This preprint develops the fifth black-hole module in the TEBAC 9D/9D+ program. Its scope is deliberately restricted to the admissible interior-core class on the static admissible branch imported from the earlier black-hole modules. The manuscript does not claim a theorem beyond that declared class, a rotating-branch theorem, a fully dynamical collapse theorem, a universal microscopic information theorem, a final Penrose--Hawking replacement theorem, or a theorem-bearing observational ringdown result. Instead, it closes the BH-V layer in theorem-bearing form on the admissible interior-core class by assembling the interior-core geometry package, the bounded-curvature chain, the smooth \ (C^\) extension package across \ (r=0\), the APS/Fredholm index layer, the topological-channel theorem, the singularity-reinterpretation theorem, and the master closure theorem. More precisely, the paper derives the admissible interior-core class from the declared field-equation, regularity, and symmetry package; constructs the horizon-to-core matching layer; proves local interior existence together with the sufficiency of the core neighborhood for the BH-V claims; proves the reduced uniqueness and energy-condition compatibility layers; establishes the boundedness of the decisive curvature invariants; proves smooth \ (C^\) extendibility across \ (r=0\) within the admissible interior-core class; constructs the APS/Fredholm index package and the associated topological information-channel layer; and proves that, in the declared scope, the black-hole center is not represented by a point singularity but by a smooth, topologically controlled core. In this precise sense, the manuscript gives a theorem-bearing nonsingular-core result for the admissible static interior branch of the TEBAC black-hole sector. The classical point-singularity picture is replaced, within the declared scope, by a smooth interior-core geometry carrying the imported topological package. The observational ringdown discussion included in the manuscript is explicitly interpretive and non-theorem-bearing; it is provided only as a conservative comparison bridge and is not part of the closure claim. This preprint should therefore be read as the interior-core and singularity module of the TEBAC 9D/9D+ black-hole architecture, closed in theorem-bearing form on the admissible interior-core class and positioned downstream of BH-I--BH-IV.
Tosho Lazarov Karadzhov (Fri,) studied this question.