Internal Spectral Package and Four-Dimensional Coefficient Export in TEBAC 9D+. Subtitle: Admissible Spectral Reduction from a 13D Ambient Geometry

This preprint studies the spectral-reduction layer of the TEBAC 9D+ framework on a frozen 13-dimensional ambient product background \ (M₁₃=M₄ K₅ F₄\), with \ (K₅=S¹ T⁴\) and \ (F₄=S⁴\). The paper isolates the internal operator-theoretic core of the framework. From the induced internal geometry on \ (M₈₍ₓ=K₅ F₄\), it defines a Dirac-type operator and the associated Laplace-type operator, derives the corresponding self-adjoint spectral package using standard elliptic theory on compact manifolds, and formulates a quotient-stable coefficient-extraction mechanism for effective four-dimensional sectorwise coefficient data. The main result is a canonical quotient-level export statement: the admissible equivalence class of the frozen spectral-reduction datum determines a unique effective four-dimensional coefficient datum in the finite target space adopted in the paper. A finite-order heat-kernel factorization of the gravitational and local correction sector is also recorded at the truncation order used here. This manuscript is a focused mathematical-physics module note. It does not claim complete low-energy numerical matching, final renormalized parameter closure, or a canonical matter--antimatter asymmetry invariant theorem. Its purpose is to provide a mathematically cleaner bridge from higher-dimensional internal operator data to exported four-dimensional coefficient sectors, while clearly identifying the remaining analytic and phenomenological burdens.