In addition to RonJohn's answer about why it doesn't show on your statement yet, I can explain why it may have gone through the third time after failing twice.
Under normal circumstances in the US, all transactions are sent to the bank immediately for authorization. In other countries (which have better EMV support and/or less always-on connections) have ways to approve a transaction without talking to the bank which are just as good as doing so, for the purposes of protecting the merchant from chargebacks and fees, and the customer from fraudulent charges. However, in the US, when that connection to the bank* is down, what's a merchant to do?
Depending on the industry, tolerance for risk, and other business factors, the merchant can choose to configure their POS to "go offline". In this setup, it simply records the transactions to replay later. In some industries, it's feasible to call the merchant's processor's voice authorization system and actually request authorization by reading the card number over the phone. In others, this is too involved a process, and they're just willing to accept the chance that the transaction won't actually go through, and so they just assume it will (most do).
Once the system comes back online, the authorization may be re-attempted, or it could simply show up in the end-of-day batch, at which point your bank won't know about it until they're asked to move your money.
So it could be that the POS, having failed to reach the bank twice, decided that it was offline and simply deemed the transaction approved. If this is the case for a $5000 transaction, I really hope the merchant knows and understands what their POS is doing...
* Technically, the transaction doesn't go straight to your bank. But it's a good enough approximation for this.