A First-Order Terminal-Closure Criterion for the Riemann Hypothesis and the Exterior-Rank Source Boundary

This record contains a Coherence Geometry manuscript formulating a strict first-order terminal-closure criterion for the Riemann Hypothesis using a matrix-valued completed explicit formula. A two-state transfer test is attached to the functional-equation pair \(α\) and \(1-α\). For an off-critical zero orbit, a negative-support one-sided test makes the prime and gamma/algebraic transfer contributions silent and yields a forced first-order transfer ledger. For the fixed test, analytic tail compactness gives an operator-norm compact-limit representative of the forced transfer target. Terminal closure is then taken in the strict first-order ordinary zero-orbit source category, while completion is performed inside the positive full-Pauli cone. The full two-state Pauli ledger is retained as accountability data. The obstruction to projective collapse of the distinguished functional-equation pair is the rank-area \[ A₁₂=Ω₁₁Ω₂₂-|Ω₁₂|^2.\] This rank-area is a second-order exterior resource. Since the first-order matrix-valued completed explicit formula has no primitive exterior-square source labels, strict first-order terminal admissibility excludes unassigned rank-area on the forced two-source channel. Hence the distinguished terminal Gram pair collapses projectively. Label-compatible source-faithfulness then gives the critical-line condition. The manuscript also identifies the boundary of the criterion. Admitting nonzero rank-area as terminal analytic data would require either a genuine second-order exterior source law or an equivalent determinant, kernel, spectral, trace, curvature, or positivity framework giving exact pair-specific control of the distinguished rank-area. No such mechanism is supplied by the first-order completed-explicit-formula framework used here. This manuscript is part of the Coherence Geometry Clay-problem research series and is categorized under the Riemann Hypothesis. It is presented as a strict first-order terminal-closure criterion and source-order boundary analysis, not as an accepted resolution of the Riemann Hypothesis.