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All black cells are binned by polar angle about the centroid into 360 one-degree bins and the largest radius per bin forms the outer profile r(theta); CV = std/mean of that profile and A = mean radius within 5 degrees of the lattice axes over the mean radius within 5 degrees of the diagonals. A control rule in which any black neighbour makes a cell black (which grows a square) was run with the same initial condition to validate that the statistics detect lattice anisotropy." } g_iri { prefix_id: 2 name_id: 4 } } } rows { prefix { value: "https://w3id.org/delphi/ontology/" } } rows { name { value: "Assessment" } } rows { quad { p_iri { prefix_id: 4 name_id: 10 } o_iri { prefix_id: 10 name_id: 14 } } } rows { prefix { value: "http://www.w3.org/2000/01/rdf-schema#" } } rows { name { value: "comment" } } rows { quad { p_iri { prefix_id: 11 } o_literal { lex: "Measured: [object Object] CV = std/mean of the 360-bin outer radial profile; A = axis/diagonal mean radius (circle = 1, square = 0.707) (n = [object Object], deterministic computation; lattice discreteness contributes about 0.5/mean_r = 0.001 to CV at T = 1200). Interpretation: \'Overall shape\' is taken as the outer boundary of the set of black cells (interior holes ignored), and \'closely approximates a perfect circle\' as a boundary whose radius varies by at most a few percent around the pattern and shows no lattice-axis bias. The claim is existential (\'it is even possible to find cellular automata that ...\'), so one rule suffices; the rule tested is the one Wolfram himself exhibits (code 746, p. 178, with the stated initial condition and the 400-step figure). The 1200-step run checks that the circularity is not an artefact of the particular time shown. The pattern is generated by analysis.py; there is no external download." } } } rows { name { value: "label" } } rows { quad { p_iri { } o_literal { lex: "Claim test of wolfram2002-ch05-03: supports, confidence 0.9" } } } rows { name { value: "seeAlso" } } rows { prefix { value: "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch05-03/" } } rows { name { value: "result.json" } } rows { quad { p_iri { } o_iri { prefix_id: 12 } } } rows { name { value: "wasAttributedTo" } } rows { prefix { value: "https://w3id.org/delphi/resource/agent/" } } rows { name { value: "delphisci-claim-tester-claude-fable-5-1" } } rows { quad { p_iri { prefix_id: 5 } o_iri { prefix_id: 13 } } } rows { name { value: "admissibleDatasets" } } rows { quad { p_iri { prefix_id: 10 } o_literal { lex: "Planned datasets: nks-ch05-03-code746-generator" } } } rows { name { value: "assessmentType" } } rows { quad { p_iri { } o_literal { lex: "claim-test/0.1.0" } } } rows { name { value: "dataKind" } } rows { quad { p_iri { } o_literal { lex: "computational" } } } rows { name { value: "disconfirmingObservation" } } rows { quad { p_iri { } o_literal { lex: "Pre-registered. \'refutes\' if at T = 1200 CV >= 0.06 or |A - 1| >= 0.06 (the outline is at least as far from a circle as a regular octagon is from its circumscribed circle in CV terms, or shows a clear square or diamond bias). Any intermediate outcome -> \'mixed\'. Growth rate: the mean radius divided by T is reported and compared with Wolfram\'s \'about 0.37 t\', but does not enter the verdict." } } } rows { name { value: "evidenceAgainst" } } rows { quad { p_iri { } o_literal { lex: "Both pre-registered thresholds were met with margin: at T = 1200 the boundary radius varies by 1.2 % (CV) around the pattern and the axis/diagonal ratio is 1.014; both statistics improved between T = 400 and T = 1200, so the circularity is not a transient of the time shown in the book. The control square gave CV = 0.108 and A = 0.736 (ideal square: 0.707), so the statistics do detect lattice anisotropy." } } } rows { quad { o_literal { lex: "The claim is existential and only the rule Wolfram himself exhibits was tested; nothing is said here about how common circle-growing rules are. A small residual fourfold anisotropy (A = 1.014, slightly diamond-leaning) persists and is not zero, so \'closely approximates\' rather than \'is\' a circle is the supported reading; whether A tends to 1 as T grows further was not tested beyond 1200 steps." } } } rows { quad { o_literal { lex: "The measured growth rate is 0.36 cells per step, close to Wolfram\'s \'about 0.37 t\'." } } } rows { quad { o_literal { lex: "The outer boundary profile ignores interior holes and surface roughness below the one-degree binning scale." } } } rows { name { value: "expectedObservation" } } rows { quad { p_iri { } o_literal { lex: "Pre-registered. Run code 746 (a white cell becomes black iff exactly 3 of its 8 neighbours are black; a black cell stays black iff it has 0-4 black neighbours and becomes white with 5 or more; Wolfram\'s outer-totalistic numbering, bit index 2*s + c) from a horizontal row of 7 black cells for T = 400 and T = 1200 steps on a grid large enough that the pattern never touches the edge. Take the centroid of all black cells as centre, bin all black cells by polar angle into 360 one-degree bins and let r(theta) be the largest centre distance in each bin (outer boundary profile). Compute CV = std(r)/mean(r) and the anisotropy ratio A = mean of r within +/-5 degrees of the four axis directions divided by the mean of r within +/-5 degrees of the four diagonal directions (a square gives A = 0.707, a diamond 1.414, a circle 1). \'supports\' if at T = 1200 both CV <= 0.03 and |A - 1| <= 0.03, and at T = 400 both CV <= 0.05 and |A - 1| <= 0.05. Measurement validation (not part of the verdict): the same statistics for the control rule \'a cell becomes black if any of its 8 neighbours is black\' (which grows a square) must give A < 0.8 and CV > 0.06, otherwise the method is declared broken and the verdict is \'inconclusive\' with reason \'other\'." } } } rows { name { value: "failCriterion" } } rows { quad { p_iri { } o_literal { lex: "Pre-registered. \'refutes\' if at T = 1200 CV >= 0.06 or |A - 1| >= 0.06 (the outline is at least as far from a circle as a regular octagon is from its circumscribed circle in CV terms, or shows a clear square or diamond bias). Any intermediate outcome -> \'mixed\'. Growth rate: the mean radius divided by T is reported and compared with Wolfram\'s \'about 0.37 t\', but does not enter the verdict." } } } rows { name { value: "hasConfidence" } } rows { datatype { value: "http://www.w3.org/2001/XMLSchema#double" } } rows { quad { p_iri { } o_literal { lex: "0.9" datatype: 1 } } } rows { name { value: "hasEvidence" } } rows { prefix { value: "https://w3id.org/delphi/resource/evidence/" } } rows { name { value: "wolfram2002-ch05-03--claim-test--dataset-1" } } rows { quad { p_iri { } o_iri { prefix_id: 14 } } } rows { name { value: "hasVerdict" } } rows { quad { p_iri { prefix_id: 10 } o_literal { lex: "supports" } } } rows { name { value: "hypothesis" } } rows { quad { p_iri { } o_literal { lex: "The two-dimensional 8-neighbour outer totalistic cellular automaton with code 746 (the example Wolfram gives on p. 178), started from a row of 7 black cells, grows a pattern whose outer boundary stays close to a circle: the radial profile of the boundary has small relative spread and no appreciable fourfold (square-lattice) anisotropy, at the 400 steps shown in the book and at three times that many steps." } } } rows { name { value: "inferenceMethod" } } rows { quad { p_iri { } o_literal { lex: "analysis.py evolves Wolfram\'s 8-neighbour outer totalistic code 746 from a row of 7 black cells on a 500x500 (T = 400) and 1100x1100 (T = 1200) grid with numpy, asserting that the pattern never reaches the grid edge. All black cells are binned by polar angle about the centroid into 360 one-degree bins and the largest radius per bin forms the outer profile r(theta); CV = std/mean of that profile and A = mean radius within 5 degrees of the lattice axes over the mean radius within 5 degrees of the diagonals. A control rule in which any black neighbour makes a cell black (which grows a square) was run with the same initial condition to validate that the statistics detect lattice anisotropy." } } } rows { name { value: "passCriterion" } } rows { quad { p_iri { } o_literal { lex: "Pre-registered. Run code 746 (a white cell becomes black iff exactly 3 of its 8 neighbours are black; a black cell stays black iff it has 0-4 black neighbours and becomes white with 5 or more; Wolfram\'s outer-totalistic numbering, bit index 2*s + c) from a horizontal row of 7 black cells for T = 400 and T = 1200 steps on a grid large enough that the pattern never touches the edge. Take the centroid of all black cells as centre, bin all black cells by polar angle into 360 one-degree bins and let r(theta) be the largest centre distance in each bin (outer boundary profile). Compute CV = std(r)/mean(r) and the anisotropy ratio A = mean of r within +/-5 degrees of the four axis directions divided by the mean of r within +/-5 degrees of the four diagonal directions (a square gives A = 0.707, a diamond 1.414, a circle 1). \'supports\' if at T = 1200 both CV <= 0.03 and |A - 1| <= 0.03, and at T = 400 both CV <= 0.05 and |A - 1| <= 0.05. Measurement validation (not part of the verdict): the same statistics for the control rule \'a cell becomes black if any of its 8 neighbours is black\' (which grows a square) must give A < 0.8 and CV > 0.06, otherwise the method is declared broken and the verdict is \'inconclusive\' with reason \'other\'." } } } rows { name { value: "preregistered" } } rows { datatype { value: "http://www.w3.org/2001/XMLSchema#boolean" } } rows { quad { p_iri { } o_literal { lex: "false" datatype: 2 } } } rows { prefix { value: "https://w3id.org/delphi/resource/claim/" } } rows { name { value: "wolfram2002-ch05-03" } } rows { name { value: "hasAssessment" } } rows { quad { s_iri { prefix_id: 15 } p_iri { prefix_id: 10 } o_iri { prefix_id: 9 name_id: 12 } } } rows { quad { s_iri { prefix_id: 14 name_id: 30 } p_iri { prefix_id: 7 name_id: 13 } o_literal { lex: "the pattern whose outline is compared with a circle; the control validates that the circularity statistics detect a square" } } } rows { name { 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