Controlled Completion of Dual-Affine Entropy Geometry: Input-Output Born Structures, Compact Holonomy, and Relative-Entropy Generating Functions

We prove a controlled completion of dual-affine entropy geometry connecting a supplied amplitude-carrier complex structure, constructed tangent-bundle Born complex structures, non-Abelian compact transport, and endpoint action through typed relative-entropy reconstruction and defects. On the supplied Hermitian carrier, Umegaki relative entropy reconstructs the entropy potential up to affine gauge and the relative logarithmic cotangent class; diagonal jets yield the Bogoliubov-Kubo-Mori (BKM) metric, Hessian cubic, and dual connections, which construct Born lifts on real doubled statistical carriers. Determinant normalization selects a principal-log traceless Gibbs representative. A quantum channel induces support-reduced output geometry and a Bregman loss whose Hessian and cubic are metric and response losses with sequential cocycles; one CPTP map recovering every state of an embedded faithful model annihilates its restricted loss and intertwines its intrinsic geometry and holonomy. Proximal families yield Rényi and Hopf-Lax identities; BKM relaxation needs mobility and scale. From a separately chosen affine-invariant metric, the compact branch constructs canonical principal SU(n) and associated tangent PSU(n) transport, compared with BKM transport by connection and curvature defects rather than conjugacy. The indexed architecture is dimension-stratified: its general branches are available for each fixed finite n≥2 under their stated inputs, full BKM tangent holonomy is proved only for n≥3, and the qutrit and independently graded AIII packages occur only at n=3 and n=5, respectively. For qutrits, the tracial BKM metric-cubic pair has oriented stabilizer PSU(3); alignment with normalized affine-invariant holonomy lets the central cubic seed a distinct, unique parallel comparison cubic defining a global tangent-frame reduction. A fixed action conversion gives an exact endpoint canonical relation; the parameterized Rényi-Moreau family gives a branch-local scale-Hamiltonian isotopy. Mechanical interpretation needs inertia and a clock. Relative entropy does not construct the amplitude-carrier complex structure or supply physical calibrations.