A one-half density theorem for the strong-Lucas 2-adic rank diagonal in the Lucas--PSW mixed regime, modulo a single equidistribution

For the affine semiprime family $n=p(2p+3)$ and a Lucas parameter pair $(P,Q)$ with$Q=-1$, in the mixed character regime $(\eps_p,\eps_q)=(-1,+1)$, the strong-Lucaspseudoprimality of $n$ is equivalent (Corollary 2 of the companion paper) to the\emph{2-adic rank diagonal} $\vt(\rk_p)=\vt(\rk_q)$, where $\rk_\ell$ is the rank ofapparition. We prove that the density of this diagonal is exactly $\tfrac12$, moduloa \emph{single} equidistribution hypothesis on the Lucas ratio $z_q=\alpha/\beta$among the squares of $\F_q^\times$ --- a $2^2$-division-field Chebotarev input. Thereduction rests on three structural facts: the rank-valuation rule$\vt(\rk_\ell)=\vt(\ell-\eps_\ell)\iff\legendre{Q}{\ell}=-1$ (``Lemma 1''), theaffine 2-adic alignment $s_q=s_p+1$, and an elementary group-counting identity --- ina finite cyclic group of order divisible by $4$, the fourth powers are exactly halfthe squares. All three, and the resulting diagonal reductions, are formalized inLean~4 / Mathlib and are axiom-clean. We confirm the theorem on $2.2\times10^6$mixed-regime pairs ($D=5$, $50.02\%$, residual $0.45\sigma$) and isolate thecomplementary $Q=+1$ column, where the diagonal is a $D$-dependent rational($\tfrac34$ for $D=5$, $\tfrac38$ for $D=21$) governed by the power structure of thefundamental unit of $\mathbb Q(\sqrt D)$. Finally we reduce the hypothesis itself: for$D=5$ it splits into a \emph{provable} Chebotarev density --- the governing field$\mathbb Q(\sqrt{i\varphi})$, of minimal polynomial $x^8+3x^4+1$, is non-abelian ($D_4$)over $\mathbb Q$, so ``$z$ is a fourth power'' is not a congruence and is equidistributedover the family's residue class --- and a single residual independence between thatFrobenius condition and the twin-prime condition $p=(q-3)/2$, the only unconditional gapremaining.