@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 evolves rule 90 from a single black cell for 2048 steps on a 4097-cell line (boundary never reached, asserted). For k = 1..11 it reconstructs the 2^k-row triangle from three translated copies of the 2^(k-1)-row triangle (top centred, bottom copies shifted by -2^(k-1) and +2^(k-1)) over a white background and counts mismatching cells against the simulated triangle, also counting black cells inside the central inverted triangle. It then downloads the OEIS A001317 b-file (3322 terms), reads each simulated row's cells at positions x = -t, -t+2, ..., t as a binary number and compares with A001317(t) for all 2048 rows."; "Measured: 0 mismatching cells in the three-copy decomposition, summed over k = 1..11 (n = 11180714, exact). Interpretation: 'Nested triangular pieces that all have exactly the same form' is operationalised as exact self-similarity under doubling: the pattern of the first 2^k steps is three translated copies of the pattern of the first 2^(k-1) steps separated by an empty (white) inverted triangle; iterated down to k = 1 this means every triangular piece is a scaled copy of the whole. The rule is the Wolfram rule 90 (new colour = XOR of the two neighbours), started from one black cell on an otherwise white infinite line, which is simulated exactly on a line wide enough that boundaries never matter. As an independent external check the rows are compared with OEIS A001317 (Sierpinski's triangle rows read as binary numbers), which is tabulated independently of any cellular-automaton code."; "Claim test of wolfram2002-ch02-01: supports, confidence 0.99"; ; prov:wasAttributedTo ; "Planned datasets: eca-rule90-single-cell-2048-steps, oeis-a001317-bfile"; "claim-test/0.1.0"; "computational"; "Pre-registered: any cell of the simulated pattern violates the three-copy decomposition at some k <= 11, or any simulated row differs from the independently tabulated A001317 value."; "Exact self-similarity was verified up to depth 2^11 = 2048 steps; the extension to all depths follows from the row formula a(t,x) = C(t,(t+x)/2) mod 2 and Lucas's theorem, which the OEIS comparison confirms independently for every simulated row but which this test does not prove.", "OEIS A001317 is itself computed from Pascal's triangle mod 2, not from a cellular automaton, so the agreement is a check against an independent construction of the same mathematical object, not against a physical dataset."; "Pre-registered: (1) for every k = 1..11 the 2^k-step triangle decomposes exactly (zero mismatching cells) into three copies of the 2^(k-1)-step triangle plus an all-white central inverted triangle; and (2) every simulated row t whose value is present in the OEIS A001317 b-file equals A001317(t) when the row's cells at positions of parity t are read as a binary number (zero mismatching rows)."; "Pre-registered: any cell of the simulated pattern violates the three-copy decomposition at some k <= 11, or any simulated row differs from the independently tabulated A001317 value."; "0.99"^^xsd:double; , ; "supports"; "Rule 90 evolved from a single black cell produces a pattern in which the triangle of the first 2^k steps is composed exactly of three copies of the triangle of the first 2^(k-1) steps (one on top, two below, with an all-white inverted triangle between them), for every k up to the depth simulated, and its rows coincide with Pascal's triangle modulo 2 as tabulated independently in OEIS A001317."; "analysis.py evolves rule 90 from a single black cell for 2048 steps on a 4097-cell line (boundary never reached, asserted). For k = 1..11 it reconstructs the 2^k-row triangle from three translated copies of the 2^(k-1)-row triangle (top centred, bottom copies shifted by -2^(k-1) and +2^(k-1)) over a white background and counts mismatching cells against the simulated triangle, also counting black cells inside the central inverted triangle. It then downloads the OEIS A001317 b-file (3322 terms), reads each simulated row's cells at positions x = -t, -t+2, ..., t as a binary number and compares with A001317(t) for all 2048 rows."; "Pre-registered: (1) for every k = 1..11 the 2^k-step triangle decomposes exactly (zero mismatching cells) into three copies of the 2^(k-1)-step triangle plus an all-white central inverted triangle; and (2) every simulated row t whose value is present in the OEIS A001317 b-file equals A001317(t) when the row's cells at positions of parity t are read as a binary number (zero mismatching rows)."; "false"^^xsd:boolean . . a ; dc:description "the pattern whose nesting is tested"; dc:license ""; "Dataset: Elementary cellular automaton rule 90 from a single black cell, 2048 steps (generated by analysis.py)" . a ; dc:description "independent tabulation of rule-90 rows from a single cell (row t read as binary)"; dc:license "CC BY-SA 4.0 (OEIS)"; "Dataset: OEIS A001317: Sierpinski's triangle (Pascal's triangle mod 2) converted to decimal, b-file"; prov:atLocation ; prov:wasDerivedFrom ; "sha256:fc61a63ffcdbfe524aee0ebf372f4a91653244cb466a94fba3b5ee48ccb04f19" . } 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 "2026-11-08T16:24:31.913Z"^^xsd:dateTime; dc:creator ; dc:license ; npx:hasNanopubType ; npx:introduces ; "Claim test of wolfram2002-ch02-01: supports, confidence 0.99"; . sub:sig npx:hasAlgorithm "RSA"; npx:hasPublicKey "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB"; npx:hasSignature "aLgjaUFn5HtaopoC2NM0+QaS+/W/qNkDJDDm2PS39IHX/qlWSnFUFkCVogVQ4+7tjsBVXXyh5dvHr0O2yPvuFG0JEe+Zw+wFf3yVwed+hPoR00Hlsmejq6ZHgR7F5blymK7lUDRwTd/vm3lW/cbpqzwVLgpqGCDfU7DVeAkVmydBf4PrPehKoG2oih5Vah6i1KEtsw9Ja40OOApOPczenGy0dOf0G+0NcitHbXsnTpldNAEflI/7+cY7vaotjxi5z4t2i+kjhrbVqItjl3jwDEby2zoLbcoHSjxOcREZk9Cw3qPsrh4IPw8h/oijnchG6Z+xEIQRceFLFFfUnz5EZQ=="; npx:hasSignatureTarget this:; npx:signedBy . }