Order 9 Tetramagic Series Impossibility

Theorem. There are no order 9 tetramagic series.
Proof.
S1 = 369
S2 = 20049
S3 = 1225449
S4 = 79895265

From the Modulo 5 Tetramagic Series Lemma,
this tetramagic series must have
 3 entries of 1 (mod 5),
 4 entries of 2 (mod 5),
 4 entries of 3 (mod 5), and
 4 entries of 4 (mod 5).

An order 9 tetramagic series must have
at least 15 entries.