Scattering-Diagram Completion from Coordinate-Free Renormalized Tail Orbits:\ Broken-Line Theta Functions, BPS Chamber Atlases, and Finite-Seed Recovery

The preceding paper of this series proved that, in a bounded phase-generic simple-branch regime, coordinate-free renormalized tail orbits determine chamberwise spectral networks, their abelianized Voros coordinates, and the simple-wall Kontsevich--Soibelman mutation factors across isolated BPS phases. That was already a substantial strengthening of the global wild Riemann--Hilbert reconstruction theorem, but a serious structural gap remained. Simple wall-crossing data do not by themselves produce a globally consistent chamber atlas: composite outgoing walls can be forced by consistency, and canonical theta coordinates arise only after one completes the initial wall data to a full scattering diagram. A strong referee can therefore ask whether the orbit determines only elementary BPS jumps, or the entire completed scattering geometry that organizes them. This paper answers that question in a sharp degree-bounded regime. We remain in the coordinate-free smooth simple-pole branch class and the phase-generic spectral-network regime of the previous paper. On a fixed compact design interval of phases we assume that only finitely many primitive simple BPS packets are active, that their charges lie in a positive cone of a finite-rank skew lattice, and that the scattering completion is degree-finite on the bounded window under study. Under those hypotheses the renormalized tail orbit determines a canonical consistent scattering diagram, its chamber atlas, and the corresponding broken-line theta functions. The first main theorem is an orbit-to-scattering closure theorem. Starting from the primitive incoming walls extracted from one-sided simple BPS packets, there exists a unique consistent completed scattering diagram \ (D\), up to the usual equivalence, whose path-ordered products coincide with the orbit continuation maps between chamberwise Voros charts. Thus the orbit determines not only elementary wall factors but the completed wall structure forced by higher-order compatibility. The second main theorem is a broken-line theta reconstruction theorem. For every integral tropical exponent \ (m\) and every chamber \ (C\) of \ (D\), the orbit determines the theta function \ (₌, ₂\) defined by the broken-line sum in the completed diagram. These theta functions transform by the diagram automorphisms, furnish canonical chamber coordinates, and separate the sparse regime from the genuinely coupled regime: when all primitive active charges commute, the theta functions collapse to monomials, whereas nontrivial broken-line corrections appear precisely when composite scattering walls survive. The third main theorem gives a scattering-atlas Torelli statement. The orbit determines, up to torus-equivariant isomorphism, the complete chamber atlas obtained by gluing algebraic tori along path-ordered products of \ (D\), together with the theta basis on each chamber. In particular, the orbit rigidly determines the finite chamber groupoid and all chamber transition maps on the compact design region. The fourth main theorem is a quantitative finite-seed recovery theorem. Fix a degree cutoff \ (K\), a finite chamber graph, and a finite set of theta labels. Then finitely many orbit probes recover all wall functions up to degree \ (K\) and the corresponding theta coefficients with error \ \! (N^-1/p_*+N+N+N+K), \ where \ (p_*\) is the maximal active Poincar\'e rank, \ (N\) is the discretization defect, \ (N\) the modeling defect, \ (N\) the measurement floor, and \ (K\) the cutoff tail. The theorem is explicitly local finite-horizon asymptotic recovery; it is not claimed as a global arbitrary-noise numerical algorithm. Conceptually, the paper upgrades chamberwise abelianization to completed scattering geometry. In the stated regime, renormalized tail orbits determine the consistent scattering diagram, the broken-line theta basis, the chamber atlas, and a finite-seed quantitative recovery scheme.