רבינו בחיי — זה יתנו... מחצית השקל

Q — Why davka half a shekel — and why is this the mitzvah that כפרה for the egel hinges on?

A — Rabbeinu Bachya reverse-engineers the arithmetic. A whole shekel is twenty gerah; half is ten. Ten what — and against what? Ten gerah against the עשרת הדברות, every single one of which was violated by the egel ha-zahav (the midrash makes this explicit). The half-shekel is therefore not a generic donation but a calibrated counter-offering: one gerah of silver for each commandment trampled. RBB layers another reading on top — they sinned at "machatzis hayom" (כי בושש משה, six hours into the day) so they must give a machatzis of currency. And the silver they bring becomes the adanim — the foundation-sockets on which the Mishkan stands, inverting their sin ("made the bottom into the top") by making bottom-of-the-system silver the literal foundation. Kapparah works here by precise counter-weighing on the שקל הקדש.