Complete the museum guide's fascinating tale about a historical feline.

Sir Paws-a-Lot, an 18th-century aristocat, is widely believed _____ the concept of the 3 PM afternoon nap.

The correct answer is to have invented.

When we use a passive reporting verb (like is believed, is said, is known) followed by an action that happened before the time of reporting, we use the perfect infinitive (to have + past participle).

"To invent" would incorrectly suggest he is inventing it now or in the future, while "that he invented" requires the impersonal structure: "It is widely believed that he invented..."