TREP–Livsic Endpoint Construction for the Riemann Xi Function. A Non-Circular Conditional Endpoint-Exclusion Criterion

Abstract We formulate a finite-part Euler–Gamma Schur–Feshbach endpoint criterion for the completed Riemann xi function in the critical spectral coordinate s = 1/2 + iz, where the function is defined through the zeta function, the Gamma function, and the standard zeta-factorization. The construction is carried out on stopped Mellin boxes with one primitive ledger, one port registry, one graph topology, one finite-part normalization, and one spectral coordinate fixed throughout the cutoff and exhaustion limits. The finite rows provide the arithmetic determinant identity, the graph-lift mechanism, port quotient control, reduced Cauchy transport, and the candidate port-free residual metric. The paper is written in a deliberately non-circular form. The zero-line conclusion is conditional on an explicit RH–TREP verification package: determinant-admissible trace-class convergence, zero-independent finite-part arithmetic identification, no unregistered Nevanlinna tail, graph-closed Fredholm zero lifting, global finite-part Mellin exhaustion without affine or coordinate drift, and a port-free Schur–Stein metric coercivity row proved independently of zero-location information. Under this package, a non-real zero of the global endpoint determinant gives a graph-closed defect vector; the registered Green identity, port coisometry, and independently verified port-free metric coercivity force that vector to vanish, contradicting Fredholm nonzero kernel production. Thus, the manuscript isolates the exact endpoint package whose independent verification would imply that all zeros of the completed xi function are real, equivalently that every non-trivial zero of the zeta function lies on the critical line where the real part of s is 1/2. It does not use a Hadamard product over non-trivial zeros, an external Hermite–Biehler or de Branges zero-line theorem, a zero-dependent chart, or a metric gap introduced after a hypothetical zero is selected.