The preceding paper in the Weinstein branch established exact closure for renormalized tail orbits in the finite-level multi-summable regime. There the alien data were still packetized: finitely many Gevrey levels, finitely many visible directions at each level, and only summable directional action packets. The natural next frontier is the failure of packetization itself. In many wild situations the accelerated Borel transforms do not have merely discrete singular points or finitely packetized rays; instead their singular support condenses onto compact rectifiable one-dimensional sets, and the jump data become operator-valued measures rather than discrete bridge residues. This paper proves an exact inverse theory in that stronger regime. On every bounded window we allow finitely many Gevrey levels \ 0<k₁<<kᵣ, \ but at each level the singular support may be a finite union of compact Lipschitz arcs and isolated atoms in the accelerated Borel plane. Along those supports the renormalized tail orbit determines finite operator-valued Radon measures of bounded variation. These are the continuous alien spectra. Their directional Laplace transforms produce the wall functions, and their ordered product integrals produce the continuation automorphisms. The total object is a transseries web. The first main theorem is an orbit-to-continuous-spectrum closure theorem. From the orbit one canonically reconstructs the stable packet cores, the level set, the spectral supports, the operator-valued bridge measures, and the multisummed wall functions on every admissible summation frame. The theorem is genuinely stronger than the finite-level multi-summable result because the spectral support is now continuous. When the measures are atomic, one recovers exactly the alien lattice of the previous paper. The second main theorem is local and categorical. To every chamber and every admissible frame we attach a completed chamber algebra obtained from the continuous spectral data. Crossing an ordinary wall gives the usual mutation functor, while crossing an oriented spectral band produces an exact continuation functor given by a product integral of the bridge-measure field. These functors satisfy a measure-valued Stokes system and glue to a measure-valued schober. A key point is that novelty no longer comes only from a hierarchy of scales: even on one fixed support band, different bridge measures produce different local transport laws. The third main theorem globalizes the local picture. Finite band discretizations of the continuous spectra form a directed system of discrete models. Their wrapped-theta categories and stopped Weinstein sectors glue under sectorial descent to a canonical exact global category and a canonical measure-stopped Weinstein sector. These are equivalent. Hence the Weinstein branch extends from discrete wild completion to a transseries wrapped-theta equivalence. The fourth main theorem is quantitative. From finitely many orbit probes up to horizon \ (N\), finitely many Borel samples on finitely many polygonal bands, and finite angular resolution separating the visible support, one reconstructs the truncated continuous spectrum, the local measure-valued schober, and the truncated global category with error bounded by \ O\! (N^-1+N+₉=₁ʳ (ⱼ+hⱼ+qⱼ+ⱼ (Rⱼ) ) ). \ Here \ (N\) is the orbit discretization defect, \ (ⱼ\) is the angular uncertainty at level \ (j\), \ (hⱼ\) is the band mesh, \ (qⱼ\) is the quadrature error on the sampled bands, and \ (ⱼ (Rⱼ) \) is the unseen variation outside spectral radius \ (Rⱼ\). If the bridge densities are \ (C^\) along the arcs, then \ (qⱼ= O (hⱼ^) \) ; for analytic densities, \ (qⱼ\) decays exponentially under spectral quadrature. The range of the theorem is stated sharply. We still assume finitely many Gevrey levels, finite rectifiable complexity of the singular support on each bounded window, and bounded total variation of the bridge measures. The paper is not a claim about arbitrary two-dimensional Borel singular surfaces, fully dense non-rectifiable spectra, or a globally conditioned numerical algorithm under arbitrary noise. Within that scope, however, it upgrades the Weinstein branch from finite packetized multisummability to exact continuous-spectrum transseries closure.
Mohammad Abu-Ghuwaleh (Wed,) studied this question.