Order 11 Tetramagic Series Impossibility

Theorem. There are no order 11 tetramagic series.
Proof.
S1 = 671
S2 = 54351
S3 = 4952651
S4 = 481386807

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

An order 11 tetramagic series must have
at least 12 entries.