Analytic Dipole Synthesis and Sectorial Slow Variation: Unbounded Variational Envelopes, Non-Uniform Divisor Realization, and Non-Local Stieltjes Convolution Mechanics

Building upon the axiomatic foundation of Complex Slowly Varying functions established in our preceding work (doi:10.5281/zenodo.23509689), this paper develops a unified forward and inverse theory of meromorphic dipole products within the category of Analytic Slow Variation ($\mathrm{ASV}$). On any interior buffered sector $\Omega_\eta \subset \mathbb{C}^*$, an analytic, eventually nowhere-vanishing function $L(z)$ belongs to $\mathrm{ASV}(\Omega_\eta)$ if its dimensionless logarithmic derivative (elasticity) $E_L(z) := z L'(z)/L(z)$ vanishes uniformly as $|z| \to \infty$. We construct a two-way conformal bridge linking discrete zero-pole divisors $\{p_n, w_n\}$ confined to a singular harbor sector $S$ with continuous $\mathrm{ASV}$ fields on the safe exterior domain $S_\eta^c = \mathbb{C} \setminus S_\eta$. Along the Forward Bridge (analysis and confinement), we extend the Primary Dipole Theorem to the Unified Sectorial Dipole Theorem, permitting unbounded polynomial intra-dipole displacements $\varepsilon_n := |w_n - p_n| \le M n^\alpha$ ($\alpha \ge 0$) governed by the master density-decay discriminant $\Delta_n^{(\alpha)} := \varepsilon_n n^{2+\alpha} / |p_n|^2 \to 0$. Deploying the Generalized Variational Envelope Method, we bypass radial sorting obstructions on unsorted two-dimensional sequences and prove that worst-case harmonic shell clumping saturates at $E_P(z) = \mathcal{O}(\sqrt{\sup \Delta_n^{(\alpha)}})$ via exact differential radius cancellation. We establish quantitative infimum remainder bounds, classify the four-regime dipole growth spectrum, and distinguish Asymptotically Constant (Plateau-Type) Slowly Varying Functions ($\mathrm{ASV}_{\mathrm{const}}$, where $L(z) \to C_\infty \in \mathbb{C}^*$) from Genuinely Slowly Varying Functions ($\mathrm{ASV}_\infty$). Furthermore, we extend the forward machinery to dynamic trumpet domains, canonical fractional branch-cut dipoles trapped in the harbor, ultra-dense quadrupole lattices, and a general Hadamard--Dipole factorization theorem that eliminates unknown exponential factors $e^{g(z)}$ via Karamata--Liouville rigidity. Along the Inverse Bridge (synthesis and divisor realization), we resolve the failure of the strict meromorphic converse by shifting from global identity to uniform sectorial asymptotic equivalence, $P(z) \sim C_\infty L(z)$ on $\Omega_\eta$. Locking the central observation bisector $\theta_0 = (\theta_1 + \theta_2)/2$ via the Canonical Backbone Protocol, we use Cauchy's integral formula on inscribed clearance disks to clamp the radial elasticity derivative $|\widetilde{E}_L'(t)| \le 1/(2t)$ without auxiliary smoothness hypotheses. We first prove the Asymptotic Dipole Realization Theorem on unit integer grids and then liberate the inverse theory via the Flexible Inverse Synthesis Engine on arbitrary non-uniform radial meshes $\{r_n\}$ satisfying the $\ell^2$-relative step summability condition $\sum (\Delta r_n / r_n)^2 < \infty$. Weighting microscopic displacements by the local mesh Jacobian ($\delta_n = \widetilde{E}_L(r_n) \Delta r_n$) preserves the continuous Stieltjes flux measure $dM(t) = \widetilde{E}_L(t)\,dt$. Finally, we prove a formal falsification theorem refuting pointwise discriminant-elasticity proportionality ($E_P \asymp \Delta$) via the non-locality of Mellin--Stieltjes convolutions, derive the exact summability criterion $\sum \varepsilon_n / r_n < \infty$ governing early flattening into $\mathrm{ASV}_{\mathrm{const}}$, and demonstrate the framework on logarithmic powers, sparse quadratic meshes, delocalized solitary dipoles, transverse phase vortices, and the digamma function $\psi(z)$.