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\[\mathrm{BH\text{-}I}\Longrightarrow\mathrm{BH\text{-}II}\Longrightarrow\mathrm{BH\text{-}III}\Longrightarrow\mathrm{BH\text{-}IV}\Longrightarrow\mathrm{BH\text{-}V}\]as a disciplined modular framework for static black-hole geometry, horizon compatibility, corrected entropy structure, information-channel bookkeeping, and admissible interior-core singularity audit. \(\mathrm{BH\text{-}I}\) 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. \(\mathrm{BH\text{-}II}\) develops the near-horizon effective geometry and first-law-compatible correction layer. \(\mathrm{BH\text{-}III}\) formulates the corrected entropy package, separating spectral, topological, and admissibility contributions. \(\mathrm{BH\text{-}IV}\) organizes the enlarged-state-space transport and information-balance interface. \(\mathrm{BH\text{-}V}\) 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\[\bigl(A_{\mathrm{BH\text{-}I}},A_{\mathrm{BH\text{-}II}},A_{\mathrm{BH\text{-}III}},A_{\mathrm{BH\text{-}IV}},A_{\mathrm{BH\text{-}V}}\bigr),\]together with the exported core package\[\bigl(\mathrm{CoreGeom},\mathrm{CoreCurv},\mathrm{CoreBnd},\mathrm{CoreDirac},\mathrm{APSCore},\mathrm{TopInfo},\mathrm{CoreExt},\mathrm{SingAudit}\bigr).\] 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 \(\mathrm{BH\text{-}V}\) 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.