Odd, Even, Fid-C: a three-state number system, deterministic prime values without exception

I'm presenting a three-state classification of the integers — **Odd, Even, Fid-C** — in which*FidC* names the state of divisibility by three, treated as a first-class numerical statealongside parity. Arranged on a repeating 3 column proof table, the two states Even and FidC claimfour of every six consecutive integers, and the two unclaimed cells — the **prime seats**,lying on a diagonal of the tile — are the only positions a prime greater than 3 may everoccupy. Every number increases by a factor of 1, 2, or 3. There is odd, even & Fid-C.