@prefix this: . @prefix sub: . @prefix np: . @prefix rdf: . @prefix prov: . @prefix npx: . @prefix dc: . @prefix xsd: . sub:Head { this: a np:Nanopublication; np:hasAssertion sub:assertion; np:hasProvenance sub:provenance; np:hasPublicationInfo sub:pubinfo . } sub:assertion { a ; dc:description "analysis.py treats a block of n cells repeated forever as an n-cell ring, builds the ring's transition map on all k^n states for each rule and computes the eventual period of every state's orbit from the functional graph (cycle detection by in-degree peeling). Families: all 256 ECA for n = 1..12 (2,096,640 rule-block pairs), all 2187 three-colour totalistic rules for n = 1..6 (2,388,204 pairs, bound 3^n), and 500 seeded random range-2 binary rules for n = 1..10 (1,023,000 pairs). It then checks the reduction itself: for all 256 ECA and all 32,256 blocks of n = 1..6, a line of repeated copies of the block is evolved for 2*2^n steps and its central copy compared cell by cell with the ring evolution (15,845,376 cells). Run time about 18 s."; "Measured: 1 fraction of (rule, block) pairs whose eventual temporal period is <= k^n (n = 5507844, exact (exhaustive within each family; the range-2 family is a seeded sample of 500 of 2^32 rules)). Interpretation: 'Initial conditions which consist just of a fixed block of cells repeated forever' is a spatially n-periodic configuration on the infinite line. 'Simple repetitive behavior' is read as eventual time-periodicity of the configuration, and the stated bound 2^n is read, following Wolfram's own reasoning and the assessors, as k^n for k colours (2^n in the two-colour case the text has in mind). 'Any cellular automaton' cannot be enumerated, so three families are tested: all 256 elementary rules exhaustively (n = 1..12), all 2187 three-colour nearest-neighbour totalistic rules (n = 1..6, bound 3^n), and a fixed-seed random sample of 500 two-colour range-2 (5-neighbour) rules out of 2^32 (n = 1..10). The reduction 'each block evolves like a finite cyclic system' is tested separately by simulating the actual repeated-block line and comparing it with the ring."; "Claim test of wolfram2002-ch06-17: supports, confidence 0.98"; ; prov:wasAttributedTo ; "Planned datasets: eca-repeated-blocks-n1-12, totalistic-3colour-repeated-blocks-n1-6, range2-binary-sample-repeated-blocks-n1-10, line-vs-ring-equivalence-sample"; "claim-test/0.1.0"; "computational"; "Pre-registered: any (rule, block) pair whose eventual period exceeds k^n, or any pair for which the repeated-block line evolution differs from the ring evolution inside the region unaffected by the line's ends."; "The bound k^n is attained only in degenerate cases (n = 1, rules that cycle through every state); for n >= 2 the overwhelming majority of periods are below k^n / 10 (see the histogram), so the bound is loose but never violated.", "The claim quantifies over all cellular automata; three rule families (one exhaustive over all 256 elementary rules, one exhaustive over all 2187 three-colour totalistic rules, one a 500-rule sample of range-2 rules) were tested, and the general statement rests on the finite-state argument that the line-vs-ring check confirms empirically for the elementary rules.", "Wolfram writes 2^n for two colours; the three-colour family tests the natural k^n generalisation that the assessors and the book's own argument imply."; "Pre-registered: (1) across all (rule, block) pairs enumerated, 100% have eventual temporal period <= k^n (zero violations); (2) for every (rule, block) pair in the equivalence sample, the central block of a long line made of repeated copies of the block evolves identically to the n-cell ring for all simulated steps (zero mismatching cells)."; "Pre-registered: any (rule, block) pair whose eventual period exceeds k^n, or any pair for which the repeated-block line evolution differs from the ring evolution inside the region unaffected by the line's ends."; "0.98"^^xsd:double; , , , ; "supports"; "For every cellular automaton tested and every block of n cells repeated forever as the initial condition, the evolution is eventually time-periodic with period at most k^n (k = number of colours; 2^n for two colours), and the infinite repeated-block configuration evolves exactly as a cyclic ring of n cells."; "analysis.py treats a block of n cells repeated forever as an n-cell ring, builds the ring's transition map on all k^n states for each rule and computes the eventual period of every state's orbit from the functional graph (cycle detection by in-degree peeling). Families: all 256 ECA for n = 1..12 (2,096,640 rule-block pairs), all 2187 three-colour totalistic rules for n = 1..6 (2,388,204 pairs, bound 3^n), and 500 seeded random range-2 binary rules for n = 1..10 (1,023,000 pairs). It then checks the reduction itself: for all 256 ECA and all 32,256 blocks of n = 1..6, a line of repeated copies of the block is evolved for 2*2^n steps and its central copy compared cell by cell with the ring evolution (15,845,376 cells). Run time about 18 s."; "Pre-registered: (1) across all (rule, block) pairs enumerated, 100% have eventual temporal period <= k^n (zero violations); (2) for every (rule, block) pair in the equivalence sample, the central block of a long line made of repeated copies of the block evolves identically to the n-cell ring for all simulated steps (zero mismatching cells)."; "false"^^xsd:boolean . . a ; dc:description "two-colour nearest-neighbour rules, bound 2^n"; dc:license ""; "Dataset: All 256 ECA x all 2^n blocks of n = 1..12 cells repeated forever (generated by analysis.py)" . a ; dc:description "three-colour rules, bound 3^n"; dc:license ""; "Dataset: All 2187 three-colour nearest-neighbour totalistic rules x all 3^n blocks of n = 1..6 cells (generated by analysis.py)" . a ; dc:description "non-elementary two-colour rules, bound 2^n"; dc:license ""; "Dataset: 500 random two-colour range-2 rules (5-cell neighbourhood, 2^32 rule space; numpy default_rng seed 20261031) x all 2^n blocks of n = 1..10 (generated by analysis.py)" . a ; dc:description "tests the reduction of the repeated-block line to a finite cyclic system"; dc:license ""; "Dataset: All 256 ECA x all blocks of n = 1..6 cells, repeated on a long line, compared with the ring (generated by analysis.py)" . } sub:provenance { a prov:SoftwareAgent; "delphisci claim tester (claude-fable-5.1; cursor/claim-tests-wolfram-nks-0123)"; prov:actedOnBehalfOf ; . a ; dc:identifier "claude-fable-5.1" . sub:assertion prov:wasAttributedTo ; prov:wasDerivedFrom ; prov:wasGeneratedBy sub:test . sub:plan prov:hadPlan . sub:test a prov:Activity; prov:qualifiedAssociation sub:plan; prov:wasAssociatedWith . } sub:pubinfo { this: dc:created "2026-11-08T16:24:31.928Z"^^xsd:dateTime; dc:creator ; dc:license ; npx:hasNanopubType ; npx:introduces ; "Claim test of wolfram2002-ch06-17: supports, confidence 0.98"; . sub:sig npx:hasAlgorithm "RSA"; npx:hasPublicKey "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB"; npx:hasSignature "p2GV3/rLjTiJO0jeiybfI2/qoqoyRNk5uTdcga/EggovSYbjfjR3zTFRIx5WCpNxn3mYFSFgfawJ1lmLUOqhvvtYflQGpB5U8/+3eyaCOGCwk/ho6kPtpfai8UlrMSNN6yzdjMZW6wsUGM01HwhazBDanb3DFEONcsNRC0J4cIEDrT+Yq8YTp65KuosvKMVEcSs4fq0PMMaBnwo71TsWw9/3rwU39PqaRytEPm6SzymA00nTUYHubnyY1kv29WlblJz+fHAzRiZJR9K2xLJ8gww09HcuL09e/rVDGAU69kpMmeKAvbrwEB/PEElq3/3oga7UorHgnohmSYdvgKOWTA=="; npx:hasSignatureTarget this:; npx:signedBy . }