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 \[ W=∑ᵢ₌₁^s λ_iϑf_i, λ_i>0, \] whose tropicalization \(φ_W\) is a convex integral piecewise-linear function determining compact rational polytopes \[ P_t=\{x∈ N_{}:φ_W(x)≤ t\}. \] 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 \[ (,Θ,W,_W,{Y},Ȳ_W), \] where \(Θ\) is the theta basis, \(_W\) the Rees algebra of the polyhedral filtration, \({Y}=()\) the formal affine mirror, and \(Ȳ_W=(_W)\) 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 \[ \!(N-1/p_*+δ_N+η_N+_N+τ_K). \] 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.