The gap almost certainly costs you money, and here is the mechanism: carrying costs (the ongoing expenses of holding a property you have not yet sold) keep running whether or not you have a certificate of occupancy, which is the document your local authority issues to confirm the home is legally habitable.
Those costs typically include construction loan interest, property taxes, insurance, and any utilities you are paying. If your loan charges, say, $1,500 a month in interest and the buyer takes six weeks to close after CO, that is roughly $2,250 in interest alone before you factor in anything else.
The CO date does not pause the clock. It is a milestone in the build, not a contract event. What actually governs your timeline with the buyer is the purchase agreement you both signed. That document will name a closing date or a closing window, and it will spell out what happens if the buyer cannot perform on time. Some agreements allow a short extension; others let you keep the earnest money deposit (the upfront money a buyer puts down to hold the deal) if they miss the deadline without cause.
The thing worth knowing right now: finishing early is genuinely good news for your schedule, and your leverage here depends almost entirely on what that purchase agreement says. Pull it out and look at the closing deadline and the buyer default clause before you decide anything.
I would also suggest looping in a real estate attorney before you send the buyer any formal notice about the timeline. One sentence from a lawyer now is cheaper than a dispute later.
What does your purchase agreement say about the closing date, and is the buyer asking for an extension or just running slow?