Much of what is known about the four mathas' founding comes not from contemporary records but from hagiographical texts called the Shankara Vijayas, composed several centuries after the philosopher's traditional lifetime. These biographies present a coherent story of Shankara traveling the length of India, debating rival teachers, and establishing each matha in turn, but historians treat the narrative with caution precisely because of the gap between the events described and the texts that describe them.
Independent evidence for the mathas' antiquity is stronger than evidence for the specific founding story: inscriptions, land grants and later administrative records confirm that all four institutions existed and held significant regional authority well before the modern period. What is harder to confirm is whether one individual personally established all four in a single organizing act, as the hagiographies describe, or whether the four mathas developed more gradually and were retrospectively unified under Shankara's name as his philosophical school gained prestige.
This is a common pattern in the history of Indian religious institutions: an influential later tradition consolidates a shared origin story around a revered founder, and the story becomes authoritative for the community long before, or even without, independent historical confirmation. Scholars generally accept Shankara's central role in shaping Advaita Vedanta while remaining more cautious about the precise mechanics and dating of the mathas' founding as usually told.