@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 sieves the integers through 400000, takes the first 16384 gaps from the prime 3, and computes the Shannon entropy of overlapping 8-bit blocks of (gap/2) mod 2. The same entropy is computed for a constant sequence of ones and for an alternating sequence. Every number in measured is written to derived/measured.json by that script."; "Measured: 7.788983 bits of overlapping 8-bit block entropy of half-gap parity (n = 16384, plugin estimator on 16377 overlapping windows; no sampling interval). Interpretation: Wolfram's remark that the primes obtained from the sieve look remarkably random is read as a statement about the gap sequence, not about the primes being odd (that parity is constant after 2). The raw gap parity after 2 is identically even, so it cannot witness randomness; the parity of the half-gap is the pre-registered randomness statistic. The repetitive controls are synthetic gap parities (all half-gaps odd; half-gaps alternating in parity), not other number sequences. 'Remarkably random' is operationalized only as high block entropy relative to those controls."; "Claim test of wolfram2002-ch04-09: supports, confidence 0.74"; ; prov:wasAttributedTo ; "The bit string must be generated by the sieve of Eratosthenes at the locked limit 400000, using only gaps from the prime 3 onward. No external prime table is consulted. The computation is not evidence about organisms."; "claim-test/0.1.0"; "computational"; "The same entropy is below 5.0 bits even though the two repetitive controls sit at or below 1.01. An entropy between 5.0 and 7.2 is mixed, not a confirmation."; "A match shows only that this computation behaves as claimed for this statistic. It says nothing about organisms.", "Every gap in the sample is even (odd_gaps_in_sample = 0). Raw gap parity is therefore a constant and was not used as the randomness statistic.", "Finite check: 16384 gaps, the largest prime used being 180533, from a sieve limited to 400000. A different bound or a different statistic can disagree.", "The ones-frequency of the half-gap parity is 0.586365 (9607 of 16384), so the sequence is biased away from a fair coin. The pre-registered statistic was block entropy, which still cleared 7.2 (measured 7.788983, all 256 blocks seen). Unbiasedness was not the pass criterion."; "Overlapping 8-bit block entropy of the first 16384 half-gap parities from the prime 3 is at least 7.2 bits, while a constant control is 0 and an alternating control is at most 1.01."; "Pre-registered. If both controls pass and H8 < 5.0, the verdict is refutes: the half-gap parity sequence is not remarkably random on this statistic. If both controls pass and 5.0 <= H8 < 7.2, the verdict is mixed. A match is only about this finite sieve and this statistic. It does not say that the primes are random in every respect, and it says nothing about organisms."; "0.74"^^xsd:double; ; "supports"; "The parity of successive half-gaps of the primes (gap/2 mod 2, gaps from the prime 3 onward) has overlapping 8-bit block entropy close to the 8-bit maximum, while a constant half-gap sequence and an alternating half-gap sequence stay at or below 1 bit."; "Shannon entropy of overlapping 8-bit blocks, compared with the two locked thresholds and with the two locked repetitive controls. The frequency of ones is reported and does not decide the verdict."; "Pre-registered. Sieve of Eratosthenes on the integers 0..400000. Take the first 16384 gaps whose left endpoint is a prime >= 3 (so every gap is even). Let b_i = ((p_{i+1}-p_i)/2) mod 2. Let H8 be the Shannon entropy, in bits, of the overlapping length-8 blocks of b. Supports if H8 >= 7.2 and both controls pass: the constant sequence of 16384 ones has H8 = 0, and the alternating sequence 1,0,1,0,... has H8 <= 1.01. If fewer than 16384 such gaps exist below 400000, or a control misses its bound, the verdict is inconclusive."; "false"^^xsd:boolean . . a ; dc:description "the prime gaps whose half-gap parity is the randomness statistic"; dc:license "generated data"; "Dataset: Primes generated by the sieve of Eratosthenes up to 400000" . } sub:provenance { a prov:SoftwareAgent; "delphisci claim tester (Grok 4.7; cursor/wolfram-claim-tests-b-355b)"; prov:actedOnBehalfOf ; . a ; dc:identifier "Grok 4.7" . 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 "2026-11-08T16:24:31.923Z"^^xsd:dateTime; dc:creator ; dc:license ; npx:hasNanopubType ; npx:introduces ; "Claim test of wolfram2002-ch04-09: supports, confidence 0.74"; . sub:sig npx:hasAlgorithm "RSA"; npx:hasPublicKey "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB"; npx:hasSignature "NY1cvqapxZyXg+p/mTDg5HxwoTkkc8tszTX8dirlFJgMHVAQMQjF9dnGRFFXglSWjgQuswy4ZC41WUadoCzwpRnKnV56wPV4lfTzr/EKA4asmyPddngWMxbuI9fsorRFVenUsPLm/9DmYuVHBocU/eASVocduxbMrSXifC1pPHYZ9jiSfbZ2TrU/4Rh7HiAW7AE2QJCnBMxlP7ckWlgFqMkwPr28YscYL3rkuzR1uSbz+8uArYre3ZYtm/h1BDw+8r0+Y8whT8PDq/R5wqQ+pFoq+XFrM/lMr+iRwfMFqZwq0m5Gm4NCJq2YhRN81e5Pbc1AoHgyRPJw7+TNBvOkUA=="; npx:hasSignatureTarget this:; npx:signedBy . }