. . . . "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." . . "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." . "Claim test of wolfram2002-ch05-03: supports, confidence 0.9" . . . "Planned datasets: nks-ch05-03-code746-generator" . "claim-test/0.1.0" . "computational" . "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." . "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." . "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." . "The measured growth rate is 0.36 cells per step, close to Wolfram's 'about 0.37 t'." . "The outer boundary profile ignores interior holes and surface roughness below the one-degree binning scale." . "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'." . "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." . "0.9"^^ . . "supports" . "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." . "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." . "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'." . "false"^^ . . "the pattern whose outline is compared with a circle; the control validates that the circularity statistics detect a square" . "generated" . . "Dataset: Two-dimensional 8-neighbour (Moore) outer totalistic cellular automaton, Wolfram code 746, generated by analysis.py; initial condition a row of 7 black cells on a white background; T = 400 and 1200 steps; grid 1100 x 1100 with the pattern kept away from the edges; control rule (code with all 'any black neighbour -> black' bits set) run with the same initial condition and T" . . . "delphisci claim tester (claude-fable-5.1; cursor/claim-tests-wolfram-nks-e5f4)" . . . "claude-fable-5.1" . . . . . . . . . "2026-11-08T16:24:31.925Z"^^ . . . . . "Claim test of wolfram2002-ch05-03: supports, confidence 0.9" . . "RSA" . "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB" . "x7E3J2RsMVUhq4kw40dtPxmpTikfYGWSZAFLihqmU23pu2EJTLy/kV0ovnpZxRG0jPhhB3lpc1etHai1ouS1IHF8XMt1NiOt01y0u5mN4abRNaTnHhEW4GTSmhqYnJE8PaZmoWUuSg1lxIMJYONtsnntqpikNF7phg5iWTDYF5Cjl/778kuGO0zhrX/QvDMs2woUq4ZJMrGJP0FYK2RdndMCLCwktC8ohJb5ovECGPA9kilSjv7F26DLwaLUbJ5knky+Gmqm00ii6p2Y+3r4teucb2w4I6KGY8npidP1QoVQLjP4pgOElqJHPyE8k3zsdaWVDf3w1tqNxowX5U7Gsw==" . . .