A Same-Source Separation Principle for the Weil--Barner Explicit Formula of the Riemann Zeta Function

We develop a same-source separation principle for the Weil--Barner explicit formula of the Riemann zeta function and apply it to the critical-line problem for its non-trivial zeros. The central point is that the zero side, the prime-power Fourier side, the polar cancellation, and the Archimedean and local principal-value terms must all be realized by one and the same admissible sequence, rather than combined from estimates of different origin. We first establish a closure criterion: for a single admissible sequence, the Weil--Barner identity cannot keep a target off-critical zero at its positive multiplicity while the remaining zero contribution and the terms off the zero side both tend to zero; the limiting identity would force that multiplicity to vanish. For an arbitrary putative off-critical zero ρ₀, we then construct a sequence Hρ₀,n realizing this same-source closure structure. In the resonance coordinate w(ρ)=(ρ-12)/i, the construction places, on the same sequence, target normalization, cancellation at ± i/2, transport and regrouping of the non-target zeros into a fixed base-shell series, finite notches at low prime-power frequencies, absolute decay with Tannery convergence at high frequencies, and local control from bounds for the completed logarithmic derivative on zero- and pole-free windows. Classical Weil--Barner, Paley--Wiener, Fourier--Mellin, zero-counting, and local ζ-estimates are used only in their stated standard forms. Applying the closure criterion to the constructed sequence rules out a putative off-critical zero; consequently all non-trivial zeros lie on the critical line.