That's all the solid work I have so far. Tons of ideas for what could be next. For example, consider convolutions on sets generated by L-systems (eg the Cantor set). I suspect those might offer continuous, finitely-supported, linearly-shifted self-replicating fns (the current article is mainly about non-linear shifts; t_1 is only piecewise linear). Maybe L-systems even completely characterize that category of self-replicating fns.

Other ideas: * Higher dimensions (eg f1 + f2 + f3 = g with domain R^2) * Shifted fns f1, f2 whose sum has surprisingly small diff from their sum: ie small |C(f1+f2)-shift(f1)|; the normal fn appears to fall into this category