@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 computes each of the six recurrences for n <= 2^20 with memoised integer arithmetic, verifies rules (e) and (c) term-by-term against the 10,000-term OEIS b-files of A005185 and A004001 (all terms match), detrends by n/2 (rules c-g) or by a log-log power-law fit for n >= 16 (rule h), lzma-compresses the increment-sign sequence of the residual, and measures sign balance and the dyadic correlation corr(r[n], r[2n])."; "Measured: 4 rules among (e)-(h) that are complex/random-like by the pre-registered criteria (n = 4). Interpretation: 'Fluctuations ... complex and in many respects random' (p. 130-131) is read as: the increment-sign sequence of the detrended fluctuations is not compressible to a small fraction of its entropy bound by a general-purpose compressor (nested/digit-based fluctuations compress to < 0.01 bits/symbol), and positive and negative fluctuations are balanced (Wolfram's own stated 'respect' for case (f)). The claim is existential ('can produce'), so one qualifying rule suffices; the book's nested rules (c) and (d) serve as internal controls that must be detected as regular. The OEIS b-files provide independent public values of two of the sequences to verify the implementation. Trend n/2 for (c)-(g) follows the book's plots; the power-law trend for (h) follows the book's caption (0.42 n^0.916)."; "Claim test of wolfram2002-ch04-08: supports, confidence 0.9"; ; prov:wasAttributedTo ; "Planned datasets: oeis-a005185-bfile, oeis-a004001-bfile, nks-recursive-sequences-p129"; "claim-test/0.1.0"; "computational"; "Pre-registered. The claim FAILS if none of rules (e)-(h) is complex/random-like (every one is undefined before N, or compresses below 0.5 bits/symbol, or is sign-unbalanced by more than 0.05). If the nested controls (c) or (d) do not compress below 0.15 bits/symbol the measure lacks discriminating power and the verdict is inconclusive (not-operationalizable). If computed terms disagree with the OEIS b-files for A005185 or A004001, the verdict is inconclusive (other) until the implementation is fixed."; "For rule (h) the power-law trend fitted over n in [16, 2^20] is 0.338 n^0.837, not the book's 0.42 n^0.916 fitted over n <= 2048; residual statistics for (h) depend mildly on the trend choice.", "Rules (g) and (h) have dyadic correlation corr(r[n], r[2n]) of 0.44 and 0.41 (vs 1.00 for nested rule c and ~0 for e, f), so part of their fluctuation is self-similar at scale 2; they still compress at > 0.97 bits/symbol, so the sign sequence itself carries no detectable regularity.", "Statistical incompressibility over 2^20 terms is evidence of complexity, not proof that no closed form exists; the test shows the fluctuations are not periodic or nested at the scales examined.", "The downloaded b-files (about 97 KB each) are not committed; analysis.py re-fetches them and the SHA-256 values above pin what was used.", "The sign-balance 'respect' is Wolfram's own stated one for case (f); other respects of randomness (e.g. spectral flatness, distribution of increments) were not pre-registered and not tested."; "Pre-registered. Rules (all with f[1]=f[2]=1, from NKS pp. 129-130): (c) f[n]=f[f[n-1]]+f[n-f[n-1]] (OEIS A004001); (d) f[n]=f[n-f[n-1]]+f[n-f[n-2]-1]; (e) f[n]=f[n-f[n-1]]+f[n-f[n-2]] (OEIS A005185); (f) f[n]=f[n-f[n-1]-1]+f[n-f[n-2]-1]; (g) f[n]=f[f[n-1]]+f[n-f[n-2]-1]; (h) f[n]=f[f[n-1]]+f[n-2f[n-1]+1]. Compute f[n] for n <= N = 2^20 (stop a rule if an index < 1 occurs). Residual r[n] = f[n] - n/2 for (c)-(g); for (h) r[n] = f[n] - a n^b with (a,b) a least-squares power-law fit in log-log. Statistics: (i) bits per symbol = 8 * lzma(preset 9|EXTREME) compressed size of the increment-sign sequence sign(r[n]-r[n-1]) in {-1,0,1} stored as one byte each, divided by N-1; (ii) sign balance |P(increment>0) - P(increment<0)|; (iii) dyadic self-similarity corr(r[n], r[2n]) for n in [N/4, N/2) (reported). A rule is 'complex/random-like' if it is well defined up to N, (i) >= 0.5 bits/symbol and (ii) <= 0.05. The claim PASSES if at least one of rules (e)-(h) is complex/random-like AND the nested controls (c) and (d) both have (i) < 0.15 bits/symbol (validity of the measure). Calibration from controls outside the family: binary digit sum s2(n) gives 0.002 bits/symbol, its partial sums 0.005, a symmetric random walk 1.10."; "Pre-registered. The claim FAILS if none of rules (e)-(h) is complex/random-like (every one is undefined before N, or compresses below 0.5 bits/symbol, or is sign-unbalanced by more than 0.05). If the nested controls (c) or (d) do not compress below 0.15 bits/symbol the measure lacks discriminating power and the verdict is inconclusive (not-operationalizable). If computed terms disagree with the OEIS b-files for A005185 or A004001, the verdict is inconclusive (other) until the implementation is fixed."; "0.9"^^xsd:double; , , ; "supports"; "At least one of Wolfram's nested recurrences using only addition and subtraction, f[n] = f[p[n]] + f[q[n]] with variable look-back such as n - f[n-1], produces fluctuations about its trend that are incompressible (not periodic or nested) and sign-balanced, while his nested examples (c) and (d) are detected as highly compressible."; "analysis.py computes each of the six recurrences for n <= 2^20 with memoised integer arithmetic, verifies rules (e) and (c) term-by-term against the 10,000-term OEIS b-files of A005185 and A004001 (all terms match), detrends by n/2 (rules c-g) or by a log-log power-law fit for n >= 16 (rule h), lzma-compresses the increment-sign sequence of the residual, and measures sign balance and the dyadic correlation corr(r[n], r[2n])."; "Pre-registered. Rules (all with f[1]=f[2]=1, from NKS pp. 129-130): (c) f[n]=f[f[n-1]]+f[n-f[n-1]] (OEIS A004001); (d) f[n]=f[n-f[n-1]]+f[n-f[n-2]-1]; (e) f[n]=f[n-f[n-1]]+f[n-f[n-2]] (OEIS A005185); (f) f[n]=f[n-f[n-1]-1]+f[n-f[n-2]-1]; (g) f[n]=f[f[n-1]]+f[n-f[n-2]-1]; (h) f[n]=f[f[n-1]]+f[n-2f[n-1]+1]. Compute f[n] for n <= N = 2^20 (stop a rule if an index < 1 occurs). Residual r[n] = f[n] - n/2 for (c)-(g); for (h) r[n] = f[n] - a n^b with (a,b) a least-squares power-law fit in log-log. Statistics: (i) bits per symbol = 8 * lzma(preset 9|EXTREME) compressed size of the increment-sign sequence sign(r[n]-r[n-1]) in {-1,0,1} stored as one byte each, divided by N-1; (ii) sign balance |P(increment>0) - P(increment<0)|; (iii) dyadic self-similarity corr(r[n], r[2n]) for n in [N/4, N/2) (reported). A rule is 'complex/random-like' if it is well defined up to N, (i) >= 0.5 bits/symbol and (ii) <= 0.05. The claim PASSES if at least one of rules (e)-(h) is complex/random-like AND the nested controls (c) and (d) both have (i) < 0.15 bits/symbol (validity of the measure). Calibration from controls outside the family: binary digit sum s2(n) gives 0.002 bits/symbol, its partial sums 0.005, a symmetric random walk 1.10."; "false"^^xsd:boolean . . a ; dc:description "independent published terms of rule (e) to validate the implementation"; dc:license "CC BY-SA 4.0 (OEIS)"; "Dataset: OEIS A005185 (Hofstadter Q-sequence) b-file"; prov:atLocation ; prov:wasDerivedFrom ; "sha256:123200bbe8c66a132f7a02ede01831b7e5bd69a35efd41bb39e9a917b6e5dde9" . a ; dc:description "independent published terms of rule (c) to validate the implementation"; dc:license "CC BY-SA 4.0 (OEIS)"; "Dataset: OEIS A004001 (Hofstadter-Conway $10000 sequence) b-file"; prov:atLocation ; prov:wasDerivedFrom ; "sha256:68bb68831fff5fb3baa056b60cb50a7800260fa79ef9902ef915b171c081154e" . a ; dc:description "program family under test"; dc:license "generated data"; "Dataset: Six nested recurrences from NKS pp. 129-130 (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 "2026-11-08T16:24:31.922Z"^^xsd:dateTime; dc:creator ; dc:license ; npx:hasNanopubType ; npx:introduces ; "Claim test of wolfram2002-ch04-08: supports, confidence 0.9"; . sub:sig npx:hasAlgorithm "RSA"; npx:hasPublicKey "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB"; npx:hasSignature "rUDa+DYYgVjhanZmdTMsinIQcJU4o6VxZhQw7mPY9ZEZsiw/9DdHtQKhYoXKNBy/hZ9oeRmt4KlBRIqpKyPklbFwxYYAW6RphCRQadOT8N7dgDzr0SEFOO7Nsz2HF7Y7n+03n4WvlqOdrxzP27W9eLreQwY9tn1aVy9D5LGO6rYM6WqOBJGHTAxbvyyBakQKIa5AI9yfV5quOKP38bSCGC9pjh1EWdAI45O8WRUABMCbV+roI7G/1AK3TzobgvSTLYHCg+aTnm3/N84Fo6LQtgRl7v4Dlo5esYsvyahLJpq1W1DJGyCargMj4Hbbk/Mno/wKk08uZKT41XlhiCjGIQ=="; npx:hasSignatureTarget this:; npx:signedBy . }