Obstructions and Necessary Conditions for the Arithmetic Lift

Research Note 50 in the "Geometry of the Critical Line" programme. This note identifies what any successful arithmetic packaging functional cannot be and what structural features it must possess, following the rigidity and open packaging problem established in Paper 50. Three no-go results are proved. First, on the Gaussian heat family, no functional acting purely on the angular/Hecke sector can yield the Weil distribution; the σ-sector must participate (factorization obstruction). Second, no spectral diagonal lift can produce the Weil prime term (diagonal no-go). Third, the ordering τ_{1/2} ∘ D ∘ E is impossible for any derivation vanishing on the fixed-point subalgebra (post-expectation derivation vanishes). Any successful prime-part evaluator must be non-diagonal and displacement-sensitive. Two further structural expectations — logarithmic-derivative character and KMS amplitude extraction — are argued but not proved as formal necessities. The σ-sector decoupling is provided by the Jacobi inversion K_{h_a}(ξ) = (ϑ_a(ξ) − 1)·Θ_σ(a) on Gaussian tests, verified to 25 decimal places. However, the naive composition Weil ∘ J is tautological and does not constitute valid packaging. At the analytic level, the prime part is governed by the logarithmic derivative −ζ'/ζ. The remaining question is precise: how does the sterile Mellin–Barnes kernel ζ(2w)·ζ(2(s−w)) collapse, under the correct packaging operation, to a single logarithmic derivative? This note does not prove the Arithmetic Lift Conjecture. It does not construct the final packaging map. It does not prove the Riemann Hypothesis. Its contribution is the identification of what fails, what is forced, and what the analytic target is.