https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/Head https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU http://www.nanopub.org/nschema#hasAssertion https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/assertion https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU http://www.nanopub.org/nschema#hasProvenance https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/provenance https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU http://www.nanopub.org/nschema#hasPublicationInfo https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/pubinfo https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.nanopub.org/nschema#Nanopublication https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/assertion https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test http://purl.org/dc/terms/description analysis.py: (A) checks phi(a XOR b) = phi(a) XOR phi(b) over all 64 neighbourhood pairs for rules 90 and 22; (B) evolves 50 random rows on a 513-cell cyclic lattice for 256 steps under rule 90 and compares every row with the XOR of the single-cell evolution shifted to each initially black cell; (C) counts black cells in the rule-90 single-cell pattern at t = 2^k and 2^k - 1; (D, E) evolve Bernoulli(p) rows of 100000 + 2*1024 cells for 1024 steps under rule 90 (p = 0.05, 0.1, 0.2, 0.5) and rule 22 (same p) with zero boundaries, measuring the density in the central 100000 cells, which the boundaries cannot influence within 1024 steps; D compares rule-90 densities at t = 2^k with 2p(1-p), E compares the rule-22 time-averaged densities over t = 512..1024 across p. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--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-ch06-15--claim-test http://www.w3.org/2000/01/rdf-schema#comment Measured: 0.00307 largest |density(t=2^k) - 2p(1-p)| over p in {0.05,0.1,0.2}, k = 0..10 for rule 90 (threshold 0.01) (n = 33, sampling: ~0.0015 (100000 correlated cells per row)). Interpretation: The claim has an algebraic clause (additivity, superposition) and an empirical clause (density persistence in rule 90 versus rule 22, the quoted sentence on p. 265). Additivity is checked exhaustively on the local rule and by exact superposition on random configurations. 'Cannot generate randomness intrinsically' is operationalized by the exact nested structure of the single-cell pattern plus the superposition identity: everything in a rule-90 pattern is a XOR of shifted copies of that nested pattern. 'Low initial density persists forever' is operationalized by the density at times 2^k, where rule 90 reduces to x_t(i) = x_0(i-2^k) XOR x_0(i+2^k) and the density must equal 2p(1-p) at arbitrarily late times; 'no long-term effect' in rule 22 is operationalized as late-time density independent of the initial density. Random initial conditions are Bernoulli(p) with fixed seeds; the lattice is a finite array measured only in a window unaffected by the boundaries within the run length. Rule numbering is Wolfram's standard code (rule 90 = l XOR r; rule 22 = exactly one of l, c, r black). https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test http://www.w3.org/2000/01/rdf-schema#label Claim test of wolfram2002-ch06-15: supports, confidence 0.95 https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--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-ch06-15/result.json https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test http://www.w3.org/ns/prov#wasAttributedTo https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1-cursor-cloud-agent https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/admissibleDatasets Planned datasets: eca-rule90-rule22-generator https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/assessmentType claim-test/0.1.0 https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/dataKind computational https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/disconfirmingObservation Pre-registered. 'Refutes' if A or B fails in any case (rule 90 is not additive / not a superposition), or if in D the rule-90 density at t = 1024 for p = 0.1 lies within 0.05 of the p = 0.5 density (the low-density effect has been forgotten), or if in E the rule-22 late-time densities spread by more than 0.05 (rule 22 remembers the initial density). Other combinations are 'mixed'. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/evidenceAgainst 'Cannot generate randomness intrinsically' is interpreted structurally (every pattern is a superposition of the nested single-cell pattern); whether XOR-superposed nested patterns count as 'random' under some statistical test is not what is being tested. Rule 90 from a p = 0.5 random row does look random and has density 0.5 at all times, consistent with 'randomness from the initial conditions'. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/evidenceAgainst Finite run: 1024 steps and times up to 2^10; 'forever' follows from the identity x_t(i) = x_0(i-2^k) XOR x_0(i+2^k) at t = 2^k, which holds for all k by additivity (verified in A and B), so the finite check is of the mechanism, not a substitute for the proof. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/evidenceAgainst Part C initially failed for a trivial reason unrelated to the claim: the single-cell array was two cells too narrow and the fixed-zero boundary cells coincided with the two black cells at t = 1024. The array was widened by one cell on each side (commit history shows the change); no threshold or criterion was changed. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/evidenceAgainst Rule 22's late-time density of about 0.351 from all four initial densities agrees with the known asymptotic density of rule 22 from random initial conditions; convergence is complete by t ~ 128 even for p = 0.05. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/evidenceAgainst The densities of 100000 cells in one row are correlated (in rule 90 at t = 2^k, cells 2^(k+1) apart share an initial cell), so the binomial standard error (0.0012 for p = 0.1) understates the sampling error; the pre-registered 0.01 tolerance was chosen to absorb this and the largest deviation observed was 0.0031. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/expectedObservation Pre-registered. (A) Additivity: the rule-90 local map phi satisfies phi(a XOR b) = phi(a) XOR phi(b) for all 64 pairs of 3-cell neighbourhoods, and the rule-22 map does not; (B) superposition: for 50 random initial conditions on a cyclic lattice of width 513 run for 256 steps, the rule-90 evolution equals the cellwise XOR of the shifted single-cell evolutions exactly (100% of 50*256 rows); (C) single-cell pattern: for rule 90 from one black cell the row at t = 2^k (k = 0..10) has exactly 2 black cells and the row at t = 2^k - 1 has 2^k black cells (nested, Pascal mod 2); (D) persistence in rule 90: with initial densities p in {0.05, 0.1, 0.2} on a measured window of 100000 cells (array padded by 1024 cells on each side so boundaries never reach the window), the measured density at t = 2^k for k = 0..10 is within 0.01 of 2p(1-p) in all 33 cases, and at t = 1024 the three densities differ from the p = 0.5 density by more than 0.1; (E) loss of memory in rule 22: with p in {0.05, 0.1, 0.2, 0.5} on the same windows, the time-averaged densities over t = 512..1024 differ from one another by less than 0.01 (max minus min). 'Supports' requires A-E. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/failCriterion Pre-registered. 'Refutes' if A or B fails in any case (rule 90 is not additive / not a superposition), or if in D the rule-90 density at t = 1024 for p = 0.1 lies within 0.05 of the p = 0.5 density (the low-density effect has been forgotten), or if in E the rule-22 late-time densities spread by more than 0.05 (rule 22 remembers the initial density). Other combinations are 'mixed'. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/hasConfidence 0.95 https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/hasEvidence https://w3id.org/delphi/resource/evidence/wolfram2002-ch06-15--claim-test--dataset-1 https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/hasVerdict supports https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/hypothesis Rule 90 is additive (linear over GF(2)): its local rule is l XOR r, so every evolution is the XOR superposition of shifted copies of the single-cell nested (Pascal mod 2) pattern. Consequently a biased random initial condition leaves a permanent trace: at every time t = 2^k each cell is the XOR of exactly two initial cells, so the density returns to 2p(1-p) forever for initial density p, and the single-cell pattern itself is nested, not random. Rule 22 (non-additive, class 3) instead forgets the initial density: from any initial density the density converges to one common value. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/inferenceMethod analysis.py: (A) checks phi(a XOR b) = phi(a) XOR phi(b) over all 64 neighbourhood pairs for rules 90 and 22; (B) evolves 50 random rows on a 513-cell cyclic lattice for 256 steps under rule 90 and compares every row with the XOR of the single-cell evolution shifted to each initially black cell; (C) counts black cells in the rule-90 single-cell pattern at t = 2^k and 2^k - 1; (D, E) evolve Bernoulli(p) rows of 100000 + 2*1024 cells for 1024 steps under rule 90 (p = 0.05, 0.1, 0.2, 0.5) and rule 22 (same p) with zero boundaries, measuring the density in the central 100000 cells, which the boundaries cannot influence within 1024 steps; D compares rule-90 densities at t = 2^k with 2p(1-p), E compares the rule-22 time-averaged densities over t = 512..1024 across p. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/passCriterion Pre-registered. (A) Additivity: the rule-90 local map phi satisfies phi(a XOR b) = phi(a) XOR phi(b) for all 64 pairs of 3-cell neighbourhoods, and the rule-22 map does not; (B) superposition: for 50 random initial conditions on a cyclic lattice of width 513 run for 256 steps, the rule-90 evolution equals the cellwise XOR of the shifted single-cell evolutions exactly (100% of 50*256 rows); (C) single-cell pattern: for rule 90 from one black cell the row at t = 2^k (k = 0..10) has exactly 2 black cells and the row at t = 2^k - 1 has 2^k black cells (nested, Pascal mod 2); (D) persistence in rule 90: with initial densities p in {0.05, 0.1, 0.2} on a measured window of 100000 cells (array padded by 1024 cells on each side so boundaries never reach the window), the measured density at t = 2^k for k = 0..10 is within 0.01 of 2p(1-p) in all 33 cases, and at t = 1024 the three densities differ from the p = 0.5 density by more than 0.1; (E) loss of memory in rule 22: with p in {0.05, 0.1, 0.2, 0.5} on the same windows, the time-averaged densities over t = 512..1024 differ from one another by less than 0.01 (max minus min). 'Supports' requires A-E. https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/ontology/preregistered false https://w3id.org/delphi/resource/claim/wolfram2002-ch06-15 https://w3id.org/delphi/ontology/hasAssessment https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/delphi/resource/evidence/wolfram2002-ch06-15--claim-test--dataset-1 http://purl.org/dc/terms/description enumerated program family whose behaviour is measured https://w3id.org/delphi/resource/evidence/wolfram2002-ch06-15--claim-test--dataset-1 http://purl.org/dc/terms/license CC0 (generated data) https://w3id.org/delphi/resource/evidence/wolfram2002-ch06-15--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-ch06-15--claim-test--dataset-1 http://www.w3.org/2000/01/rdf-schema#label Dataset: Elementary cellular automata 90 and 22 (generated by analysis.py) https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/provenance https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1-cursor-cloud-agent 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-cursor-cloud-agent http://www.w3.org/2000/01/rdf-schema#label delphisci claim tester (Claude Fable 5.1 (Cursor cloud agent); cursor/claim-tests-wolfram-nks-f0e1) https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1-cursor-cloud-agent 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-cursor-cloud-agent https://w3id.org/delphi/ontology/usesModel https://w3id.org/delphi/resource/model/claude-fable-5-1-cursor-cloud-agent https://w3id.org/delphi/resource/model/claude-fable-5-1-cursor-cloud-agent http://purl.org/dc/terms/identifier Claude Fable 5.1 (Cursor cloud agent) https://w3id.org/delphi/resource/model/claude-fable-5-1-cursor-cloud-agent http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/LanguageModel https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/assertion http://www.w3.org/ns/prov#wasAttributedTo https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1-cursor-cloud-agent https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/assertion http://www.w3.org/ns/prov#wasDerivedFrom https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch06-15/result.json https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/assertion http://www.w3.org/ns/prov#wasGeneratedBy https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/test https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/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/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/test http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.w3.org/ns/prov#Activity https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/test http://www.w3.org/ns/prov#qualifiedAssociation https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/plan https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/test http://www.w3.org/ns/prov#wasAssociatedWith https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1-cursor-cloud-agent https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/pubinfo https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU http://purl.org/dc/terms/created 2026-11-08T16:24:31.928Z https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU http://purl.org/dc/terms/creator https://orcid.org/0009-0007-4948-9192 https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU http://purl.org/dc/terms/license https://creativecommons.org/licenses/by/4.0/ https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU http://purl.org/nanopub/x/hasNanopubType https://w3id.org/delphi/ontology/Assessment https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU http://purl.org/nanopub/x/introduces https://w3id.org/delphi/resource/assessment/wolfram2002-ch06-15--claim-test https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU http://www.w3.org/2000/01/rdf-schema#label Claim test of wolfram2002-ch06-15: supports, confidence 0.95 https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU https://w3id.org/delphi/ontology/domain https://w3id.org/delphi/resource/domain/wolfram-nks-2002 https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/sig http://purl.org/nanopub/x/hasAlgorithm RSA https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/sig http://purl.org/nanopub/x/hasPublicKey MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/sig http://purl.org/nanopub/x/hasSignature p90yBwUUtgEHcrjT1By7R7uMyG+esuWAaaXQrErCO7yVAj8JC+21gJR0xpI/un93cs4ufIkkweqg1D/hYsXM/jzaFANHD9hn5ahJVFK/OexVUI4mYfzFeb2l8jd5sB5fpPZtJP4B6hbaPSIFn3U72xK7sZZwoigpY2AQdiVCOsZOGMflNjd/R6rvDnKi6qIirNl1WQoOWMPLOr2Mwn1dENbuyUGJB69NErmZTjxcQ9is1+67it44ipF1y2wCHEe25d+dHgKEF30t4DQP71GMp3qWaj+w79Cmv4NwMly4U6YAueGKyqM+ygd5Cyv68oAA7TPmxXojSBR2JMsxjZrw6Q== https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/sig http://purl.org/nanopub/x/hasSignatureTarget https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU https://w3id.org/np/RAy7uHpP99hBjwxnD--fvnN6T1kbTAv-5aRk7iUXeyBHU/sig http://purl.org/nanopub/x/signedBy https://orcid.org/0009-0007-4948-9192