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 12 modulo a singleindependence statement (E2) for the quadratic character χL of the D₄-fieldL=(√iφ) along the Bateman--Horn set. Here E2 is identified exactly:χL(q)=12aq(f) for the unique weight-one CM newform f=θ_ψ₁ⁿᵉʷ(320,-20·), so that for q=a²+b² primary theE2 sign is the elementary Gaussian spin (-1)(a-1)/2a+3b5, 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 X2/3 barrier. We also close theEisenstein checkpoint by Zagier's shell formalism, and prove the local identityuntwisted conic Fourier sums are Kloosterman sums,Sₚ(k;c)=χ₋₄(p)S(c,4\|k\|²;p), collapsing thesplit/nonsplit dichotomy to a sign. The residual core is a single A1/2-coherence gain, and we map itsobstruction: published trilinear Kloosterman-fraction exponents(Duke--Friedlander--Iwaniec, Bettin--Chandee) fall short by A3/8; 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 P₂statement with both spins is therefore conditional on one named estimate,Hypothesis~TP(A) --- a triple-product/bottom-spectrum second moment for thefamily L(12,θ_ψ⊗ FA⊗ uⱼ) --- with all otheranalytic inputs either proved here, quoted from standard sources, or reduced tofinite certificates. At X=10⁹ ($2.2$M prime pairs) the E2 density is$0.499437$ (-1.67σ).