Graph-Series Semantics and Abel Regularization for Recursive Hybrid Quantum Programs

We introduce a graded graph-series semantics for recursive hybrid quantum programs interpreted in the quantum orchestra monad. Finite terminating executions are represented by directed paths whose edges carry normal completely positive subunital maps and whose terminal vertices carry classical results. Path concatenation defines a graded execution category, while continuation grafting models outcome-dependent sequential composition. We construct a semantic evaluation from admissible execution-graph series to quantum orchestras and prove that it is compatible with both channel composition and Kleisli composition. For finitary recursive programs, the truncation of the execution series at degree $n$ is shown to coincide with the $n$-th Kleene approximant of the associated Scott-continuous recursion functional. Consequently, evaluation of the complete graph series recovers the ordinary least-fixed-point denotation. Weighting a graph of degree $n$ by $q^n$, with $0<q<1$, yields an Abel-regularised semantics whose Scott limit as $q\to 1^{-}$ is the unregularised recursive denotation. Equivalently, the parametrisation $q=e^{-t}$ exponentially suppresses long executions and reconstructs the denotation as $t\to 0^{+}$. In a supplementary linear feedback sector, repeated recursion is represented by the execution resolvent $(I-qST)^{-1}$. We identify $I-qST$ with an algebraic cross-ratio of graph subspaces. Under Hilbert--Schmidt assumptions, the associated return operator is trace class and defines the Fredholm feedback determinant $\operatorname{det}_{F}(I-qST),$ whose zeros detect singular feedback configurations and whose logarithmic expansion records closed loop traversals.