Perfect-Power Quadratic-Trace Lucas Towers

Let r,s∈ be coprime, with r odd and s≠0, and put z=r+s√-2, D=r²-2s², Q=r²+2s², Bₙ=z²ⁿ+ z²ⁿ2. Assume that, for an odd integer f≥3, one has a perfect-power endpoint Bf=σ Aᶠ with σ∈\±1\ and $A>0$. For each pᵉ f we construct a normalized prime-power tower and prove an exact coprimality splitting of its layers. If p A, the tower forces the depth-e Wieferich condition pᵉ⁺¹ Qᵖ⁻¹-1. If p A, every prime q in the lower layer carries a residue packet modulo the full square modulus q^2vq(Bf/pᵉ). Its Teichm\"uller component satisfies an exact phase relation; a nontrivial packet implies pᵉ q-1, vₚ(q(Q))=vₚ(q-1), vₚ(q(p))=vₚ(q-1)-e+1. These conditions define a weighted acyclic support graph and yield a terminal classification at the largest prime divisor of f. The unit lower-layer alternative collapses to the pure prime-power boundary by Mih{a}ilescu's theorem. Using the primitive-divisor theorem of Bilu--Hanrot--Voutier, we also obtain a distinguished lower-support prime of rank $2h$ whenever the reduced index h exceeds $15$. The results are necessary structural conditions; they do not prove the Beal conjecture.