https://w3id.org/np/RAppC40zoPDSJE-f_pX4bVfECzS3NsZceJkqubLXWnfPE/Head https://w3id.org/np/RAppC40zoPDSJE-f_pX4bVfECzS3NsZceJkqubLXWnfPE http://www.nanopub.org/nschema#hasAssertion https://w3id.org/np/RAppC40zoPDSJE-f_pX4bVfECzS3NsZceJkqubLXWnfPE/assertion https://w3id.org/np/RAppC40zoPDSJE-f_pX4bVfECzS3NsZceJkqubLXWnfPE http://www.nanopub.org/nschema#hasProvenance https://w3id.org/np/RAppC40zoPDSJE-f_pX4bVfECzS3NsZceJkqubLXWnfPE/provenance https://w3id.org/np/RAppC40zoPDSJE-f_pX4bVfECzS3NsZceJkqubLXWnfPE http://www.nanopub.org/nschema#hasPublicationInfo https://w3id.org/np/RAppC40zoPDSJE-f_pX4bVfECzS3NsZceJkqubLXWnfPE/pubinfo https://w3id.org/np/RAppC40zoPDSJE-f_pX4bVfECzS3NsZceJkqubLXWnfPE http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.nanopub.org/nschema#Nanopublication https://w3id.org/np/RAppC40zoPDSJE-f_pX4bVfECzS3NsZceJkqubLXWnfPE/assertion https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test http://purl.org/dc/terms/description analysis.py sieves all primes below 10^8 (pi(10^8) = 5761455, matching OEIS A006880; first 10000 primes match the OEIS A000040 b-file), simulates Cramer's model on the same range with seed 20261031, and evaluates the seven pre-registered respects on both: coarse gap CDF versus exponential, consecutive-gap correlation, equidistribution over residues coprime to 12, Poisson fit of counts in intervals of length 18, Fourier spectrum of the prime indicator over 2^22 odd numbers below 10^8, lzma compressibility of the half-gap sequence relative to its shuffle, and the scaled fluctuation of pi(x) - Li(x). All seven thresholds were satisfied by the Cramer simulation, so all seven respects were calibrated. Runtime about 3 min. https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/Assessment https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test http://www.w3.org/2000/01/rdf-schema#comment Measured: 3 respects (of 7 calibrated) in which the primes below 10^8 are random-like at the pre-registered thresholds (n = 2760321, with 2.76 million primes in [5x10^7, 10^8] every reported deviation from the Cramer model is far outside sampling error (the Cramer simulation gives the sampling scale)). Interpretation: The claim's testable core is 'their distribution seems in many respects random'; the historical remark about what mathematics neglected is not tested. 'Random' is given the book's own operational meaning (p. 1067 ff.: statistical indistinguishability from a random model in specified respects) with Cramer's model as the declared random model; 'many respects' is read as a majority of a list fixed in advance, which includes respects where structure is known to exist (Fourier peaks from small moduli, finite-size Poisson deviations) so that the test can fail. Trivial sieve structure (no even primes) is removed only in R3 by restricting to classes coprime to 12; all other statistics are applied to the raw sequences. https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test http://www.w3.org/2000/01/rdf-schema#label Claim test of wolfram2002-ch02-17: refutes, confidence 0.55 https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test http://www.w3.org/2000/01/rdf-schema#seeAlso https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch02-17/result.json https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test http://www.w3.org/ns/prov#wasAttributedTo https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1 https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/admissibleDatasets Planned datasets: primes-sieve-and-cramer-generator, oeis-a000040-bfile, oeis-a006880-bfile https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/assessmentType claim-test/0.1.0 https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/dataKind computational https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/disconfirmingObservation Pre-registered. FAIL (refutes) if the primes are random-like in at most K - 4 of the K calibrated respects (3 of 7), or if the gap sequence is plainly regular (lzma ratio of the raw half-gap byte sequence below 0.1, or a single Fourier frequency carrying more than half the non-DC spectral power). MIXED for exactly K - 3 random-like respects (4 of 7). If the OEIS validation fails the generator is wrong and the result is 'inconclusive' (other). If the OEIS files cannot be fetched, validation falls back to the published pi(10^8) value only and this is reported. https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/evidenceAgainst Only the random-model comparison is tested; the claim's historical remark about mathematics neglecting this is outside the test. https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/evidenceAgainst Respects in which the primes are random-like: residue classes coprime to 12 are equidistributed to 6e-5 (Chebyshev bias invisible at this scale), consecutive-gap correlation is below 0.05 in magnitude, and pi(x) - Li(x) stays within 1.3 Cramer-model standard scales (in fact the primes fluctuate less than the random model: -754 at 10^8 versus a model scale of about 2300). https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/evidenceAgainst The fourth failure is large: the prime indicator has a Fourier peak at period 3 (and smaller ones at other small primes) 87600 times the mean power, versus 15.5 for the Cramer model. This is the periodic sieve structure the book itself acknowledges as a regularity. https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/evidenceAgainst The verdict follows the pre-registered rule (random-like in at most 3 of 7 calibrated respects). It refutes the claim as operationalised here; it does not show that primes are 'plainly regular' (the pre-registered regularity triggers did not fire: raw lzma ratio 0.55, largest Fourier peak carries 4.2% of the power). https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/evidenceAgainst Three of the four failures are small in magnitude and are the known arithmetic corrections to Cramer's model: the coarse gap CDF deviates from exponential by 0.036 (threshold 0.03; Hardy-Littlewood clustering of small gaps), counts in intervals of length 18 have Fano factor 0.76 instead of 1 (total variation from Poisson 0.078; Montgomery-Soundararajan variance deficit), and lzma compresses the gap sequence 3.8% better than its shuffle (threshold 2%; consecutive gaps are correlated, rho = -0.035, the Lemke Oliver-Soundararajan type bias). With thresholds of 0.05 for R1 and 0.95 for R6 the count would be 5 of 7 and the verdict 'supports'; the result is therefore threshold-sensitive and the pre-registered thresholds were strict. https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/expectedObservation Pre-registered. Generate all primes below N = 10^8 by sieve; validate against the OEIS A000040 b-file (first 10000 primes must match exactly) and the published value pi(10^8) = 5761455 (OEIS A006880). Simulate the Cramer model on the same range with numpy seed 20261031 and run every statistic on both. Seven respects, each with a threshold for 'random-like': R1 coarse gap distribution: for primes in [N/2, N], maximum over j of |F_emp(30 j) - (1 - exp(-30 j / mean gap))| < 0.03, where F_emp is the empirical CDF of gaps; R2 Pearson correlation of consecutive gaps in [N/2, N]: |rho| < 0.05; R3 equidistribution over the four residue classes coprime to 12: maximum deviation of a class share from 1/4 below 0.005 (the Cramer model is evaluated on its own coprime-to-12 events); R4 counts of primes in consecutive disjoint intervals of length H = round(ln N) in [N/2, N]: total variation distance between the empirical count distribution and Poisson with the same mean < 0.02; R5 discrete Fourier transform of the indicator of primes among the 2^22 odd numbers just below N: ratio of the largest non-zero-frequency power to the mean power < 30 (a Gumbel-scale maximum for random indicators is about ln(2^21) ~ 15); R6 sequential compressibility: lzma size of the sequence of half-gaps (clipped at 255) in [N/2, N] divided by the lzma size of a seeded random permutation of the same sequence >= 0.98; R7 maximum over x <= N of |pi(x) - Li(x)| / sqrt(x / ln x) < 3. A respect is calibrated only if the Cramer simulation satisfies its threshold; miscalibrated respects are dropped and reported. PASS (supports) if the primes are random-like in at least (K - 2) of the K calibrated respects (so 5 of 7 if all are calibrated). https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/failCriterion Pre-registered. FAIL (refutes) if the primes are random-like in at most K - 4 of the K calibrated respects (3 of 7), or if the gap sequence is plainly regular (lzma ratio of the raw half-gap byte sequence below 0.1, or a single Fourier frequency carrying more than half the non-DC spectral power). MIXED for exactly K - 3 random-like respects (4 of 7). If the OEIS validation fails the generator is wrong and the result is 'inconclusive' (other). If the OEIS files cannot be fetched, validation falls back to the published pi(10^8) value only and this is reported. https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/hasConfidence 0.55 https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/hasEvidence https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-1 https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/hasEvidence https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-2 https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/hasEvidence https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-3 https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/hasVerdict refutes https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/hypothesis If the claim holds, the primes below 10^8, generated by the sieve of Eratosthenes (the 'simple rule'), look random in most of a pre-declared list of statistical respects when compared with Cramer's random model (independent events with probability 1/ln n), while possibly showing structure in a minority of respects. https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/inferenceMethod analysis.py sieves all primes below 10^8 (pi(10^8) = 5761455, matching OEIS A006880; first 10000 primes match the OEIS A000040 b-file), simulates Cramer's model on the same range with seed 20261031, and evaluates the seven pre-registered respects on both: coarse gap CDF versus exponential, consecutive-gap correlation, equidistribution over residues coprime to 12, Poisson fit of counts in intervals of length 18, Fourier spectrum of the prime indicator over 2^22 odd numbers below 10^8, lzma compressibility of the half-gap sequence relative to its shuffle, and the scaled fluctuation of pi(x) - Li(x). All seven thresholds were satisfied by the Cramer simulation, so all seven respects were calibrated. Runtime about 3 min. https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/passCriterion Pre-registered. Generate all primes below N = 10^8 by sieve; validate against the OEIS A000040 b-file (first 10000 primes must match exactly) and the published value pi(10^8) = 5761455 (OEIS A006880). Simulate the Cramer model on the same range with numpy seed 20261031 and run every statistic on both. Seven respects, each with a threshold for 'random-like': R1 coarse gap distribution: for primes in [N/2, N], maximum over j of |F_emp(30 j) - (1 - exp(-30 j / mean gap))| < 0.03, where F_emp is the empirical CDF of gaps; R2 Pearson correlation of consecutive gaps in [N/2, N]: |rho| < 0.05; R3 equidistribution over the four residue classes coprime to 12: maximum deviation of a class share from 1/4 below 0.005 (the Cramer model is evaluated on its own coprime-to-12 events); R4 counts of primes in consecutive disjoint intervals of length H = round(ln N) in [N/2, N]: total variation distance between the empirical count distribution and Poisson with the same mean < 0.02; R5 discrete Fourier transform of the indicator of primes among the 2^22 odd numbers just below N: ratio of the largest non-zero-frequency power to the mean power < 30 (a Gumbel-scale maximum for random indicators is about ln(2^21) ~ 15); R6 sequential compressibility: lzma size of the sequence of half-gaps (clipped at 255) in [N/2, N] divided by the lzma size of a seeded random permutation of the same sequence >= 0.98; R7 maximum over x <= N of |pi(x) - Li(x)| / sqrt(x / ln x) < 3. A respect is calibrated only if the Cramer simulation satisfies its threshold; miscalibrated respects are dropped and reported. PASS (supports) if the primes are random-like in at least (K - 2) of the K calibrated respects (so 5 of 7 if all are calibrated). https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/ontology/preregistered false https://w3id.org/delphi/resource/claim/wolfram2002-ch02-17 https://w3id.org/delphi/ontology/hasAssessment https://w3id.org/delphi/resource/assessment/wolfram2002-ch02-17--claim-test https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-1 http://purl.org/dc/terms/description the sequence the claim is about and the declared random model https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-1 http://purl.org/dc/terms/license generated by this test https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-1 http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/EvidenceItem https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-1 http://www.w3.org/2000/01/rdf-schema#label Dataset: Primes below 10^8 from a sieve of Eratosthenes and a Cramer-model simulation on the same range, generated by analysis.py https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-1 http://www.w3.org/ns/prov#wasDerivedFrom https://www.wolframscience.com/nks/p48--why-these-discoveries-were-not-made-before/ https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-2 http://purl.org/dc/terms/description independent published list of the first primes to validate the sieve https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-2 http://purl.org/dc/terms/license CC BY-SA 4.0 (OEIS) https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-2 http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/EvidenceItem https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-2 http://www.w3.org/2000/01/rdf-schema#label Dataset: OEIS A000040, the prime numbers (b-file) https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-2 http://www.w3.org/ns/prov#atLocation https://oeis.org/A000040/b000040.txt https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-2 http://www.w3.org/ns/prov#wasDerivedFrom https://oeis.org/A000040 https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-2 https://w3id.org/delphi/ontology/hasChecksum sha256:ef3beecd3bca4a02ca5053e5a3afc3e0d69e43b49b9aaa91073fc4fb9e4d50de https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-3 http://purl.org/dc/terms/description published value of pi(10^8) to validate the sieve count https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-3 http://purl.org/dc/terms/license CC BY-SA 4.0 (OEIS) https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-3 http://www.w3.org/1999/02/22-rdf-syntax-ns#type https://w3id.org/delphi/ontology/EvidenceItem https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-3 http://www.w3.org/2000/01/rdf-schema#label Dataset: OEIS A006880, number of primes < 10^n (b-file) https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-3 http://www.w3.org/ns/prov#atLocation https://oeis.org/A006880/b006880.txt https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-3 http://www.w3.org/ns/prov#wasDerivedFrom https://oeis.org/A006880 https://w3id.org/delphi/resource/evidence/wolfram2002-ch02-17--claim-test--dataset-3 https://w3id.org/delphi/ontology/hasChecksum sha256:8111c06f2093b8cf138480a458945bf2eb1a01ee783d64041a7b11a102cd08a3 https://w3id.org/np/RAppC40zoPDSJE-f_pX4bVfECzS3NsZceJkqubLXWnfPE/provenance https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1 http://www.w3.org/1999/02/22-rdf-syntax-ns#type http://www.w3.org/ns/prov#SoftwareAgent https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1 http://www.w3.org/2000/01/rdf-schema#label delphisci claim tester (claude-fable-5.1; 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