Operator Package 11: Final RH Theorem via Modules A--C (Operator Packages 7--10)

Operator Package 11: Final RH Theorem via Modules A--C (Operator Packages 7--10)

This package is the terminal wrapper in the Riemann Function Operators program. It collects the three modules developed across Operator Packages 7--10 into a single, audit-ready theorem statement: once the Module B handoff is discharged (existence of a canonically normalized window-removal determinant limit $\Delta_{s,\infty}$ together with the entire-function identification $\Delta_{s,\infty}\equiv \Xi$) and the Module C Herglotz/Hermite--Biehler certificate is stable under window growth, the zeros of the classical Riemann $\Xi$-function are forced to lie on the real axis in the $t$-plane, which is equivalent to the Riemann Hypothesis.

What this wrapper does (and does not do).

Package 11 does not introduce new analytic machinery; it provides a clean terminal interface for readers and reviewers:

• Module A (Package 7): identification closure $\Xi_{\mathrm{APM}}\equiv \Xi$ as entire functions.
• Module B (Package 8 + Package 10): determinant closure and explicit discharge form for the remaining even-jet condition at the origin, yielding $\Delta_{s,\infty}\equiv \Xi$.
• Module C (Package 9): positivity forcing via a Herglotz ratio and Hermite--Biehler / de Branges structure on growing windows, implying the limiting determinant has only real zeros.

Under these explicit hypotheses, the final theorem concludes that $\Xi(t)$ has only real zeros; equivalently, every nontrivial zero of $\zeta(s)$ lies on the critical line $\Re(s)=\tfrac12$.

Added terminal corollary.

For clarity and downstream citation, the package records the standard corollary translating real-axis zeros of $\Xi(t)=\xi(\tfrac12+i t)$ into the critical-line statement for nontrivial zeros of $\zeta(s)$.

How to use this record.

Readers can treat Package 11 as the one-page theorem endpoint of the program, while consulting Packages 7--10 for the definitions, hypotheses ledger, and verification checkpoints (analytic and numerical) associated to the Module B handoff and the Module C Herglotz forcing conditions.