Canonical Cluster-Mirror Reconstruction from Coordinate-Free Renormalized Tail Orbits:\ Theta Algebras, Tropical Potentials, and Finite-Polytope Recovery

The preceding paper of this series proved that, in a bounded phase-generic simple-branch regime, coordinate-free renormalized tail orbits determine the completed scattering diagram, its chamber atlas, and its broken-line theta functions. The missing step was the mirror object itself: one still needs the theta multiplication law, the positive potential, and the polyhedral filtration controlling compactification. This paper provides that step in a positive finite-type regime. Assuming positive wall normalization, a finite-rank charge lattice, a finite seed-frozen generating system, and degree-finite broken-line completion on the observation window, we show that the orbit determines the full truncated cluster-mirror package. Our first theorem is an orbit-to-theta-algebra closure theorem: for every cutoff \ (K\), the orbit determines a unique commutative associative algebra on the truncated theta basis, and the inverse limit yields a complete theta algebra \ (\). The second theorem constructs from the frozen packets a canonical positive potential \ W=₈=₁ˢ ᵢ₅㶁, ᵢ>0, \ whose tropicalization \ (W\) is a convex integral piecewise-linear function determining compact rational polytopes \ Pₜ=\x N_{: W (x) t\}. \ This isolates a sharp sparse-versus-coupled dichotomy: commuting primitive charges give toric polyhedra, while noncommuting charges force bends and new facets. Our third theorem is a canonical cluster-mirror Torelli theorem. The orbit determines, up to torus-equivariant isomorphism, the package \ (, , W, W, Y, YW), \ where \ (\) is the theta basis, \ (W\) the Rees algebra of the polyhedral filtration, \ (Y= () \) the formal affine mirror, and \ (YW= (W) \) the associated compactification. Our fourth theorem is a quantitative finite-polytope recovery statement: finitely many orbit probes recover the truncated theta structure constants, potential coefficients, and facet data with error \ \! (N^-1/p_*+N+N+N+K). \ If a facet-separation margin is present, the combinatorial type of the truncated polytope stabilizes exactly for all sufficiently large \ (N\). Thus, in the stated regime, renormalized tail orbits determine not only the scattering diagram but the canonical cluster-mirror algebra and its positive compactification geometry.