Automorphic Numbers with 10 Digits

Theorem
The only $10$-digit automorphic numbers are:
 * $1 \, 787 \, 109 \, 376$
 * $8 \, 212 \, 890 \, 625$

Proof
We have:

thus demonstrating they are automorphic.

By Automorphic Numbers in Base 10, there are no others.