Final Consolidation: Identification, Determinant Closure, and Positivity Forcing for the Riemann Function Operators Program

Final Consolidation: Identification, Determinant Closure, and Positivity Forcing for the Riemann Function Operators Program This note consolidates the Riemann Function Operators program into a single, audit-ready theorem chain. The series organizes the route to the Riemann Hypothesis (RH) into three modules and makes explicit the remaining analytic hypotheses required for the terminal conclusion. Module A (Identification). The APM-derived Fourier multiplier $\Xi_{\mathrm{APM}}(t)$ is identified with the classical Riemann $\Xi(t)$ as an entire function (Operator Package 7). Module B (Determinant closure). A canonically normalized, cutoff-removed Fredholm determinant on growing log-scale windows admits a window-removal limit $\Delta_{s,\infty}$, and under three explicit hypotheses (B1) trace-norm cutoff removal, (B2) locally uniform window-removal convergence after canonical normalization, and (B3) even-jet matching at the origin, one obtains the entire-function identity $\Delta_{s,\infty}\equiv \Xi$ (Operator Package 8). Module C (Positivity forcing). If, on a sequence of growing windows, the determinant ratio $r_I(z)=\Delta^{-}_{s,I}(z)/\Delta^{+}_{s,I}(z)$ is Herglotz on $C_+$ and remains stable under window growth (C1--C3), then Hermite--Biehler / de Branges forcing implies each windowed $\Delta^{+}_{s,I}$ has only real zeros, and the real-zero property propagates to the limit $\Delta_{s,\infty}$ by local-uniform convergence. Since $\Delta_{s,\infty}\equiv \Xi$, this yields RH under the stated hypotheses (Operator Package 9). What this consolidation provides. A single self-contained Hypothesis Ledger for Modules B and C (B1--B3, C1--C3), stated in the same trace-class and analytic language used throughout the series. A one-page modular implication theorem: Module A + (B1--B3) + (C1--C3) $\Rightarrow$ RH. A short set of operational checkpoints mapping directly to the remaining hypotheses, including Nystrom/Fredholm determinant numerics, trace-ideal stability tests, and Pick-matrix/Herglotz positivity tests. Scope and intent. This document is a consolidation and interface note: it does not re-derive all analytic estimates from the component packages, but instead presents the final theorem chain and isolates the exact points requiring verification or discharge to convert the program into a complete RH proof within the framework. References included in the manuscript. Standard background sources include Titchmarsh--Heath-Brown (The Theory of the Riemann Zeta-Function), Simon (Trace Ideals and Their Applications), and Bornemann (numerical evaluation of Fredholm determinants).