Operator Package 8: Canonical Determinant Closure (Module B) for the Riemann Function Operators Series

This paper completes Module B in the Riemann Function Operators proof program: the analytic “determinant closure” step that stabilizes a finite-window, cutoff-regularized determinant generator and identifies its canonically normalized limit with the classical Riemann $\Xi$-function. Core construction. On a finite log-window $I=[q_-,q_+]$ (length $L_q$), define a windowed convolution operator $W_I$ induced by the APM coherence kernel $\Phi_{\mathrm{APM}}$, and its log-phase twist $W_I(t)=U_t W_I U_t^{-1}$ where $(U_t f)(q)=e^{itq}f(q)$. Introduce a Sobolev renormalization\[A_s=(I+H_{\mathrm{APM}})^{-s/2},\qquad s>\tfrac12,\]and the trace-class family\[K_{s,I}(t)=A_s\,W_I(t)\,A_s\in \mathsf{S}_1.\]The Fredholm determinant generator is\[\Delta_{s,I}(t)=\det\!\bigl(I+\lambda K_{s,I}(t)\bigr).\] Promoted analytic tools. This package promotes two previously deferred bottlenecks into full, reusable lemmas: Double-limit stability (cutoff $E\to\infty$, window $L_q\to\infty$).Using trace-norm continuity of Fredholm determinants and an explicit $\mathsf{S}_1$-topology epsilon-chase, the paper proves order-independence and local-uniform convergence of determinants under the normalized double limit. Canonical normalization + jet pipeline. A unique scalar normalization is fixed by\[C(L_q,\vartheta;s,\lambda)=\frac{\Xi(0)}{\Delta_{s,I}(0)},\qquad\widetilde{\Delta}_{s,I}(t)=C\,\Delta_{s,I}(t),\]eliminating hidden scalar gauges. A Banach-space Cauchy principle for $\mathsf{S}_1$-valued analytic maps, combined with Frechet analyticity of the Fredholm determinant on $\mathsf{S}_1$, yields a clean transfer from trace-class operator jet convergence to determinant jet convergence. Purely internal closure criterion (no external uniqueness set). Because both $\widetilde{\Delta}$ and $\Xi$ are even entire functions, equality reduces to equality of the even Maclaurin jets at the origin:\[\Delta_{\infty}^{(2k)}(0)=\Xi^{(2k)}(0)\quad (k\ge 0)\;\Longrightarrow\;\Delta_{\infty}(t)\equiv \Xi(t).\]The target jets are computable directly from kernel moments,\[\Xi^{(2k)}(0)=(-1)^k\int_{\mathbb{R}} q^{2k}\Phi_{\mathrm{APM}}(q)\,dq,\]and therefore define falsifiable, numerically testable checkpoints. Concrete commutator-jet mechanism (high-value lemma + toy model). The paper includes an explicit commutator-kernel identity for windowed convolutions:\[(\mathrm{ad}_Q)^k(W_I)\ \text{has kernel}\ (q-u)^k\Phi(q-u),\]hence $W_I^{(k)}(0)$ and $K_{s,I}^{(k)}(0)$ are explicitly computable from $(q-u)^k\Phi(q-u)$ sandwiched by $A_s$. A worked toy model (free window Laplacian with periodic holonomy) is included to illustrate the mechanism and connect the analytic pipeline directly to numerics. Position in the series. Module A fixes the identification target (the APM multiplier equals the classical $\Xi$ under the canonical theta partition). This paper closes Module B by stabilizing the determinant generator in trace-class topology and reducing the global entire-function identification to internal jet matching. Module C (next) addresses the positivity / canonical-system conditions that force zeros onto the critical line. Practical checkpoints (falsifiable). The paper specifies three operational diagnostics:(i) rank stability under $E\uparrow$ (cutoff removal), (ii) window stability under $L_q\uparrow$ with canonical normalization at $t=0$, and (iii) even-jet stabilization via Cauchy differentiation or trace expansions, compared against $\Xi$-moment targets.