It's not "make N antibodies". It's a bunch of cells in parallel creating cells specialized to each create a single random antibody. Cells that create antibodies

It's not "make N antibodies". It's a bunch of cells in parallel creating cells specialized to each create a single random antibody. Cells that create antibodies that don't work don't reproduce (linear decay). Cells that create antibodies that worked reproduce (exponential growth). This process stops when the infection is gone. For the mRNA vaccines, the antibodies only target the some protein, but on the other hand they all target the spike protein. Natural immunity products antibodies that could match want part oft the virus. The amount of protection you get against a new variant is related to how well your existing antibodies match the new virus. With natural immunity it is likely that only some of the antibodies match (and since each antibody exists in random amounts, the matching antibodies may be the ones that you have much less of). With vaccine immunity,all of the antibodies produced will work against the new variant if the spike protein is the same, offering nearly the same immunity to the variant as the original virus (assuming the spike protein doesn't change significantly). We know that the spike protein is significantly less likely to change than other parts of the virus so that's a reasonable assumption.