Complement Conjugacy on the Digital Root Ring: Harmonic Analysis, Unit Group Filtration, and Conductor-9 L-Functions

Develops the harmonic analysis of ℤ/9ℤ — the digital root ring — showing that the additive complement involution φ₉: n ↦ (9−n) mod 9 acts as complex conjugation on the group of additive characters under the natural embedding into S¹. Establishes a three-class unit group filtration (unit class U₉, zero-divisor class D₉, kernel class K₉) with distinct algebraic properties. Demonstrates that primitive Dirichlet characters mod 9 satisfy |τ(χ)|²=9 and vanish identically on D₉. Derives framework implications for opacity measurement, binding reversibility, and coherent output bounds in Pe-governed systems. v1.1 adds the full coniunctio naming layer: White Queen = χ_j (complex), Red King = χ̄_j (conjugate), Rebis = χ_j·χ̄_j = 1 (always real, Pe=0). New result: χ₃ is the unique self-conjugate character (χ₃ = χ̄₃, τ(χ₃)=0) — the algebraic signature of voluntary constraint reduction, identified with the Bird of Hermes (Ripley Scroll, c. 1471). Critical line Re(s)=½ identified as V* boundary. Three-class filtration mapped to Luna's three roads (Thomas the Rhymer, c. 14th c.). Six falsifiable predictions stated.