. . . . "analysis.py ran all 2187 three-colour totalistic codes from a single gray cell for T = 3000 steps on an array of width 2*(T+128)+1 (boundaries cannot reach the centre), then classified the central 128-column strip: time-periodic with p <= 256 over rows 2250..2999, or lzma compressibility rho over rows 1500..2999 below tau. Calibration before the census: nested references rho = 0.017..0.037 (Pascal mod 2 / mod 3, Thue-Morse and period-doubling outer products), irregular references rho = 0.099..0.136 (elementary rules 110, 124, 137, 193) and 1.11..1.12 (rules 30, 45); classes separated, tau = sqrt(0.0367 * 0.0985) = 0.0601. Result: 1706 codes periodic, 145 non-periodic but nested-compressible (rho <= 0.042), 336 irregular (rho 0.12..1.11, median 0.90; 169 with rho >= 0.9); regular fraction 1851/2187 = 0.8464, inside the pre-registered [0.80, 0.90] band." . . "Measured: 0.8464 fraction of the 2187 codes classified regular (late-time strip periodic with p <= 256, or rho < tau) (n = 2187, deterministic given tau = 0.060; a few long-transient codes may be misclassified at T = 3000). Interpretation: 'Ultimately quite regular' is read as 'repetitive or nested at late times', operationalised as late-time periodicity of the central strip or compressibility at the level of classical nested patterns; 'the remainder' as everything compressing like random or localised-structure patterns, with elementary class-3 and class-4 rules from a different family used to fix the threshold. 'About 85 %' is given a +/- 5 percentage-point band. Long transients (Wolfram mentions code 1599 resolving only after thousands of steps) may leave a few ultimately regular codes labelled irregular at T = 3000, which biases the regular fraction downward slightly. All patterns are generated by analysis.py; there is no external download." . "Claim test of wolfram2002-ch03-05: supports, confidence 0.75" . . . "Planned datasets: nks-ch03-05-totalistic3-generator" . "claim-test/0.1.0" . "computational" . "Pre-registered. 'refutes' if the regular fraction is below 0.70 or above 0.95. 'mixed' if it lies in [0.70, 0.80) or (0.90, 0.95]. 'inconclusive' with reason 'other' if the calibration fails to separate the reference classes (largest nested rho >= smallest irregular rho). The number of irregular codes that are random-looking (rho above the smallest random reference) versus structured-irregular is reported but not decisive." . "'About 85 %' is Wolfram's rounded figure; the pre-registered band [0.80, 0.90] was chosen to allow +/- 5 points of classification ambiguity. The measured 84.6 % lands within 0.4 points of the quoted figure." . "Codes are classified from a single gray cell only, as in the book's figure; a black-cell start would give a different census." . "Periodicity and compressibility are measured on the central 128 columns only; patterns whose irregularity is confined to the edges of the growing region (and whose centre is periodic) count as regular." . "T = 3000 steps: codes with transients longer than about 750 steps in the periodicity window, or whose late-time structure first appears beyond 3000 steps (NKS mentions code 1599 at thousands of steps), may be misclassified; code 1599 is counted irregular here (rho 0.55)." . "The pre-registered 'random-like' sub-count (rho above the smallest 2-colour random reference, 1.11) is not comparable across alphabets and came out as 0; the post-hoc quantiles added to measured.extra show 169 of the 336 irregular codes have rho >= 0.9, i.e. compress essentially like random 3-colour data. This sub-count was pre-registered as non-decisive." . "The regular/irregular split depends on tau, which is calibrated on elementary (2-colour) reference rules; the gap between the nested references (max 0.037) and the structured-irregular references (min 0.099) is wide and the census has no codes between 0.042 and 0.119, so the fraction is insensitive to the exact value of tau within that gap." . "Pre-registered. Each code c in 0..2186 (Wolfram's convention: base-3 digit k of c is the new colour for neighbourhood sum k in 0..6; colours 0 = white, 1 = gray, 2 = black) is run for T = 3000 steps from a single gray cell on a white background with an array wide enough that boundary effects cannot reach the central 128 columns. Statistics on the central strip (columns -64..63): (i) periodic: rows 2250..2999 are time-periodic with some period p <= 256; (ii) rho = 8 * lzma_bytes / (cells * log2(3)) over rows 1500..2999. Calibration (performed before the codes are run): nested references are the same-size strips of Pascal's triangle mod 3 (3 symbols), Pascal's triangle mod 2, and the Thue-Morse and period-doubling outer-product patterns (normalised by log2 of their own alphabet size); irregular references are the strips of elementary rules 110, 124, 137, 193 (localised structures) and 30, 45 (random) run from a single black cell with the same geometry. tau = geometric mean of the largest nested-reference rho and the smallest irregular-reference rho. A code is 'regular' iff it is periodic or rho < tau. 'supports' if the regular fraction lies in [0.80, 0.90]." . "Pre-registered. 'refutes' if the regular fraction is below 0.70 or above 0.95. 'mixed' if it lies in [0.70, 0.80) or (0.90, 0.95]. 'inconclusive' with reason 'other' if the calibration fails to separate the reference classes (largest nested rho >= smallest irregular rho). The number of irregular codes that are random-looking (rho above the smallest random reference) versus structured-irregular is reported but not decisive." . "0.75"^^ . . "supports" . "When all 2187 three-colour nearest-neighbour totalistic cellular automata are run from a single gray cell for a few thousand steps, about 85 % produce patterns whose late-time central region is time-periodic or compresses like classical nested patterns, and the remaining 15 % or so compress like random or class-4 (localised-structure) patterns." . "analysis.py ran all 2187 three-colour totalistic codes from a single gray cell for T = 3000 steps on an array of width 2*(T+128)+1 (boundaries cannot reach the centre), then classified the central 128-column strip: time-periodic with p <= 256 over rows 2250..2999, or lzma compressibility rho over rows 1500..2999 below tau. Calibration before the census: nested references rho = 0.017..0.037 (Pascal mod 2 / mod 3, Thue-Morse and period-doubling outer products), irregular references rho = 0.099..0.136 (elementary rules 110, 124, 137, 193) and 1.11..1.12 (rules 30, 45); classes separated, tau = sqrt(0.0367 * 0.0985) = 0.0601. Result: 1706 codes periodic, 145 non-periodic but nested-compressible (rho <= 0.042), 336 irregular (rho 0.12..1.11, median 0.90; 169 with rho >= 0.9); regular fraction 1851/2187 = 0.8464, inside the pre-registered [0.80, 0.90] band." . "Pre-registered. Each code c in 0..2186 (Wolfram's convention: base-3 digit k of c is the new colour for neighbourhood sum k in 0..6; colours 0 = white, 1 = gray, 2 = black) is run for T = 3000 steps from a single gray cell on a white background with an array wide enough that boundary effects cannot reach the central 128 columns. Statistics on the central strip (columns -64..63): (i) periodic: rows 2250..2999 are time-periodic with some period p <= 256; (ii) rho = 8 * lzma_bytes / (cells * log2(3)) over rows 1500..2999. Calibration (performed before the codes are run): nested references are the same-size strips of Pascal's triangle mod 3 (3 symbols), Pascal's triangle mod 2, and the Thue-Morse and period-doubling outer-product patterns (normalised by log2 of their own alphabet size); irregular references are the strips of elementary rules 110, 124, 137, 193 (localised structures) and 30, 45 (random) run from a single black cell with the same geometry. tau = geometric mean of the largest nested-reference rho and the smallest irregular-reference rho. A code is 'regular' iff it is periodic or rho < tau. 'supports' if the regular fraction lies in [0.80, 0.90]." . "false"^^ . . "the finite rule space whose regular fraction the claim reports" . "generated" . . "Dataset: All 2187 three-colour nearest-neighbour totalistic cellular automata (Wolfram code numbering) from a single gray cell, 3000 steps, generated by analysis.py, plus the calibration reference patterns described in pass_criterion" . . . "delphisci claim tester (claude-fable-5.1; cursor/claim-tests-wolfram-nks-e5f4)" . . . "claude-fable-5.1" . . . . . . . . . "2026-11-08T16:24:31.919Z"^^ . . . . . "Claim test of wolfram2002-ch03-05: supports, confidence 0.75" . . "RSA" . "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB" . "eoM3Ry3xwAX/fQE0aI7q1O1J+MepIfHHccKR2Jvw5jLD35y2WnJEDXYckjKwSdvDpryzWpMw4ojYH3Z1lNTWOYBkoZbapaixyOf6wD5QBwkvWg+9WfcYDhe7iAef7jB7pEn7Rvstn6Tac7K5u3UTFVHstbc2W/wFhBCNcBMmyfnS7QyCD+gmg4aaXtPFdGATcqY86bWUf9hibjG7rtv2hjEXs/uwzy6Rko/oflzb8kk8RB2Z9yXAKjAYk/T98Cn3GE/OuvlEtgagCNKOP8wfxaptnbmZlF7IqUBR6826GNyYrDeszpCdb52d5S4BuFv8x8FwJGWALsz1GE9+kd/IqQ==" . . .