Fixed-Shift Paired-Main Rigidity, Kuznetsov Source-Ancestry Exhaustion, and a Conditional Consistency Obstruction

This paper develops a fixed-physical-shift source analysis for Mellin-neutralized symmetric-square covariance in arithmetic quantum observables. Its main purpose is not to assert a complete crossover law, but to isolate rigorously which parts of the fixed-shift covariance mechanism are genuinely forced by paired main terms and which parts remain dependent on a separate full-covariance promotion. The first result is an exact paired-main fixed-shift wave $$F_\xi(z) = M(\xi) \left( e^{i\xi z} + U(\xi) e^{-i\xi z} \right), \qquad \vert{}U(\xi)\vert{} = 1,$$ together with its finite-$L$ paired-main kernel. The coalescent limit recovers the previously identified sine kernel with coefficient $3\kappa_w$. The proof keeps the complete two-piece approximate-functional-equation coefficient through the natural-length transition, so that the large-$m$ tail is controlled before residue extraction. The second part gives an exact source-ancestry analysis of the twisted symmetric-square first moment. In particular, the ordinary Kuznetsov diagonal, the discriminant-zero boundary $n=2m$, continuous analytic-continuation residues, the special $L_{-4m^2}$ branch, and the $n<2m$ and $n>2m$ Zagier sectors are separated explicitly. Within the standard Kuznetsov architecture, the Kloosterman off-diagonal is exhausted by $$\mathcal K_{\mathrm{std}} \longleftrightarrow ET_2 + MT_2 + S_1 + S_2,$$ while $ET_1$ has continuous-contour ancestry rather than standard Kloosterman ancestry. A separate contour audit repairs the continuous source. The exact spectral dependence of the archimedean AFE coefficient is retained throughout the contour shift. The central crossed residues are shown to be lower order, $$\mathcal O_A \left( X^{1/2} G (\log X)^C \right) = o(X^G),$$ while the moving side residues are rapidly small. Hence the previously identified continuous contour omission does not furnish a main-scale restoring carrier. Finally, the paper proves a conditional consistency obstruction. If one simultaneously assumes (i) the frozen logarithmic plateau from the integrated Mellin-neutralization parent, (ii) promotion of the paired-main fixed-shift kernel to the full covariance on every fixed physical-shift disk, and (iii) a pure-polylogarithmic growing-shift positive-packet envelope, then Gevrey Fourier localization forces the plateau to equal the paired-main sine functional. Their certified constants differ: $$P_{q,w} = \frac{\pi}{48} \kappa_w \Vert{}q\Vert{}_2^2, \qquad F_{q,w}^{\mathrm{diag}} = \frac{3}{2} \kappa_w \Vert{}q\Vert{}_2^2.$$ Therefore these inputs cannot all hold simultaneously. Claim boundary. This paper does not prove the full fixed-shift covariance identity $$\mathcal C_R = K_{L,w}^{\mathrm{FPS}} + o(1),$$ does not promote the proof-core pure-polylogarithmic growing-shift packet estimate to an unconditional headline theorem, and does not identify an intrinsic crossover scale or a complete crossover law. These questions remain open. The accompanying bundle contains the LaTeX source, theorem-status and provenance notes, SHA256 checksums, PDF preflight information, and a lightweight audit script checking constant identities, source bookkeeping, and the presence of selected proof-repair structures. The audit script is not a formal proof verifier. Version v0.2r2 — Internal Technical-Freeze Pass / Release Candidate.