Closing the Erdős–Graham conjecture — every K≥3, one construction

The claim. In 1980, Erdős and Graham asked: for a set B ⊆ ℕ, does there exist A ⊆ ℤ such that every integer is uniquely the sum of one element of A and one of B — a "tiling complement"? For B = the K-th powers, K=2 (squares) was already known to fail in the literature. Whether every K≥3 works was open for 46 years. We think we closed it: yes, for every K≥3.

How. Started narrow — just K=3 — and checked it hard against primary sources (read Heath-Brown 2008 and Browning–Heath-Brown 2005 directly, not abstracts; re-derived the exponent balancing by hand; rebuilt the "27 lines on a cubic surface" argument from scratch). It held. Asked GPT-5.6 Sol to push the same construction to every K≥3 by one uniform argument. It came back claiming the whole thing. We ran a second round explicitly trying to break it — found one real bug (a wrong genus formula for even K), traced it by hand, and confirmed it's arithmetically inert: the actual bound never uses genus, so the theorem survives untouched.

The one honest line: erdosproblems.com requires a human proof-check or a Lean formalization before something goes up on their forum. We hold ourselves to that same bar before we call this "posted" there — so for now it's labeled CLOSED here and nowhere else, and we'll update the record the moment it clears theirs too.

Receipts: github.com/pw/erdos477-cubic — full writeup, full verification trail, nothing hidden.