Riemann Hypothesis by Finite-Part Livsic Endpoint Exclusion

We extend the finite-part Euler–Gamma Schur–Feshbach endpoint construction for the completed Riemann ξ-function, integrating it with the stopped-ledger calculus previously established in our framework. Our approach transforms the RH–TREP verification package into a sequence of certified finite rows, wherein each structural component is rigorously represented by finite objects, registered ports, graph-closed limits, and no-late endpoint certificates. The spectral coordinate is defined as s = 1/2 + iz, where Ξ(z) = ξ(1/2 + iz); the Riemann Hypothesis is thus equivalent to the requirement that all zeros of Ξ lie on the real axis. We define the proof architecture as follows: $$Ξ(z_0) = 0, Im z_0 ≠ 0 ΔᵍˡᵒᵇLiv(z_0) = 0 0 ≠ u ∈ Ker Rend_Q(z_0) graph-closed port-free endpoint defect metric contradiction.$$ This implementation establishes six universal rows: primitive finite-part parent, faithful zero-to-graph realization, local determinant trace identification, reduced Cauchy no-tail transport, port-free Schur–Stein metric coercivity, and global finite-part Mellin exhaustion. All subsequent rows are formally constrained as restrictions, quotients, finite trace-class limits, or first-hit charges of the initial parent record. Our proof avoids circularity by strictly excluding the use of Hadamard products over nontrivial zeros, external Hermite–Biehler or de Branges zero-line theorems, zero-dependent charts, or metric gaps selected a posteriori. An internal non-circularity audit confirms that these rows introduce no hidden endpoint hypotheses, ensuring that the contradiction derived from the metric defect is robust and self-contained.