Sonin–Euler compressions: phase localization and an arithmetic remainder

For a Sonin space and a finite Euler product, either one-sided subpolynomial bound on a single centered trace is equivalent to the Riemann hypothesis. The growth exponent on each side is the supremum of the real parts of nontrivial zeros minus one half. We prove quantitative localization of the leakage at small Gram eigenvalues and a growing range Fourier estimate without a zero-free hypothesis. The inverse of the original transport is not uniformly bounded on the localized sector. We give a trace-class spectral shift formulation and a rank obstruction to pointwise nonnegativity after a finite-rank polar correction. Floating-point diagnostics and a separate finite support Weil certificate accompany the article. The one-sided bound remains open.