OEIS A209723: Barker's conjectures on distinct clockwise edge differences

We prove the Conjectures of Colin Barker in the FORMULA section of OEIS A209723, dated July 12, 2018: the parity formulas and ordinary generating function for ternary arrays with distinct clockwise edge differences. We also prove Hardin's empirical recurrence. A decomposition of the row-compatibility graph into two three-vertex stars and isolated edges gives a formula for every rectangular size. Its corollaries include the companion formulas in A209721–A209726, Beck's first A209721 conjecture, Peterson's A209721 recurrence and triangle interpretation, and the column recurrences and diagonal identification in A209727.