The Arithmetic Lift: Rigidity and the Open Packaging Problem

Paper 50 in the "Geometry of the Critical Line" programme. This paper proves that after the sterile carrier (Paper 49) and the prime-power amplitudes (Paper 47) are fixed, the remaining arithmetic-lift problem is uniquely determined and reducible to a single operator-valued packaging problem in the Bost–Connes crossed product. The main results are: (1) A rigidity theorem showing that at most one admissible arithmetic extension exists on the sterile kernel class — once the arithmetic support, reduced identity functional, and prime-power amplitudes are frozen, the scalar target is unique. (2) A reduction theorem showing that any packaging map satisfying the operator-valued packaging property (identity preservation, prime-sector evaluation, non-arithmetic annihilation, continuity) yields the Arithmetic Lift Conjecture. (3) The precise formulation of the Open Packaging Problem as the single remaining verification target. A scalar Mellin–Barnes representation of the sterile spectral zeta is included as analytic motivation for the form of the desired operator-valued lift. The class structure relevant to arithmetic packaging is shown to belong to the operator-valued level rather than to scalar displacement data alone, because scalar summation collapses distinct multiplicative words onto the same displacement. This paper does not prove the Arithmetic Lift Conjecture. It does not construct the lifted kernel explicitly. It does not verify the packaging property. It does not prove the Riemann Hypothesis. Its contribution is narrower: after sterility, the arithmetic target is rigid and the remaining open work is exactly the operator-valued packaging problem.