Black Holes in TEBAC 9D/9D+: BH-III Spectral-Topological Entropy Package, Corrected Entropy Variation, and Static-Branch Export Readiness

This preprint develops the third black-hole module in the TEBAC 9D/9D+ program. Its scope is deliberately restricted to the static admissible branch imported from BH-I and refined by BH-II. The manuscript does not claim a general black-hole thermodynamics theory, a full observational fit, a rotating-branch theorem, a Hawking-radiation theorem, an information-transport theorem, or an interior-core theorem. Instead, it closes the BH-III layer in theorem-bearing form by constructingthe spectral-topological entropy package, the corrected entropy functional, the leading area-law decomposition, the admissible entropy-variation package, the corrected entropy-variation theorem, and the export-ready entropy layer for downstream BH-IV use. More precisely, the paper defines an intrinsic entropy-density package on the horizon, with area, spectral, topological, and renormalized slots, and constructs the corrected entropy functional as an integral over the horizon cross-section. It proves that the entropy functional is well defined on the static admissible branch, proves the intrinsicity of the topological correction class, derives the leading decompositionH^ren=AreaH4G₄₅₅+SH^spec+SH^top+SH^ct, the admissible entropy-variation family, and proves the corrected entropy-variation theorem together with the theorem-level closure of the spectral-topological entropy package and the BH-IV export-readiness corollary. In addition to the theorem-bearing layer, the preprint contains an explicitly marked interpretive physical bridge. This bridge is non-export and non-theorem-bearing. It is used only to relate the static-branch entropy package conservatively to black-hole phenomenology and to visualize the TEBAC bulk-to-horizon reading. The manuscript also includes an artist's rendering of blurred Dirac-bulk resonance dressings around the horizon throat, interpreted heuristically as smeared time-resonance structures with dark-matter-like phenomenological appearance. This image has no evidentiary role in the formal proofs. Within the larger TEBAC 9D/9D+ black-hole program, the present BH-III module is intended to serve as the entropy-side bridge toward two later modules. The planned BH-IV module will address horizon-to-bulk information transport and dynamicalblack-hole response. Its intended scope is to define an admissible information-transport channel class between the effective \ (4\) D horizon sector and the higher-dimensional bulk sector, to formulate a theorem-bearing transport package on the static admissible branch, and to derive controlled correction targets for quasinormal-mode spectra that may later be compared with ringdown observations in a conservative way. In that future module, external constraint layers from established black-hole uniqueness, stability, and boundary-condition results are intended to be used as admissibility filters rather than as decorative citations. The planned BH-V module will address the interior-core geometry and the singularity question inside the TEBAC framework. Its intended scope is not merely philosophical reinterpretation, but the construction of a controlled core-regularity package: an admissible interior transition datum, a compactification-compatible core geometry, and boundedness/regularity targets for the effective interior sector as \ (r 0\). In this intended reading, the central black-hole region is to be studied not as a formal point of infinite density, but as a candidate transition regime toward the compact internal sector \ (K₅\), subject to theorem-bearing regularity tests. These future directions are programmatic and are stated here only to indicate the downstream role of BH-III inside the full TEBAC black-hole architecture. No claim is made in the present paper that BH-IV or BH-V has already been completed, nor thatthe information paradox or the singularity problem has already been solved here. The paper should therefore be read as a static-branch BH-III module inside the larger TEBAC 9D/9D+ black-hole program and as a noncircular export layer for the laterBH-IV information-transport package.