Joint Spectral Gaps for the Arithmetic Marchenko–Weil Translation Form

We study the joint spectral-gap problem arising from the localized arithmetic Marchenko–Weil (AMW) representation of the Weil quadratic form for the Riemann zeta function. Rather than estimating the prime-power translations separately, the analysis retains their joint interference. Three exact reductions are obtained. First, the two pole-neutral moment conditions characterize the compactly supported range of the one-dimensional operator. Second, a weighted continuum of translation differences has an exact rank-two defect from the identity, which vanishes on the pole-neutral subspace and yields an isometric translation frame. Third, comparison of the discrete prime-power translation operator with a continuum Volterra operator produces an exact first-order factorization. Writing the prime-power operator as isolates arithmetic discreteness in a bounded discrepancy operator. This gives an exact renormalized representation Q_W(f)=a_R[f]-2\langle H_Rf,f\rangle, where is a closed coercive positive form and is bounded and self-adjoint. The associated normalized operator K_R=A_R^{-1/2}H_RA_R^{-1/2} is compact and self-adjoint, and localized Weil positivity is equivalent to \lambda_{\max}(K_R)\leq \frac12. The construction also shows that summing the optimal spectral gaps of the individual prime-power translations necessarily loses a factor two asymptotically, while the continuum joint factorization recovers the missing contribution. A convergent Ritz hierarchy with an explicit operator-norm tail bound is obtained. No zeta-zero ordinates enter the construction.