. . . . "analysis.py encodes each constraint system as a bitmask of allowed length-w words. Satisfiability is decided by iterating a reachability vector over the k^(w-1) de Bruijn vertices for 4V + 1 edge steps (existence of a valid word of length 4V + w). Independently, every period word u with |u| <= V is enumerated (sum of k^p over p = 1..V candidates, each stored as the bitmask of its cyclic length-w windows) and u^infinity is accepted iff its window mask is contained in the allowed set. The two indicators are compared for all 138,856 systems (exhaustive for (k,w) = (2,2), (2,3), (2,4), (3,2), (4,2); seeded samples of 2000 and 5000 for (2,5) and (3,3)); minimal periods are tabulated. Run time about 3 s." . . "Measured: 1 fraction of satisfiable constraint systems that have a periodic solution with period <= k^(w-1) (n = 129939, exact for the exhaustive families; the two sampled families are 2000 and 5000 seeded draws). Interpretation: Wolfram's 'local set of constraints' on a one-dimensional array of coloured cells (p. 210-211: each cell's neighbourhood must be one of a given set of allowed configurations) is a one-dimensional subshift of finite type: a set of allowed words of a fixed window length w over k colours. 'Can be satisfied at all' is read as the existence of a bi-infinite arrangement all of whose windows are allowed; by compactness this is equivalent to the existence of arbitrarily long valid finite words, and operationally it is tested by the existence of a valid word of length 4V + w (any valid word longer than V must revisit a de Bruijn vertex, so this length is far beyond the threshold, and the check is done by a reachability iteration, not a cycle search). 'Simple and purely repetitive arrangement' is read as a periodic configuration, found by brute-force enumeration of all period words u with |u| <= V = k^(w-1) and checking all cyclic windows. All constraint systems are enumerated exhaustively for (k, w) in {(2,2), (2,3), (2,4), (3,2), (4,2)} and sampled with a fixed seed for (2,5) and (3,3)." . "Claim test of wolfram2002-ch05-15: supports, confidence 0.97" . . . "Planned datasets: sft-k2-w2-all, sft-k2-w3-all, sft-k2-w4-all, sft-k3-w2-all, sft-k4-w2-all, sft-k2-w5-sample, sft-k3-w3-sample" . "claim-test/0.1.0" . "computational" . "Pre-registered: at least one constraint system admits a valid word of length 4V + w but no periodic arrangement with period <= V (this would refute the claim as stated, since the shortest possible periodic solution is bounded by V); or the two satisfiability indicators disagree in the other direction, which would indicate a broken test." . "Satisfiability was operationalised by the existence of a valid word of length 4V + w; by compactness this is equivalent to the existence of a bi-infinite arrangement, but the equivalence itself relies on finiteness of the vertex set, so the two indicators are not logically independent even though they are computed by different algorithms." . "The claim is a theorem (a non-empty one-dimensional shift of finite type contains a periodic point, from a cycle in its de Bruijn graph); the computation confirms it exhaustively for all two-colour constraints with windows up to 4 and all four-colour constraints with window 2, and by sampling for windows 5 (two colours) and 3 (three colours). The maximal minimal period found (8 for k = 2, w = 4; exactly the two de Bruijn-cycle constraint sets) matches the bound V = k^(w-1)." . "Wolfram's 'simple' is read as 'periodic'; the observed minimal periods are small (at most V), but 'simple' is not otherwise quantified." . "Pre-registered: across all enumerated or sampled constraint systems, every system that admits a valid word of length 4V + w (the operational test of satisfiability, independent of any cycle argument) also admits a periodic arrangement u^infinity with |u| <= V whose every cyclic window of length w is allowed; and every system with no such long valid word has no periodic arrangement either. Zero violations in both directions." . "Pre-registered: at least one constraint system admits a valid word of length 4V + w but no periodic arrangement with period <= V (this would refute the claim as stated, since the shortest possible periodic solution is bounded by V); or the two satisfiability indicators disagree in the other direction, which would indicate a broken test." . "0.97"^^ . . . . . . . . "supports" . "For every one-dimensional local constraint system (a set of allowed length-w colour words over k colours) that admits any bi-infinite satisfying arrangement, there is a purely periodic satisfying arrangement, and its period can be taken at most V = k^(w-1)." . "analysis.py encodes each constraint system as a bitmask of allowed length-w words. Satisfiability is decided by iterating a reachability vector over the k^(w-1) de Bruijn vertices for 4V + 1 edge steps (existence of a valid word of length 4V + w). Independently, every period word u with |u| <= V is enumerated (sum of k^p over p = 1..V candidates, each stored as the bitmask of its cyclic length-w windows) and u^infinity is accepted iff its window mask is contained in the allowed set. The two indicators are compared for all 138,856 systems (exhaustive for (k,w) = (2,2), (2,3), (2,4), (3,2), (4,2); seeded samples of 2000 and 5000 for (2,5) and (3,3)); minimal periods are tabulated. Run time about 3 s." . "Pre-registered: across all enumerated or sampled constraint systems, every system that admits a valid word of length 4V + w (the operational test of satisfiability, independent of any cycle argument) also admits a periodic arrangement u^infinity with |u| <= V whose every cyclic window of length w is allowed; and every system with no such long valid word has no periodic arrangement either. Zero violations in both directions." . "false"^^ . . "exhaustive" . "" . . "Dataset: All 16 allowed-word sets, 2 colours, window 2 (generated by analysis.py)" . "exhaustive" . "" . . "Dataset: All 256 allowed-word sets, 2 colours, window 3 (generated by analysis.py)" . "exhaustive" . "" . . "Dataset: All 65536 allowed-word sets, 2 colours, window 4 (generated by analysis.py)" . "exhaustive" . "" . . "Dataset: All 512 allowed-word sets, 3 colours, window 2 (generated by analysis.py)" . "exhaustive" . "" . . "Dataset: All 65536 allowed-word sets, 4 colours, window 2 (generated by analysis.py)" . "sample" . "" . . "Dataset: 2000 allowed-word sets sampled uniformly from the 2^32 sets, 2 colours, window 5 (numpy default_rng seed 20261031; generated by analysis.py)" . "sample" . "" . . "Dataset: 5000 allowed-word sets sampled uniformly from the 2^27 sets, 3 colours, window 3 (same seed; generated by analysis.py)" . . "delphisci claim tester (claude-fable-5.1; cursor/claim-tests-wolfram-nks-0123)" . . . "claude-fable-5.1" . . . . . . . . . "2026-11-08T16:24:31.926Z"^^ . . . . . "Claim test of wolfram2002-ch05-15: supports, confidence 0.97" . . "RSA" . "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB" . "LPykeK/KjqRcRJfXnrzY7o1kmML6e9xf9Jo651mhaMsQE9c1ocovoJsekTFC71I+hVO5LKoZ08EbdB0DsNoRXHlJKHDA1rB4Fwsc0J2LbwF0ZcEP52aNkaQV3MLtJCbxhlqfFc+LNKcUr8p26SzoHHMPpQF4MIe+PxiDbGzWjZGDwjJ+Vbl+RT8GfP4h5uqNKUSEZHn1OZetVAgiG+5J40Mdy19XrFTc4iE1vs5cMwD5QubKY9ZdhMtTf16Zdb4aEpaH8uz3J+hafzWIq/0h7s7QCJqdH0ciGyLlfrg3QHmle/HSlU0c8ZNKq8f5AlO0+GEzjyf98RtbsjneZmnM0g==" . . .