TEBAC 9D/9D+: A Revised Foundational Framework for Higher-Dimensional Geometry, Defect Dynamics, Spectral Sources, and Dark-Matter Phenomenology

This is a revised, expanded, and theorem-strengthened Zenodo version of the first foundational preprint on the TEBAC 9D/9D+ framework. The manuscript reformulates the original short foundation paper as a claim-safe mathematical-physics framework for higher-dimensional geometry, defect-supported four-dimensional dynamics, internal spectral sources, topological admissibility, and dark-sector phenomenology. The revised version replaces the early “complete theory” framing with a more precise foundational architecture: TEBAC 9D/9D+ is presented as a structured research programme with conditional theorems, effective-field reductions, spectral-source bookkeeping, and phenomenological test templates. The canonical geometric architecture is written in terms of₉=M₄ K₅, ₅=S^1 T^4, with the lifted realization₁₃=M₄ K₅ F₄, ₄=S^4. \ The revised paper incorporates a defect-metric layer\_=^*G₀₁, conditional dimensional-reduction theorem, a defect-source theorem, an equivariant gauge-sector emergence proposition, a topological invariant theorem based on characteristic classes and third-Chern pairings, an internal spectral-source theorem, a Fredholm determinant stability layer, and a dark-sector rotation-curve theorem. The dark-matter/core-cusp discussion is retained from the original paper but reformulated in a claim-safe way. The NGC 1560 rotation-curve figure and the core-radius/halo-mass scaling relation are presented as phenomenological templates and test diagnostics, not as final dataset-level statistical confirmation. Full external validation requires systematic fitting across public rotation-curve datasets such as SPARC and LITTLE THINGS, with comparison against standard dark-matter and modified-gravity baselines. The revised version also includes a methodological spectral-determinant appendix connecting the foundation module with the broader TEBAC 9D/9D+ programme, including the RH and BSD modules. This connection is structural and non-deductive: RH/BSD determinant methods are used only as part of the common TEBAC operator-spectral methodology, not as physical inputs for deriving dark matter, gravity, or experimental predictions. This preprint does not claim final experimental confirmation of TEBAC 9D/9D+, complete numerical extraction of all effective constants, or closure of all phenomenological sectors. It is intended as a public-release, referee-readable foundation module for the broader TEBAC 9D/9D+ research programme.