Moreover, a careful reading of Genesis seems to indicate that God considers relevant “time” to have begun with spiritual man, or even, civilized man, when God provided the earth with a gardener. Which is why the “age” of the rocks, fossils, created matter is largely irrelevant to any spiritual creation
The fact that this is the year 5770 from creation is not merely a tradition, but according to the late Lubavitcher Rebbe (zt"l) a halakhah as well. That's as authoritative as it can get.
This 5770 year calendar began when Adam first spoke after being created on the Sixth Day. Only then did the world begin to "function," so nothing prior to that time can truly be said to have any age at all.
Oral Torah also gives the exact time of the first new moon in history, as well as the time of a theoretical new moon a year earlier than this--a year that never actually occurred but existed only in the Mind of G-d as an archtype. This time is encoded in the opening passage of Genesis at equidistant letter sequences.