Operator Package 10: Hypothesis Discharge (Module B3 + Module C Handoff) for the Riemann Function Operators Series

Operator Package 10: Hypothesis Discharge (Module B3 + Module C Handoff) for the Riemann Function Operators Series This package closes the interface between Modules B and C in the Riemann Function Operators program. Module C (positivity forcing) is already operationally packaged in Operator Package 9 via a Herglotz ratio certificate and the Hermite--Biehler / de Branges real-zero forcing mechanism on growing windows. The remaining analytic bottleneck for passing from determinant closure to positivity forcing is the Module B jet condition (B3): matching the even Maclaurin jets at the origin of the canonically normalized determinant limit $\Delta_{s,\infty}$ with those of the classical Riemann $\Xi$. Main contribution. Package 10 converts (B3) into a fully explicit, trace-class “audit target” by splicing together: Package 7 (Module A splice): the kernel-moment identity expressing $\Xi^{(2k)}(0)$ as the $2k$-th moment of the APM $\Xi$-kernel $\Phi_{\mathrm{APM}}$;Package 8 (Module B splice): commutator/jet formulas for the twisted windowed convolution family $W_I(t)=U_t W_I U_t^{-1}$ and the induced trace-class family $K_{s,I}(t)=A_s W_I(t) A_s$, together with determinant-jet continuity under trace-class jet convergence. The result is an explicit discharge checklist: (B3) reduces to verifying that the window-grown determinant jets converge to the same even-jet targets determined by the kernel moments. New operational checkpoint (B3-N). To support falsifiability on finite windows, the package includes a numerical checkpoint comparing two independent evaluations of the even jets at the origin under window growth: Cauchy jet extraction of $\Delta^{g}_{s,I_L}$ via discretized contour integration on a small circle in the complex $t$-plane; and moment targets for $\Xi^{(2k)}(0)$ computed from $\int q^{2k}\Phi_{\mathrm{APM}}(q)\,dq$. The mismatch $\varepsilon_k(L)=\bigl|\Delta^{g\,(2k)}_{s,I_L}(0)-\Xi^{(2k)}(0)\bigr|$ is expected to decay as the window length $L_q(L)\to\infty$ if (B3) holds. Role in the series. Once the Module B handoff is discharged (existence of a canonically normalized window-removal limit and the identity $\Delta_{s,\infty}\equiv \Xi$), Package 9 applies unchanged: the Herglotz/Hermite--Biehler forcing mechanism then implies $\Delta_{s,\infty}$ has only real zeros, hence $\Xi$ has only real zeros. Package 10 therefore functions as the explicit bridge enabling Module C to be invoked without restating its full machinery. References included in the manuscript. Background tools and numerics rely on Simon (Trace Ideals and Their Applications) and Bornemann (Nystrom evaluation of Fredholm determinants), while the positivity forcing invokes the de Branges framework as developed in Operator Package 9.