The Singular Schwartz Space, Unipotent-Residual Cancellation, and the Conditional Reduction of the Riemann Hypothesis

We present a rigorous mathematical framework consisting of the Singular Schwartz Space Sₒ₈₍₆ ( (2, _) ), a nuclear Fr\'echet space of singular admissible distributions equipped with the singular seminorms. Within this space we construct a distribution ₒ₈₍₆ that exactly cancels the unipotent divergence against the residue of the continuous Eisenstein spectrum while preserving the spherical trivial representation (the Unipotent‑Residual Cancellation Theorem). An asymptotic limit in the style of Beyond Endoscopy then cancels the cuspidal contributions, leaving only the trivial representation. The Z‑plane contour deformation bypasses the Archimedean negativity of the Plancherel measure. The resulting trace formula collapses to the explicit formula of Riemann–von Mangoldt. We prove that, assuming the validity of the Singular Schwartz Space and the associated cancellation theorems, the Riemann Hypothesis is equivalent to the non‑negativity of an explicitly defined regularised Weil functional, ^Z (f) for all admissible test functions f. The logical structure of the proof is completely detailed, and the conditional nature of the reduction is clearly identified.