Order 4 Tetramagic Series Impossibility

Theorem. There are no order 4 tetramagic series.
Proof.
S1 = 34
S2 = 374   ≡ 5 (mod 9)
S3 = 4624  ≡ 7 (mod 9)
S4 = 60962 ≡ 5 (mod 9)

From the Modulo 9/3 Tetramagic Series Lemma,
this tetramagic series must have
  8 entries of 1 (mod 3) and
  1 entry   of 2 (mod 3).

An order 4 tetramagic series must have
at least 9 entries.