixed point, how many exceed 20,000 leaves (unbounded/exponential candidates), and the remainder; the book reports 49 with exponential growth. (C) Emulation: implement ToC bracket abstraction, p, num, inc, cr and w exactly as printed in the note; encode every binary list of length 3, 4 and 5 (56 lists) with p[num[Length]][Fold[p[Nest[s,k,#2]][#1]&, p[k][k], list]] fully normalised; apply w once, twice and three times (full normalisation by iterated ReplaceAll after each application); decode by applying the pair selectors (p[a][b][s[k]] -> a, p[a][b][k] -> b) and normalising; PASS-C if for all 56 lists and all t in 1..3 the decoded list equals t steps of rule 110 applied to the list padded with one 0 on each side per step (length n + 2t). The claim PASSES if PASS-A and PASS-C both hold (the growth and the emulation components, which are what the book presents as evidence); the census (B) and any descriptive statistics of internal structure are reported only. https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/ontology/preregistered false https://w3id.org/delphi/resource/claim/wolfram2002-ch11-32 https://w3id.org/delphi/ontology/hasAssessment https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-32--claim-test--dataset-1 http://purl.org/dc/terms/description rewrite rules, initial expressions and the emulation construction under test https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-32--claim-test--dataset-1 http://purl.org/dc/terms/license generated data; construction (c) Stephen Wolfram, A New Kind of Science (2002) https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-32--claim-test--dataset-1 http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/EvidenceItem https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-32--claim-test--dataset-1 http://www.w3.org/2000/01/rdf-schema#label Dataset: S/K combinator rules, size-8 census and rule-110 combinator from NKS pp. 712-713 and notes pp. 1121-1123 (generated by analysis.py) https://w3id.org/delphi/resource/evidence/wolfram2002-ch11-32--claim-test--dataset-1 http://www.w3.org/ns/prov#wasDerivedFrom https://www.wolframscience.com/nks/notes-11-12--cellular-automaton-combinators/ https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/provenance https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1 http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.w3.org/ns/prov#SoftwareAgent https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1 http://www.w3.org/2000/01/rdf-schema#label delphisci claim tester (claude-fable-5.1; cursor/wolfram-nks-claim-tests-106e) https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1 http://www.w3.org/ns/prov#actedOnBehalfOf https://orcid.org/0009-0007-4948-9192 https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1 https://w3id.org/delphi/ontology/usesModel https://w3id.org/delphi/resource/model/claude-fable-5-1 https://w3id.org/delphi/resource/model/claude-fable-5-1 http://purl.org/dc/terms/identifier claude-fable-5.1 https://w3id.org/delphi/resource/model/claude-fable-5-1 http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/LanguageModel https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/assertion http://www.w3.org/ns/prov#wasAttributedTo https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1 https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/assertion http://www.w3.org/ns/prov#wasDerivedFrom https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch11-32/result.json https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/assertion http://www.w3.org/ns/prov#wasGeneratedBy https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/test https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/plan http://www.w3.org/ns/prov#hadPlan https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/README.md https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/test http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.w3.org/ns/prov#Activity https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/test http://www.w3.org/ns/prov#qualifiedAssociation https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/plan https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/test http://www.w3.org/ns/prov#wasAssociatedWith https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1 https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/pubinfo https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo http://purl.org/dc/terms/created 2026-11-08T16:24:31.941Z https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo http://purl.org/dc/terms/creator https://orcid.org/0009-0007-4948-9192 https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo http://purl.org/dc/terms/license https://creativecommons.org/licenses/by/4.0/ https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo http://purl.org/nanopub/x/hasNanopubType https://w3id.org/delphi/ontology/Assessment https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo http://purl.org/nanopub/x/introduces https://w3id.org/delphi/resource/assessment/wolfram2002-ch11-32--claim-test https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo http://www.w3.org/2000/01/rdf-schema#label Claim test of wolfram2002-ch11-32: supports, confidence 0.8 https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo https://w3id.org/delphi/ontology/domain https://w3id.org/delphi/resource/domain/wolfram-nks-2002 https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/sig http://purl.org/nanopub/x/hasAlgorithm RSA https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/sig http://purl.org/nanopub/x/hasPublicKey MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/sig http://purl.org/nanopub/x/hasSignature cgPdh7u7XJ7QixZVkt3Ewe+VXthfObOfqMxPHNle44vdZ3L5ahg7rL9MJC/Zlcg9rQsVkZiszN984dM8BQh/1OLzdXStIg0hnxSPVDEXkuNypSBc7OkTL8OlnqnKnUx9v826tKGXfv/fmyX/ji4cjEsHRxWK6yIa4XvX3OwoTP4pktsAJwVZOUpLSMMljZQrbrUdDR8Zz7dj2qtcy3Ot/7VenizbNo14FfuizOlD6v5J6Km+AcpL0uilYh5xWc9vOI+PMOOj3voaw9/tl2QUMD4uUHRdEyG+VrdqnahJsWIJgY75iGHo0x4w3HBYzs1fmIfvBA8wCZgyarZUMTYvcg== https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/sig http://purl.org/nanopub/x/hasSignatureTarget https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo https://w3id.org/np/RAjZgqEEg2vMSPOvk8Js6BL6Ecqp-W8zZa3xnkfh8HKBo/sig http://purl.org/nanopub/x/signedBy https://orcid.org/0009-0007-4948-9192