Spectral-Network Abelianization from Coordinate-Free Renormalized Tail Orbits:\ Voros Coordinates, BPS Wall-Crossing, and Finite-Graph Recovery

The preceding paper of this series proved that, in a bounded coordinate-free smooth simple-pole regime over ADE miniversal bases, renormalized tail orbits determine the global filtered symplectic Stokes local system on an admissible irregular cover together with a finite wild character atlas. That result solved the global irregular Riemann--Hilbert problem in the orbit category, but it still left open the next geometric question. A wild character atlas is not yet a canonical coordinate system: one still wants abelianized Darboux coordinates, a phase-dependent spectral network description, and a precise wall-crossing law when the phase meets a BPS wall. A strong referee can therefore ask immediately whether the orbit remembers only a nonabelian character point, or also the finer spectral-network geometry that resolves it into Voros-type coordinates and cluster-like mutations. This paper answers that question in a sharp but robust generic regime. We remain within bounded ADE families and the coordinate-free smooth simple-pole branch class from the previous paper, and we impose the additional phase-generic assumptions that the spectral cover has only simple branch points on the chosen compact design region, that active anti-Stokes directions are separated, that there are no higher-valent joints or simultaneous multi-BPS collisions on the phase interval under study, and that only finitely many simple BPS phases occur there. In that setting the renormalized tail orbit determines not only the global wild character point but also a canonical spectral-network package. The first main theorem constructs, for every non-BPS phase \ (\), an orbit-invariant admissible spectral network \ (_\) on the oriented blow-up. Its isotopy class depends only on the orbit-determined formal exponential type and Stokes local system, and it varies locally constantly away from the finite BPS set. The second main theorem proves a canonical abelianization theorem: after choosing a finite graph dual to \ (_\), the global wild character atlas admits rational symplectic coordinates \ (X_\) indexed by a relative-cycle lattice on the spectral cover. These are orbital Voros coordinates; they are recovered functorially from the orbit and satisfy the expected log-symplectic Poisson law. The third main theorem establishes a simple-wall BPS wall-crossing formula. When the phase crosses a simple BPS value \ (_*\), the one-sided coordinate systems are related by a finite Kontsevich--Soibelman product, equivalently by a cluster mutation when only one primitive flippable charge is active. The active charges and their multiplicities are themselves orbit invariants read off from the resonant packet structure of the tail orbit. The fourth main theorem gives a quantitative finite-graph recovery scheme. On a fixed finite graph of rays, streets, and loops away from unresolved collisions, finitely many orbit probes over a bounded observation window recover the truncated Voros coordinates and, when present, the one-sided wall-crossing factors with error \ \! (N^-1/p_*+N+N+N), \ where \ (p_*\) is the maximal active Poincar\'e rank, \ (N\) is the discretization defect, \ (N\) the modeling defect, and \ (N\) the measurement floor. The detector is explicitly local and finite-horizon; it is not claimed as a globally conditioned arbitrary-noise algorithm. Conceptually, the paper upgrades the previous global wild Riemann--Hilbert theorem from a nonabelian character description to a phase-sensitive abelianized one. In the stated generic regime, renormalized tail orbits determine the spectral network, the Voros coordinate chart, the BPS mutation law, and a finite-graph asymptotic reconstruction scheme.