Convergence, Density, and Final Weil Positivity Paper 4

This paper closes a four-part programme reducing the Weil positivity criterionto verifiable functional-analytic conditions. The preceding papers establish uniformcommutator coercivity on the kernel (Paper 1), a restricted bridge inequality withdefect (Paper 2), and modal concentration together with commutator regularityfor a dense regular Weil subclass W0 (Paper 3). The present paper addressesthe two remaining gaps: (i) the extension from W0 to the full Weil test class Wvia an explicit density argument in a logarithmic Schwartz topology, and (ii) thelimit passage Λ → ∞ via a three-term approximation argument combining density,continuity of the Weil functional, and convergence on the regular subclass.No new structural hypotheses are introduced. The paper consolidates the resultsof Papers 1–3 into a single final theorem: under the four master hypotheses (uniformcoercivity, bridge inequality, regularity on W0, and density with limit stability),the Weil functional satisfies W (g) ≥ 0 for all g in the full Weil class, which by theclassical Weil equivalence implies the Riemann Hypothesis.