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 L of the D₄-fieldL= (i) along the Bateman--Horn set. Here E2 is identified exactly: L (q) =12aq (f) for the unique weight-one CM newform f=_₁^new (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 X^2/3 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) =-₄ (p) S (c, 4\|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 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, _ FA uⱼ) --- with all otheranalytic inputs either proved here, quoted from standard sources, or reduced tofinite certificates. At X=10⁹ (2. 2M prime pairs) the E2 density is0. 499437 (-1. 67).