. . . . "analysis.py generates the two LFSR sequences over one full period (periods verified maximal: 2047 and 1,048,575), the first 2^20 bits of the binary Champernowne word (checked bit for bit against the 10,000 terms of OEIS A030190 after accounting for its leading n = 0 term) and seeded random references. For each sequence it counts all overlapping blocks of lengths 1..8 (1..r-1 for the LFSRs), reports the maximum absolute and relative deviation from uniform, the chi-square spread of block counts against the empirical distribution over 200 random sequences of the same length, the Berlekamp-Massey linear complexity of the first 4000 bits, and the lzma compression ratio of the packed bits. For Champernowne the maximum relative deviation is also computed on the first N/4 and N/2 bits. Run time about 17 s." . . "Measured: 0 chi-square of length-8 block counts of the 20-bit LFSR over its full period (random reference median 250, 1st percentile 173) (n = 1048575, exact for the sequences; random-reference percentiles from 200 draws). Interpretation: The claim is the negation of 'uniform block frequencies for all lengths is a universal test for randomness', argued by two explicit counterexamples on NKS p. 594: sequence (e), 'generated by a linear feedback shift register (essentially an additive cellular automaton) with tap positions {2, 11}', and sequence (f), 'formed by concatenating base 2 digits of successive integers'. The LFSR is implemented as a[n] = a[n-2] XOR a[n-11] from a register containing a single 1 (polynomial x^11 + x^2 + 1), and a longer one with a[n] = a[n-3] XOR a[n-20]; maximal period is verified by computation rather than assumed. 'Accurate block frequencies' is given two operational readings: exact (counts within 1 of uniform, which an m-sequence achieves by construction for L <= r - 1) and coarse (every block frequency within a factor 1.5 of uniform and converging with N). 'Non-random' is operationalised by linear complexity (LFSR), by lzma compressibility (Champernowne) and by the block-count spread that Wolfram names on p. 595, measured as the chi-square statistic of block counts compared with the empirical distribution over random sequences of the same length (block counts of overlapping windows are dependent, so no analytic chi-square p-value is used)." . "Claim test of wolfram2002-ch10-14: mixed, confidence 0.8" . . . "Planned datasets: lfsr-taps-2-11-full-period, lfsr-taps-3-20-full-period, binary-champernowne-2pow20, oeis-a030190-bfile, random-reference-pcg64" . "claim-test/0.1.0" . "computational" . "Pre-registered: the LFSR block counts deviate by more than 1 from uniform for some L <= r - 1, or its chi-square spread is not below the 1st percentile of random sequences for some L in 4..8, or its linear complexity is not r; for Champernowne, a block frequency deviation that does not decrease with N or exceeds 0.5 at L <= 8, or a compression ratio above 0.9 (no regularity detected by compression)." . "Part (1) holds in full: both maximal-length LFSR sequences have every block of length up to r-1 within one count of exact uniformity, linear complexity 11 and 20 (random reference about 2000 for 4000 bits), and block-count chi-square essentially zero for all lengths 4-8, below every one of 200 random references; this is exactly Wolfram's 'abnormally small spread', and it alone establishes that uniform block frequencies do not certify randomness. lzma does not detect the LFSR regularity (ratio 1.0), so the 'other test' has to be linear complexity or spread." . "Part (2) fails as pre-registered: the Champernowne sequence's block-frequency deviations at N/4, N/2, N do not decrease monotonically for L <= 6 (the known 1/log N bias oscillates with the bit-length of the integers being concatenated), its chi-square spread is abnormally LARGE rather than small (far above all random references at every L), its compression ratio 0.78 falls between the pre-registered pass (<= 0.5) and fail (> 0.9) thresholds, and Berlekamp-Massey sees nothing. Its block frequencies are only coarsely uniform (every block within a factor 1.46 of uniform at L <= 8; 52.5% ones). At any finite length a chi-square frequency test rejects it, so Wolfram's sequence (f) is a counterexample only in the limiting (normal-number) sense, not in the finite-sample sense his picture suggests." . "The verdict 'mixed' follows the pre-registered rule: the core proposition (block frequencies are not a universal test) and the LFSR example with its small spread are confirmed; the characterisation of the concatenated-integer sequence as having accurate block frequencies with a small spread is not." . "Pre-registered: (1) LFSR (taps {2,11} as in NKS p. 594, period 2047, and a 20-bit register with period 2^20 - 1): over one full period every block of length L <= r - 1 (r = register length) occurs a number of times within 1 of the uniform expectation (N - L + 1)/2^L; Berlekamp-Massey linear complexity equals r (a random sequence of the same length has about N/2); and the chi-square statistic of its block counts for each L in 4..8 lies below the 1st percentile of 200 random sequences of the same length (the 'abnormally small spread'). (2) Binary Champernowne sequence, N = 2^20 bits: for every L <= 8 the maximum relative deviation of any block frequency from 2^-L decreases from the first N/4 bits to the first N/2 bits to all N bits and is <= 0.5 at N (coarse accuracy: every block within a factor 1.5 of uniform); its lzma compression ratio is <= 0.5 while that of the random reference is >= 0.95 (detectable regularity); and its bits agree with OEIS A030190 for every term in the b-file. Verdict 'supports' if (1) and (2) both hold; 'mixed' if (1) holds but (2) fails; 'refutes' if (1) fails." . "Pre-registered: the LFSR block counts deviate by more than 1 from uniform for some L <= r - 1, or its chi-square spread is not below the 1st percentile of random sequences for some L in 4..8, or its linear complexity is not r; for Champernowne, a block frequency deviation that does not decrease with N or exceeds 0.5 at L <= 8, or a compression ratio above 0.9 (no regularity detected by compression)." . "0.8"^^ . . . . . . "mixed" . "Wolfram's two example sequences, a maximal-length linear feedback shift register output and the concatenated binary digits of successive integers, have block frequencies that are uniform (the LFSR exactly so over its period, the concatenation increasingly so with length) yet are detectably non-random by other tests, in particular an abnormally small spread of block counts for the LFSR, so uniform block frequencies cannot be a universal randomness test." . "analysis.py generates the two LFSR sequences over one full period (periods verified maximal: 2047 and 1,048,575), the first 2^20 bits of the binary Champernowne word (checked bit for bit against the 10,000 terms of OEIS A030190 after accounting for its leading n = 0 term) and seeded random references. For each sequence it counts all overlapping blocks of lengths 1..8 (1..r-1 for the LFSRs), reports the maximum absolute and relative deviation from uniform, the chi-square spread of block counts against the empirical distribution over 200 random sequences of the same length, the Berlekamp-Massey linear complexity of the first 4000 bits, and the lzma compression ratio of the packed bits. For Champernowne the maximum relative deviation is also computed on the first N/4 and N/2 bits. Run time about 17 s." . "Pre-registered: (1) LFSR (taps {2,11} as in NKS p. 594, period 2047, and a 20-bit register with period 2^20 - 1): over one full period every block of length L <= r - 1 (r = register length) occurs a number of times within 1 of the uniform expectation (N - L + 1)/2^L; Berlekamp-Massey linear complexity equals r (a random sequence of the same length has about N/2); and the chi-square statistic of its block counts for each L in 4..8 lies below the 1st percentile of 200 random sequences of the same length (the 'abnormally small spread'). (2) Binary Champernowne sequence, N = 2^20 bits: for every L <= 8 the maximum relative deviation of any block frequency from 2^-L decreases from the first N/4 bits to the first N/2 bits to all N bits and is <= 0.5 at N (coarse accuracy: every block within a factor 1.5 of uniform); its lzma compression ratio is <= 0.5 while that of the random reference is >= 0.95 (detectable regularity); and its bits agree with OEIS A030190 for every term in the b-file. Verdict 'supports' if (1) and (2) both hold; 'mixed' if (1) holds but (2) fails; 'refutes' if (1) fails." . "false"^^ . . "Wolfram's sequence (e)" . "" . . "Dataset: LFSR a[n] = a[n-2] XOR a[n-11], one full period (generated by analysis.py)" . "longer m-sequence for block lengths up to 12" . "" . . "Dataset: LFSR a[n] = a[n-3] XOR a[n-20], one full period (generated by analysis.py)" . "Wolfram's sequence (f)" . "" . . "Dataset: Concatenated binary digits of 1, 2, 3, ..., first 2^20 bits (generated by analysis.py)" . "independent tabulation of sequence (f) for cross-checking the generator" . "CC BY-SA 4.0 (OEIS)" . . "Dataset: OEIS A030190: binary Champernowne sequence, b-file" . . . "sha256:b5d6fbe62cc958bd03bd039450354c72d3bb2695e06772fc3900be0613e11f5f" . "reference distribution of block-count spread and of compression ratio" . "" . . "Dataset: Random reference bit sequences from numpy PCG64, seed 20261031 (generated by analysis.py)" . . "delphisci claim tester (claude-fable-5.1; cursor/claim-tests-wolfram-nks-0123)" . . . "claude-fable-5.1" . . . . . . . . . "2026-11-08T16:24:31.935Z"^^ . . . . . "Claim test of wolfram2002-ch10-14: mixed, confidence 0.8" . . "RSA" . "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB" . "KgPQYE1+XwgrA+0wlZ4T9U3lCwCYhAMvYVxQGoBzC7HEsICI7+IsUfOpdrg0A0ZPIrj4Ay1Ll/7VHGWxXzhSCfJ9nocJp54CI5HK2A+qTl/wrIzucNFg1xRTZlQku46bIa6df/JAtGZdleJC69yi+fMqLCYBP1/iOo8RjfYk3DdPmhoGOb65ukGoWCulWtNUn1v1iL840XB9pxXzvY195+HtvZ2ZRjFne2DlTG36wTrPam4q+OS7d+f0mNAsoLivMgYlpImM9u0DCqWq/43YCPGv6P8XL297kVCs9xioi44Ehae1Q9F0VvM4oilYXKMSdX1j3+KSh6YrnsC1neLmGA==" . . .