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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)." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch04-08: supports, confidence 0.9" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch04-08/result.json" } ], "http://www.w3.org/ns/prov#wasAttributedTo": [ { "@id": "https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1" } ], "https://w3id.org/delphi/ontology/admissibleDatasets": [ { "@value": "Planned datasets: oeis-a005185-bfile, oeis-a004001-bfile, nks-recursive-sequences-p129" } ], "https://w3id.org/delphi/ontology/assessmentType": [ { "@value": "claim-test/0.1.0" } ], "https://w3id.org/delphi/ontology/dataKind": [ { "@value": "computational" } ], "https://w3id.org/delphi/ontology/disconfirmingObservation": [ { "@value": "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." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "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." }, { "@value": "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." }, { "@value": "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." }, { "@value": "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." }, { "@value": "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." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "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." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "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." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.9", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-08--claim-test--dataset-1" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-08--claim-test--dataset-2" }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-08--claim-test--dataset-3" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "supports" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "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." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "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])." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "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." } ], "https://w3id.org/delphi/ontology/preregistered": [ { "@value": "false", "@type": "http://www.w3.org/2001/XMLSchema#boolean" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-08--claim-test--dataset-1", "http://purl.org/dc/terms/description": [ { "@value": "independent published terms of rule (e) to validate the implementation" } ], "http://purl.org/dc/terms/license": [ { "@value": "CC BY-SA 4.0 (OEIS)" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: OEIS A005185 (Hofstadter Q-sequence) b-file" } ], "http://www.w3.org/ns/prov#atLocation": [ { "@id": "https://oeis.org/A005185/b005185.txt" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://oeis.org/A005185" } ], "https://w3id.org/delphi/ontology/hasChecksum": [ { "@value": "sha256:123200bbe8c66a132f7a02ede01831b7e5bd69a35efd41bb39e9a917b6e5dde9" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-08--claim-test--dataset-2", "http://purl.org/dc/terms/description": [ { "@value": "independent published terms of rule (c) to validate the implementation" } ], "http://purl.org/dc/terms/license": [ { "@value": "CC BY-SA 4.0 (OEIS)" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: OEIS A004001 (Hofstadter-Conway $10000 sequence) b-file" } ], "http://www.w3.org/ns/prov#atLocation": [ { "@id": "https://oeis.org/A004001/b004001.txt" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://oeis.org/A004001" } ], "https://w3id.org/delphi/ontology/hasChecksum": [ { "@value": "sha256:68bb68831fff5fb3baa056b60cb50a7800260fa79ef9902ef915b171c081154e" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch04-08--claim-test--dataset-3", "http://purl.org/dc/terms/description": [ { "@value": "program family under test" } ], "http://purl.org/dc/terms/license": [ { "@value": "generated data" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: Six nested recurrences from NKS pp. 129-130 (generated by analysis.py)" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://www.wolframscience.com/nks/p130--recursive-sequences/" } ] }, { "@id": "https://w3id.org/delphi/resource/claim/wolfram2002-ch04-08", "https://w3id.org/delphi/ontology/hasAssessment": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch04-08--claim-test" } ] } ] } ]