. . . . "analysis.py enumerates all 196 two-colour (blocks 1-3) and 1728 three-colour (blocks 1-2) neighbour-independent substitution rules, iterates each from a single cell of colour 1 until the word has 131072 symbols (or 60 steps), and counts distinct factors of length 1..14 with a sorted rolling-code method. It generates rule 30 from a single black cell with big-integer bit operations, taking the central 40001 cells of row 20000 and the first 65536 centre-column cells, and applies the same count. It also measures what share of the centre column's length-14 factors appear in any binary substitution word, and runs the one-cell-per-cell neighbour-dependent substitution systems for rules 90 and 30 against the direct CA for 1000 steps. Runtime about 19 s." . . "Measured: 157 maximum number of distinct length-14 factors over all 1924 neighbour-independent substitution words (threshold 225) (n = 1924, exact combinatorial counts). Interpretation: 'Repetitive' is operationalised as eventually periodic (bounded subword complexity, Morse-Hedlund), 'nested' as a self-similar word, which for fixed points of morphisms has subword complexity O(n^2) at most (Pansiot); the threshold (n+1)^2 at n = 14 is therefore a generous upper bound for repetitive-or-nested and about a tenth of what a random-like binary sequence of this length shows. 'Cannot emulate rule 30' is tested within the enumerated family by the complexity gap (a word with p(14) <= 225 cannot contain the rule 30 row, whose own p(14) is measured); it is not a proof over all encodings. The neighbour-dependent half is tested exactly as the book describes it (rules 'always yielding just one cell and corresponding quite directly to cellular automata', p. 666)." . "Claim test of wolfram2002-ch11-09: supports, confidence 0.85" . . . "Planned datasets: substitution-systems-and-rule30-generator" . "claim-test/0.1.0" . "computational" . "Pre-registered. FAIL (refutes) if any neighbour-independent rule in the family has p(14) >= 2000 (rule-30-like complexity), or if the neighbour-dependent construction does not reproduce rule 90 or rule 30. If some neighbour-independent rule has 225 < p(14) < 2000 the result is 'mixed' and the rule is reported with its complexity profile." . "The impossibility half is tested on the two complete enumerated families named in the book's own discussion (p. 83), not for arbitrarily long replacement blocks or alphabets; the complexity gap (at most 157 distinct length-14 factors versus 14922-16034 for rule 30) is a finite proxy for the theorem that fixed points of morphisms have polynomially bounded subword complexity." . "The length-14 factor share result (at most 0.5% of rule 30's centre-column factors occur in any binary substitution word) is reported as a direct check that none of the enumerated systems reproduces rule 30's sequence under the identity encoding; other encodings were not explored." . "The neighbour-dependent half is near-definitional: a substitution system whose single output cell depends on the cell and its two neighbours is a cellular automaton; the test confirms it reproduces rules 90 and 30 exactly for 1000 steps, which is what the book's figure shows." . "The subword-complexity criterion treats any word with p(14) <= 225 as repetitive-or-nested; the words with the largest complexity (e.g. 0 -> 0, 1 -> 011, p(n) = 2,4,6,9,13,...,81) are non-primitive morphic words whose complexity grows faster than linearly but far below that of rule 30, consistent with the known n log n / n^2 classes for such words." . "Pre-registered. For each of the 196 two-colour rules {0 -> w0, 1 -> w1} with |w0|, |w1| in {1,2,3} and each of the 1728 three-colour rules with block lengths in {1,2}, start from a single cell of colour 1 and iterate until the word has >= L = 131072 symbols (or 60 steps if it never does), truncate to L, and compute the number p(n) of distinct length-n factors for n = 1..14. For rule 30 from a single black cell compute p(14) of (i) the central 40001 cells of the row at step 20000 and (ii) the first 65536 cells of the centre column. Build the neighbour-dependent substitution system whose rule maps each cell to one new cell given by the elementary-CA table of its (left, self, right) neighbourhood, and run it for 1000 steps from a single black cell. PASS (supports) if (a) every neighbour-independent rule has p(14) <= 225 = (14+1)^2 (eventually periodic words are those with p(14) = p(13); the rest are counted as nested), (b) both rule 30 sequences have p(14) >= 2000, and (c) the neighbour-dependent systems for rules 90 and 30 reproduce the cellular-automaton rows exactly at every one of the 1000 steps." . "Pre-registered. FAIL (refutes) if any neighbour-independent rule in the family has p(14) >= 2000 (rule-30-like complexity), or if the neighbour-dependent construction does not reproduce rule 90 or rule 30. If some neighbour-independent rule has 225 < p(14) < 2000 the result is 'mixed' and the rule is reported with its complexity profile." . "0.85"^^ . . "supports" . "If the claim holds, every neighbour-independent substitution system in a complete enumerated family (2 colours with replacement blocks of length 1-3; 3 colours with blocks of length 1-2) produces, from a single cell, a sequence whose subword complexity stays bounded (repetitive) or grows at most polynomially slowly (nested), far below the near-exponential subword complexity of rule 30's rows and centre column, so none of them can reproduce rule 30; while a neighbour-dependent substitution system (one output cell per cell, chosen from the cell and its two neighbours) reproduces rules 90 and 30 exactly." . "analysis.py enumerates all 196 two-colour (blocks 1-3) and 1728 three-colour (blocks 1-2) neighbour-independent substitution rules, iterates each from a single cell of colour 1 until the word has 131072 symbols (or 60 steps), and counts distinct factors of length 1..14 with a sorted rolling-code method. It generates rule 30 from a single black cell with big-integer bit operations, taking the central 40001 cells of row 20000 and the first 65536 centre-column cells, and applies the same count. It also measures what share of the centre column's length-14 factors appear in any binary substitution word, and runs the one-cell-per-cell neighbour-dependent substitution systems for rules 90 and 30 against the direct CA for 1000 steps. Runtime about 19 s." . "Pre-registered. For each of the 196 two-colour rules {0 -> w0, 1 -> w1} with |w0|, |w1| in {1,2,3} and each of the 1728 three-colour rules with block lengths in {1,2}, start from a single cell of colour 1 and iterate until the word has >= L = 131072 symbols (or 60 steps if it never does), truncate to L, and compute the number p(n) of distinct length-n factors for n = 1..14. For rule 30 from a single black cell compute p(14) of (i) the central 40001 cells of the row at step 20000 and (ii) the first 65536 cells of the centre column. Build the neighbour-dependent substitution system whose rule maps each cell to one new cell given by the elementary-CA table of its (left, self, right) neighbourhood, and run it for 1000 steps from a single black cell. PASS (supports) if (a) every neighbour-independent rule has p(14) <= 225 = (14+1)^2 (eventually periodic words are those with p(14) = p(13); the rest are counted as nested), (b) both rule 30 sequences have p(14) >= 2000, and (c) the neighbour-dependent systems for rules 90 and 30 reproduce the cellular-automaton rows exactly at every one of the 1000 steps." . "false"^^ . . "the computational objects the claim is about" . "generated by this test" . . "Dataset: Enumerated neighbour-independent substitution systems (2 colours, blocks 1-3; 3 colours, blocks 1-2), rule 30 evolutions, and neighbour-dependent substitution emulations of rules 90 and 30, generated by analysis.py" . . . "delphisci claim tester (claude-fable-5.1; cursor/claim-tests-wolfram-nks-9241)" . . . "claude-fable-5.1" . . . . . . . . . "2026-11-08T16:24:31.937Z"^^ . . . . . "Claim test of wolfram2002-ch11-09: supports, confidence 0.85" . . "RSA" . "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB" . "rym7XaeFu7XN9/+ZGrGl7K7uT6Y8X9hg+Zt2ETU5FTvyWhngX84yDQFbFoNG/IwcpmrKHw0xmQhpBrZ1+/CtDOe9KiRk/NHD1Cq9Pcp7aUNZb7+2hiu7Vt87XLATAppXzfEaTDe2OdkRy24oapYOK2225hvLt4sXrwR1gEw9j2xtcj4Cwyi2hzaqjUNlasr5xSYnQNfa10QO20HNAuB1y8KcfDpmwE0TvVqZunw8L540R9uJ7G9o4HR66P6hpvO5ieejQdKPLRwrYdmW3dB8nV4iR/kjoqEiUGOsp65ev9BxUUu5w2fCqoAimVe+R60fLvXshX2Cg7Tfnj0ovKHNhA==" . . .