Points paid at closing are:

Points and the down payment are entirely separate financial obligations — the down payment is the buyer's equity contribution toward the purchase price, while points are a fee paid to the lender to reduce the interest rate; conflating the two reflects a fundamental misunderstanding of mortgage cost structures.

Points are not applied to reduce the principal balance of the loan; they are paid directly to the lender as compensation for offering a lower interest rate, and they have no effect on the loan's starting principal — the loan amount remains the same regardless of how many points are paid.

Discount points are non-refundable under any circumstances, including early payoff or refinancing; once paid, they are earned by the lender as prepaid interest, and a borrower who refinances or sells before the break-even point simply loses the benefit of the points they paid without any refund.