Operator Package 9: Positivity Closure (Module C) for the Riemann Function Operators Series

This paper completes Module C of the Riemann Function Operators proof program by supplying a positivity mechanism that forces the zeros of the canonically normalized limiting determinant to lie on the real axis. Combined with the prior handoffs (Module A: identification of the APM-derived multiplier with the classical $\Xi$ function; Module B: determinant closure under cutoff/window removal), Module C becomes the decisive “critical-line forcing” step for the Riemann Hypothesis. Core construction (internal certificate). On each finite log-window $I$, the program builds an analytic trace-class family $K_{s,I}(z)=A_s W_I(z) A_s$ and the two Fredholm determinants $\Delta^{\pm}_{s,I}(z)=\det(I\pm \lambda K_{s,I}(z))$. The promoted positivity certificate is the ratio\[m_I(z)=\frac{\Delta^{-}_{s,I}(z)}{\Delta^{+}_{s,I}(z)},\]together with the Hermite--Biehler packaging\[E_I(z)=\Delta^{+}_{s,I}(z)-\mathrm{i}\,\Delta^{-}_{s,I}(z).\]If $m_I$ is Herglotz on the upper half-plane, then $E_I$ is Hermite--Biehler, the associated de Branges kernel is positive semidefinite, and the windowed determinant $\Delta^{+}_{s,I}$ has only real zeros. Window-stable persistence of this certificate under $L_q\to\infty$ transfers real-zero forcing to the limit$\Delta_{s,\infty}\equiv \Xi$. High-value proof-closing tool (added in this package). A key promoted addition is a Gram-factorization / resolvent-realization theorem: if the Pick kernel induced by $m_I$ admits a Gram factorization, or if $m_I$ admits a Nevanlinna-type resolvent model $m_I(z)=a_I+b_I z+\langle(\mathbb{T}_I-z)^{-1}g_I,g_I\rangle$ with $\mathbb{T}_I$ self-adjoint and $b_I\ge 0$, then $m_I$ is automatically Herglotz, yielding a proof-grade positivity certificate. This converts the positivity step from an abstract requirement into a concrete operator-model construction. Operational, falsifiable checkpoints. The paper records multiple numerical diagnostics that map directly onto the proof obligations:(i) Pick-matrix PSD tests for point clouds in $\mathbb{C}_+$,(ii) de Branges kernel Gram PSD tests,(iii) Hermite--Biehler phase monotonicity checks,(iv) dissipation-sign tests for the normalized imaginary-part operators derived from the complex-twist kernel,and (v) stability of these certificates under increasing window size. Position in the series. This is the capstone package that closes the positivity fork laid out in the proof-program roadmap. It is intended to be read after Packages 5--8, which provide the determinant generator, identification, and determinant-closure modules. Related records (series context). Operator Package 1 (Zenodo DOI: 10.5281/zenodo.18158307).Operator Package 2 (Zenodo DOI: 10.5281/zenodo.18158379).Operator Package 3 (Zenodo DOI: 10.5281/zenodo.18185891).Operator Package 4 (Zenodo DOI: 10.5281/zenodo.18189297).Operator Package 5 (Zenodo DOI: 10.5281/zenodo.18190604).Proof Program Roadmap (Zenodo DOI: 10.5281/zenodo.18193427).Operator Package 7 (Zenodo DOI: 10.5281/zenodo.18194020).Operator Package 8 (Canonical Determinant Closure / Module B).QADI Community DOI Overview: 10.5281/zenodo.17479314.