Digital Root Arithmetic in Z/9Z √ 5: Order, Orbits, and the Flag Graph of P2(F3)

We study the multiplicative structure of the golden ratio φ = (1+√5) /2 in the ring Z/9Z√5. We prove ord (φ) = 24 and φ¹² = −1, establishing a mirror symmetry in the Fibonacci sequence modulo 9. The Pisano period π (9) = 24 generates a tower with π (3ᵏ) = 8·3^ (k−1). We classify the three orbits of the Fibonacci recurrence on (Z/9Z) ² as cosets of 1, 8 in (Z/9Z) *, with quotient Z/3Z. We prove a Universal Conservation Theorem: for all 84 divisor classes of the period-24 cycle, the digital root sum equals 0 mod 9. We compute the exact autocorrelation r (12) = −107/173. Finally, we analyze the flag graph of the projective plane P² (F₃), proving it has eigenvalue multiplicities 1, 12, 12, 27 and satisfies the Ramanujan property.