The analytic frontier of the Lucas-PSW rank diagonal: CM theta twists, conic Kloosterman kernels, and the triple-product obstruction

The companion theorem note proves that the strong-Lucas $2$-adic rank diagonal forthe family $n=p(2p+3)$, $Q=-1$, $D=5$ has density exactly $\tfrac12$ modulo a singleindependence statement (E2) for the quadratic character $\chi_L$ of the $D_4$-field$L=\Q(\sqrt{i\varphi})$ along the Bateman--Horn set. Here E2 is identified exactly:$\chi_L(q)=\tfrac12a_q(f)$ for the unique weight-one CM newform $f=\theta_\psi\inS_1^{\mathrm{new}}(320,\legendre{-20}{\cdot})$, so that for $q=a^2+b^2$ primary theE2 sign is the elementary Gaussian spin $(-1)^{(a-1)/2}\legendre{a+3b}{5}$, and E2becomes a prime-pair sum twisted by a fixed CM theta form. We prove here, and quote from the analytic companion where indicated, acollection of unconditional approximations to E2: sieve-majorant andcolumn-averaged half-density statements, a Bombieri--Vinogradov theorem for theCM coefficients, a Chen-style almost-prime replacement with both spins, and asmooth conic estimate reaching the natural $X^{2/3}$ barrier. We also close theEisenstein checkpoint by Zagier's shell formalism, and prove the local identity\emph{untwisted conic Fourier sums are Kloosterman sums},$\mathcal S_p(k;c)=\chi_{-4}(p)S(c,\overline4\|k\|^2;p)$, collapsing thesplit/nonsplit dichotomy to a sign. The residual core is a single $A^{1/2}$-coherence gain, and we map itsobstruction: published trilinear Kloosterman-fraction exponents(Duke--Friedlander--Iwaniec, Bettin--Chandee) fall short by $A^{3/8}$; thearbitrary-coefficient spectral large sieve saturates at the orthogonality scale(a positive-diagonal theorem, measured sharp); the CM theta coefficient vectorprojects onto the rigorously computed bottom Maass spectrum at the random-vectorscale (measured, with pre-registered controls); and by the local identity thefaithful conic kernel carries no Sali\'e mechanism --- the finite local componentof the spin character is trivial on the local conic torus. The full Chen $P_2$statement with both spins is therefore conditional on one named estimate,Hypothesis~TP($A$) --- a triple-product/bottom-spectrum second moment for thefamily $L(\tfrac12,\theta_\psi\otimes F_A\otimes u_j)$ --- with all otheranalytic inputs either proved here, quoted from standard sources, or reduced tofinite certificates. At $X=10^9$ ($2.2$M prime pairs) the E2 density is$0.499437$ ($-1.67\sigma$).