@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 iterates x -> m x from 1 for 3000 steps for each of the 64 (m, b) pairs using Python integers, converts each power to base-b digits, keeps the interior 25%-75% of digit positions per row, concatenates (one byte per digit) and lzma-compresses; it also computes digit-value frequencies in the window. Per-pair results are in derived/pairs.csv. Runtime about 4 minutes."; "Measured: 43 pairs (m,b) with m not a power of b whose interior digit array is 'complex' by the pre-registered criteria (n = 53). Interpretation: Wolfram's 'even rather simple systems based on numbers can lead to highly complex behavior' (p. 115) is illustrated by the digit sequences of successive powers (p. 116-120: 2^t in base 2 is trivial, 3^t in base 2 looks random). 'Highly complex behavior' is read as digit arrays that a general-purpose compressor cannot reduce below 80% of their entropy bound and whose digit frequencies are uniform to within 2%, in explicit contrast to the trivial members of the same family that compress to below 10%. The claim is existential; the family is the smallest natural one containing the book's opening examples. The window excludes the leading digits (biased by Benford-type effects) and trailing digits (eventually periodic in t at fixed position, and all zero when b divides m)."; "Claim test of wolfram2002-ch04-02: supports, confidence 0.9"; ; prov:wasAttributedTo ; "Planned datasets: nks-powers-digit-arrays"; "claim-test/0.1.0"; "computational"; "Pre-registered. The claim FAILS if no pair with m not a power of b is 'complex' (every such digit array compresses below 0.8 log2(b) bits per digit or has digit frequencies deviating from uniform by more than 0.02). If any trivial pair (m a power of b) is not 'simple', the measure lacks discriminating power and the verdict is inconclusive (not-operationalizable)."; "Incompressibility by lzma over about 3.6 million interior digits (for 3^t in base 2: 1.086 bits/digit, digit frequencies within 0.0004 of 1/2) is evidence against periodic or nested structure at these scales, not a proof of normality; the digits of 3^t in base 2 are not proven to be normal.", "The 10 non-complex non-trivial pairs split into two explicable groups: (2,4), (2,8), (4,8), (8,4), (3,9) where m and b are powers of a common integer (the pre-registered 'm is a power of b' control missed this symmetric case; they compress to < 0.001 of the bound, i.e. are trivially regular), and the five m = 6 pairs in bases 2, 3, 4, 8, 9 where the trailing zeros of 6^t = 2^t 3^t occupy 1/log_b(6) of each row and intrude into the fixed 25%-75% window, lowering compression ratios to 0.30-0.83 and biasing digit frequencies.", "The test covers the book's opening family (powers in positional notation) only; the claim's other illustrations in Chapter 4 (e.g. the 3n/2 system, iterated maps) were not tested."; "Pre-registered. Family: all pairs (m, b) with multiplier m in 2..9 and base b in 2..9, iterating x_{t+1} = m x_t from x_0 = 1 for t = 1..T = 3000 (i.e. the digits of m^t). For each pair build the digit array of rows m^t written in base b; take the interior window of each row consisting of digit positions between 25% and 75% of that row's length (least significant digit = position 0), concatenate all interior digits in row order into one byte string (one digit per byte). Measure (i) bits per digit = 8 * lzma(preset 9|EXTREME) compressed size / number of digits; (ii) maximum absolute deviation of any digit-value frequency from 1/b. A pair is 'complex' if (i) >= 0.8 * log2(b) and (ii) <= 0.02; a pair is 'simple' if (i) < 0.1 * log2(b). The claim PASSES if at least one pair with m not a power of b is 'complex' AND every pair with m a power of b (m = b^j, j >= 1, e.g. (2,2), (4,2), (8,2), (9,3), (3,3), ...) is 'simple' (validity check that the measure separates trivially regular digit arrays). Reported, not decisive: how many of the 64 pairs are complex, and whether the pairs with m and b sharing all prime factors (e.g. (6,2), (4,8), (2,4)) behave differently."; "Pre-registered. The claim FAILS if no pair with m not a power of b is 'complex' (every such digit array compresses below 0.8 log2(b) bits per digit or has digit frequencies deviating from uniform by more than 0.02). If any trivial pair (m a power of b) is not 'simple', the measure lacks discriminating power and the verdict is inconclusive (not-operationalizable)."; "0.9"^^xsd:double; ; "supports"; "Within the family of the simplest arithmetic iterations, repeated multiplication by a fixed small integer m viewed as a base-b digit array (Wolfram's opening example, pp. 116-120), some members produce digit arrays whose interior is incompressible with digit frequencies indistinguishable from uniform, while the trivial members (m a power of b) are detected as highly compressible."; "analysis.py iterates x -> m x from 1 for 3000 steps for each of the 64 (m, b) pairs using Python integers, converts each power to base-b digits, keeps the interior 25%-75% of digit positions per row, concatenates (one byte per digit) and lzma-compresses; it also computes digit-value frequencies in the window. Per-pair results are in derived/pairs.csv. Runtime about 4 minutes."; "Pre-registered. Family: all pairs (m, b) with multiplier m in 2..9 and base b in 2..9, iterating x_{t+1} = m x_t from x_0 = 1 for t = 1..T = 3000 (i.e. the digits of m^t). For each pair build the digit array of rows m^t written in base b; take the interior window of each row consisting of digit positions between 25% and 75% of that row's length (least significant digit = position 0), concatenate all interior digits in row order into one byte string (one digit per byte). Measure (i) bits per digit = 8 * lzma(preset 9|EXTREME) compressed size / number of digits; (ii) maximum absolute deviation of any digit-value frequency from 1/b. A pair is 'complex' if (i) >= 0.8 * log2(b) and (ii) <= 0.02; a pair is 'simple' if (i) < 0.1 * log2(b). The claim PASSES if at least one pair with m not a power of b is 'complex' AND every pair with m a power of b (m = b^j, j >= 1, e.g. (2,2), (4,2), (8,2), (9,3), (3,3), ...) is 'simple' (validity check that the measure separates trivially regular digit arrays). Reported, not decisive: how many of the 64 pairs are complex, and whether the pairs with m and b sharing all prime factors (e.g. (6,2), (4,8), (2,4)) behave differently."; "false"^^xsd:boolean . . a ; dc:description "program family under test"; dc:license "generated data"; "Dataset: Digit arrays of m^t in base b for m, b in 2..9, t = 1..3000 (generated by analysis.py)"; prov:wasDerivedFrom . } sub:provenance { a prov:SoftwareAgent; "delphisci claim tester (claude-fable-5.1; cursor/wolfram-nks-claim-tests-106e)"; prov:actedOnBehalfOf ; . a ; dc:identifier "claude-fable-5.1" . 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 "2026-11-08T16:24:31.921Z"^^xsd:dateTime; dc:creator ; dc:license ; npx:hasNanopubType ; npx:introduces ; "Claim test of wolfram2002-ch04-02: supports, confidence 0.9"; . sub:sig npx:hasAlgorithm "RSA"; npx:hasPublicKey "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB"; npx:hasSignature "vvghdkKrT/B/AZgida2bCbGxx8E3Xcqgm4X2JC11HkWrDaUvcJ/4MywADbUYeJIg9MUQTq1wWQJmbeO5vuCv2IdgVaXkSydOhcxS+NoLQnAiANz/GyX89N99wlwxOKw+vx3Vm58CF5PfTtJUIqHjcYllTbRT3aXU3LkeGKIGESjnVxecIY5yWY0SR+YhwGFXZcAtXhoyvVG3N0v+G651fVACRwkpFIzc3tuiWcmGqcYM5CTrmSQoRj5PlA/dxRDFFsnWFhEhFwvmCGcXR+qLBIsjwSgeUqMMODeJvwYTg+U2rT+B6E+rnRbuOUTTURNHGAigJ0MKxzq/NUe/Wfd4sg=="; npx:hasSignatureTarget this:; npx:signedBy . }