TEBAC 9D/9D+ Black-Hole Sector V: Topological Core Geometry, APS-Protected Information Channels, and a Candidate Resolution of the Singularity Problem

This preprint develops the fifth black-hole module in the TEBAC 9D/9D+ program. Its scope is deliberately restricted to the static admissible branch already organized in BH-I, refined geometrically in BH-II, entropy-structured in BH-III, and transport-extended in BH-IV. Within that precise regime, the manuscript constructs a theorem-bearing interior-core package in which the standard point-singularity picture is replaced by an admissible topological core geometry with bounded curvature invariants, smooth core extendibility, a compact quaternionic rotational sector, and an APS-type boundary index package. A central result of the paper is the construction of a topological information-conservation framework at the level of index-protected zero-mode channels. In this formulation, the core transition is governed not by uncontrolled collapse to a singular point, but by a controlled higher-dimensional interior package whose boundary spectral asymmetry and APS index remain stable under admissible branch-preserving deformations. The manuscript also introduces a strict singularity audit. Rather than claiming a blanket refutation of the classical Penrose–Hawking singularity theorems in their native hypothesis class, the paper argues that the admissible BH-V core branch lies outside that standard hypothesis package and therefore supports a different interior outcome. In this precise sense, the work develops a mathematically controlled candidate route to singularity resolution in the black-hole sector of TEBAC 9D/9D+. This version further includes a conservative physical comparison layer, connecting the BH-V core picture to horizon-scale imaging constraints, ringdown-scale strong-field data, and external uniqueness/comparison filters of Yazadjiev type. The result is an intermediate but ambitious theorem-bearing preprint that suggests TEBAC 9D/9D+ may provide a viable path toward resolving the black-hole singularity problem without sacrificing mathematical discipline.