@prefix this: . @prefix sub: . @prefix np: . @prefix rdf: . @prefix prov: . @prefix npx: . @prefix dc: . @prefix xsd: . sub:Head { this: a np:Nanopublication; np:hasAssertion sub:assertion; np:hasProvenance sub:provenance; np:hasPublicationInfo sub:pubinfo . } sub:assertion { a ; dc:description "analysis.py generates S1 (26 s with S2) and S2 and checks them against the OEIS b-files (0 mismatches in 100,001 and 10,000 terms). (A) For each sequence it computes the NIST serial increments del-psi^2_L for L = 1..12 from cyclic overlapping block counts and their chi-square p-values. (B) For each of the 88 elementary CA rules minimal under reflection and complement it runs 64 steps on the first 16,384 bits (cyclic boundary) of every sequence and of 100 PCG64 random inputs, and z-scores the mean black density and the zlib-9 size of the 64 x 16,384 output against the random inputs. Verdict thresholds and controls are those pre-registered."; "Measured: [object Object] chi-square p-values (A) and z-scores against 100 random inputs (B) (n = 1048576, deterministic sequences; z-scores use a 100-sample reference (sd estimated to about 7 %)). Interpretation: Wolfram's sequences (g) and (h) on p. 594-596 are the rule 30 centre column and the base-2 digits of pi; 'statistical analysis, even with some generalization' refers on those pages to block-frequency analysis and to running simple cellular automata on the sequence and examining the result. (A) is the standard overlapping-block frequency test (NIST serial increments), which covers block frequencies of all lengths up to 12 without double-counting. (B) generalizes by feeding each sequence as initial condition to every elementary CA (the 'simple cellular automata' class is fixed to the 88 inequivalent elementary rules, a pre-declared class as the assessors required) and checking two summary statistics of the output against random inputs. 'No detectable deviation' is bounded by these tests, these lengths, and this CA class; it is not a proof of randomness. A rule 30 centre-column NIST battery already exists (wolfram2002-ch02-05); this test is complementary and adds pi."; "Claim test of wolfram2002-ch10-15: supports, confidence 0.8"; ; prov:wasAttributedTo ; "The exact rule 30 centre column and binary digits of pi, generated by committed code and cross-checked against independent tabulations (OEIS); random references from a seeded standard PRNG; known non-random positive controls."; "claim-test/0.1.0"; "computational"; "A block-frequency p-value below 1e-6, 4 or more below 0.01, or an elementary CA whose output density or compressibility on either sequence is more than 6 standard deviations from its value on random input."; "All pre-registered pass conditions hold: no block-frequency p-value below 0.01 for either sequence (smallest 0.032, rule 30 at L = 7), and no elementary CA moves density or compressibility more than 3.45 standard deviations from its value on random input, while the PRNG negative control is clean (max |z| 2.81) and the Champernowne and Thue-Morse positive controls are flagged by 101 and 148 of the 176 CA statistics.", "Bounded non-detection only: 2^20 bits per sequence, block lengths <= 12, one window of 16,384 bits for the CA analysis, 64 steps, and the 88 elementary rules. Wolfram's statement is about what statistical analysis can 'readily recognize'; more powerful tests or longer samples are not covered, and randomness is not proved.", "The CA analysis is not blind to subtle structure: it detects the m-sequence LFSR (exploratory control) through the linear rules 60, 90, 105 and 150, even though the LFSR passes every block-frequency test with p > 0.998. For these linear rules zlib cannot compress random input at all (stored blocks, reference sd = 0), so any compressibility counts as an infinite z; rule 30 and pi showed none.", "The rule 30 block p-values are all between 0.03 and 0.26 (4 of 12 below 0.05), a mild clustering below 0.5 that the pre-registered rule does not count and that, given the strong correlation between successive del-psi^2 increments computed on the same sequence, is not by itself significant; ch02-05's NIST battery on 10^6 bits of the same sequence also found nothing."; "Both sequences give block-frequency p-values consistent with uniformity (none < 1e-4, at most 2 of 24 < 0.01) and elementary-CA output statistics within 5 standard deviations of those for random input, while the CA analysis flags the Champernowne and Thue-Morse sequences."; "Pre-registered. 'refutes' if any (A) p-value is below 1e-6, or 4 or more of the 24 are below 0.01, or any (B) |z| > 6 for S1 or S2. 'mixed' if neither the pass nor the refute condition holds. If the negative control fails, the verdict is 'inconclusive' (reason 'other'). If a positive control is not detected, the CA analysis is underpowered and the verdict can be at most 'mixed'. If a computed sequence disagrees with its OEIS b-file, the verdict is 'inconclusive' (reason 'other')."; "0.8"^^xsd:double; , , , , ; "supports"; "The first 2^20 bits of the rule 30 centre column and the first 2^20 bits of the binary expansion of pi show no detectable deviation from randomness under (A) overlapping block-frequency statistics for block lengths 1-12 and (B) analysis by all 88 inequivalent elementary cellular automata run on the sequence, while the same CA analysis does detect known non-random sequences."; "NIST serial-test increments del-psi^2_L for L = 1..12 with chi-square p-values; z-scores of CA output density and zlib size against 100 random inputs for all 88 inequivalent elementary rules; thresholds and controls as pre-registered."; "Pre-registered. Sequences: S1 = rule 30 centre column from a single black cell, steps 0 .. 2^20 - 1 (computed; must agree with the OEIS A051023 b-file); S2 = binary digits of pi after the binary point, first 2^20 (computed by Chudnovsky series; must agree with the OEIS A004601 b-file). (A) Block frequencies: for each L = 1..12 the NIST SP 800-22 serial increment del-psi^2_L = psi^2_L - psi^2_{L-1} (cyclic overlapping L-blocks), p-value from chi-square with 2^(L-1) df; 12 p-values per sequence, 24 in total. (B) CA analysis: for each of the 88 elementary rules that are minimal under left-right reflection and black-white complement, take the first 16384 bits as a cyclic initial condition, run 64 steps and record two statistics of steps 1-64: mean black density and zlib (level 9) size of the bit-packed 64 x 16384 pattern. Compare each with the same statistic for 100 random initial conditions (numpy PCG64, seed 20261031): z = (x - mean) / sd (if sd = 0, z = 0 when x equals the reference value and infinite otherwise). 176 z-scores per sequence. Validity controls: (i) a fresh PCG64 sequence of 2^20 bits must give max |z| < 5 and pass (A) under the same rule as below (negative control); (ii) the binary Champernowne sequence (concatenated binary integers 1, 2, 3, ...) and the Thue-Morse sequence must each give max |z| > 5 (positive controls, showing the CA analysis can detect non-randomness). 'supports' if, for both S1 and S2, no (A) p-value is below 1e-4, at most 2 of the 24 (A) p-values are below 0.01 (P(>= 3 of 24) ~ 0.2 % under the null), every (B) |z| <= 5, and the controls hold."; "false"^^xsd:boolean . . a ; dc:description "sequence S1 (Wolfram's sequence g)"; dc:license ""; "Dataset: Centre column of rule 30 from a single black cell, steps 0 .. 2^20 - 1 (generated by analysis.py, bit-packed light-cone evolution)"; prov:wasDerivedFrom . a ; dc:description "sequence S2 (Wolfram's sequence h)"; dc:license ""; "Dataset: First 2^20 binary digits of pi after the binary point (generated by analysis.py; Chudnovsky series, exact integers, 64 guard bits)" . a ; dc:description "independent tabulation validating S1"; dc:license "CC BY-SA 4.0 (OEIS)"; "Dataset: OEIS A051023, centre column of rule 30, b-file (100,001 terms)"; prov:atLocation ; prov:wasDerivedFrom ; "sha256:4d0c4a0272bc2b0fb5c1e91281d6dffc757d67cd6ffd8aa9f7486dac4685c1a9" . a ; dc:description "independent tabulation validating S2"; dc:license "CC BY-SA 4.0 (OEIS)"; "Dataset: OEIS A004601, expansion of pi in base 2, b-file (10,000 terms)"; prov:atLocation ; prov:wasDerivedFrom ; "sha256:504677b5536df5a24b4b2b47880562a52fc02b48f7d6fdde5cdabff9f5cb708d" . a ; dc:description "negative and positive controls and the random reference for the CA analysis"; dc:license ""; "Dataset: Control sequences of 2^20 bits: numpy PCG64 (seed 20261031), binary Champernowne, Thue-Morse, and LFSR a[n] = a[n-3] XOR a[n-20] (exploratory); 100 PCG64 random CA inputs of 16384 bits (generated by analysis.py)" . } sub:provenance { a prov:SoftwareAgent; "delphisci claim tester (claude-opus-5.5; cursor local subagent)"; prov:actedOnBehalfOf ; . a ; dc:identifier "claude-opus-5.5" . sub:assertion prov:wasAttributedTo ; prov:wasDerivedFrom ; prov:wasGeneratedBy sub:test . sub:plan prov:hadPlan . sub:test a prov:Activity; prov:qualifiedAssociation sub:plan; prov:wasAssociatedWith . } sub:pubinfo { this: dc:created "2026-11-08T16:24:31.935Z"^^xsd:dateTime; dc:creator ; dc:license ; npx:hasNanopubType ; npx:introduces ; "Claim test of wolfram2002-ch10-15: supports, confidence 0.8"; . sub:sig npx:hasAlgorithm "RSA"; npx:hasPublicKey "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB"; npx:hasSignature "ZSPN7syyatLe9T9dyo2danluBECWwwXf2lOvw4udrUwdHeZEqioiuV08mPjTAF2cjiDaAR9E/hfi4AmBpTYv3ZdQE4gIbGV7LpTmU4/8J2TwwMr48QRp8O3yz0OSZDmw/FMqSugFVjveyXM4gFAeuRuQv+M7rJ4tcgsl5bEiT+boLUmN9iuYWfOUGyT3nJX2h6oVmUYcKsRRXl9ioL32SLcOlQTiy9E4f6AbZk91oDgmEfsEksnHue1CuipULgfN9a3PmdoeeoFjiAZBdwKaA1whsi29QourcqGXTzDV8UOmf0J5tX31bTcLf4W8L6DWHxZiIogtRII9YrEdTCRbjA=="; npx:hasSignatureTarget this:; npx:signedBy . }