A Robinhood Chain stock token pays no cash dividend. Dividends, and stock splits with them, are folded into an on-chain multiplier — ERC-8056. A balance is raw and never moves on a corporate action; only the multiplier does, and shares are raw times that number.
That single fact is the whole design. hdfi splits a token into the shares it will settle for at maturity and the shares the multiplier adds before then, and runs a pool where the two trade. There is nothing to distribute, so nothing is distributed: yield is settled against an index, and the vault stays solvent by construction rather than by accounting.