TEBAC 9D/9D+ Black-Hole Master Paper I--V: Static Admissible Branches, Horizon Mechanics, Entropy Corrections, Information Transport, and Admissible Interior-Core Singularity Audit

This master paper assembles the first five modules of the TEBAC 9D/9D\ (^+\) black-hole program into a single integrated theorem-bearing document on the static admissible branch. The manuscript organizes the chain-I-II-III-IV-V a disciplined modular framework for static black-hole geometry, horizon compatibility, corrected entropy structure, information-channel bookkeeping, and admissible interior-core singularity audit. \ (BH-I\) constructs the canonical admissible static branch, the admissible horizon notion, the near-horizon normal form, the horizon spectral package, and the non-circular export interface. \ (BH-II\) develops the near-horizon effective geometry and first-law-compatible correction layer. \ (BH-III\) formulates the corrected entropy package, separating spectral, topological, and admissibility contributions. \ (BH-IV\) organizes the enlarged-state-space transport and information-balance interface. \ (BH-V\) supplies the admissible static interior-core module: a reduced core system, bounded-curvature and smooth-extendibility criteria on the declared admissible continuation domain, a compact core-boundary Dirac/APS package, and an index-protected topological channel count. The main output is the master static-branch export package\ (A₁₇-₈, A₁₇-₈₈, A₁₇-₈₈₈, A₁₇-₈ₕ, A₁₇-ₕ), with the exported core package\ (CoreGeom, CoreCurv, CoreBnd, CoreDirac, APSCore, TopInfo, CoreExt, SingAudit). \ The paper is deliberately scope-controlled. It does not claim a rotating-branch theorem, a fully dynamical collapse theorem, a Page-curve theorem, evaporation unitarity, direct observational confirmation, or a universal microscopic information-recovery theorem. The theorem-bearing claim is restricted to the declared static admissible branch and its exported module interfaces. Within that regime, the classical \ (r=0\) locus admits a no-point-singularity reading through the admissible \ (BH-V\) core audit: bounded curvature, smooth admissible core extension, and an APS-protected diagnostic channel count. This version includes a normalized bibliography, cross-linked references throughout the text, cleaned module integration, a global theorem-bearing/interpretive/observational layer statement, and TikZ-based explanatory diagrams.