Saturated M\"obius-dual kernels]{Saturated M\"obius-Dual Kernels in 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, α=z², β= z², Uₙ=αⁿ-βⁿ/α-β, Bₙ=αⁿ+βⁿ/2. Assume that f≥3 is odd and Bf=σ Aᶠ with σ∈\±1\ and $A>0$. Fix a prime power pᵉ f, put h=f/pᵉ, and assume $h>1$ and the clean alternative p A. Then p Bₕ, |Bₕ|=Rᶠ, and Rpᵉ<Q. For the M\"obius dual MₙU=∏d nUdμ(n/d), we prove an exact divisor-lattice balance. For every d h, $d>1$, an explicit correction factor Cd, a squarefree defect Ed∈\1,P⁺(d)\, and an integer Sd>0 satisfy Cd|M2dU|=EdSdᶠ. On the support-saturated sublattice (h) d, the correction disappears and, with λ=P⁺(h), |M2dU|=ληdSdᶠ, ηd∈\0,1\, where the external kernels Sd are pairwise coprime, satisfy (Sd,2fQΔ)=1, and the defect positions form at most one complete λ-power chain. Applying the theorem of Bilu--Hanrot--Voutier at every saturated $d>15$ yields pairwise distinct external primes qd fQΔ with zU(qd)=2d, 4d qd--2qd, qdᶠ U_qd--2qd. Their product gives an explicit lower bound for Q. Exponent-supported lower primes simultaneously carry a near-depth-f Lucas--Wieferich condition and a deep ordinary Q-Wieferich condition. These are necessary structural conditions and do not prove the Beal conjecture.