Z₉ and the Standard Model Gauge Group

The first three papers established that Z9 arithmetic determines 32 Standard Model parameters. All three assumed the Standard Model gauge group as given. We show that the gauge group itself is constrained by the Z9 ring. The multiplicative unit group is isomorphic to the product of centers of the non-abelian gauge factors. The ring decomposes as units, ideal, and identity, with generator counts 8 + 3 + 1 = 12 matching the gauge boson spectrum exactly. A uniqueness scan of all Z_n for n = 2 to 500 confirms that Z9 is the only modular ring exhibiting this complete gauge correspondence.