The preceding paper of this series showed that, in a bounded resurgent-wild regime, a coordinate-free renormalized tail orbit determines a bounded alien character point classifying the corresponding resurgent quantum packet. That result is already an exact moduli theorem for individual packets, but it still lives at representation level. A character point is the shadow of a deeper object: the tensor category of all bounded packets carried by the same local singular geometry and the affine group scheme acting on every realization simultaneously. The purpose of the present paper is to reconstruct that deeper object directly from the orbit. We work on admissible ramified charts of bounded ADE-type and smooth simple-pole branch models, under six standing restrictions: finitely many active blocks; an ambient finite action monoid in the Borel plane closed under the observable tensor operations; simple algebraic-logarithmic singular jets of bounded order; one-summability away from finitely many singular directions; bounded geometric wall complexity; and exponential-scale separation sufficient to order the observed bridge contributions. These hypotheses are deliberately strong. They exclude dense Borel webs, unbounded tensor growth, and globally conditioned arbitrary-noise numerics, but they still allow genuinely nonabelian alien transport. The first main theorem proves tensor closure. The category of bounded packets is rigid, exact, and -linear, and it admits canonical median and formal fiber functors. The second main theorem is an exact Tannakian reconstruction theorem: the packet category is neutral Tannakian and canonically equivalent to the category of finite-dimensional algebraic representations of an orbit Galois group scheme \ (), \ where is the prounipotent alien radical generated by orbit-recovered bridge symbols, is the bounded geometric transport group, and is the formal weight torus encoding irregular and logarithmic weights. The third main theorem identifies the comparison object between the formal and median fibers as a principal alien period torsor \ _=^ (_, _). \ Its representation quotients recover exactly the alien character varieties of the preceding paper, so the latter become concrete shadows of a universal Tannakian object rather than isolated moduli spaces. The fourth main theorem is quantitative. For every fixed depth M, finitely many orbit probes, wall probes, and Borel contour samples reconstruct the truncated Hopf algebra ₎ₑ₁^ M and the truncated comparison torsor point up to error \ (N^-1+N+N+N+₍, ₌). \ The detector is explicitly local and finite-horizon; it is not claimed to be a globally conditioned numerical algorithm under arbitrary noise. Conceptually, the paper upgrades the orbit theory from exact classification by character points to exact reconstruction of the full Galois symmetry object. Within the bounded regime treated here, coordinate-free renormalized tail orbits determine not only the packet and its alien transport, but the neutral tensor category to which it belongs, its wild fundamental group scheme, and its comparison torsor.
Mohammad Abu-Ghuwaleh (Tue,) studied this question.