Einstein--Hilbert Dynamics in the TEBAC 9D/9D+ Defect Formalism: Effective Field Architecture, Spectral Sources, and Closure Audit

This is a revised, expanded, and theorem-strengthened Zenodo version of the preprint on Einstein--Hilbert dynamics in the TEBAC 9D/9D+ defect formalism. The manuscript develops a claim-safe effective-field architecture in which the observed four-dimensional spacetime is interpreted as a defect-supported sector inside a higher-dimensional TEBAC background. The canonical geometric data include\[M₉=M₄× K₅,₅=S¹× T⁴,\]and the lifted realization\[M₁₃=M₄× K₅× F₄,\]with the observable defect metric written as\[γμν=ι*GAB.\] The central effective defect Einstein equation is formulated in the schematic form\[Gμν[γ]+Λeff\,γμν={8π Geff}{c⁴}\,Tˡᵒᶜμν+Eᵇᵘˡᵏμν+Qᵉᵐᵇᵉᵈμν.\] Compared with the original March 2026 version, this revised version substantially strengthens the mathematical structure, theorem map, non-circularity discipline, and public-release claim envelope. It adds a conditional defect-reduction theorem, a sector-separated variational uniqueness theorem, a variable-coupling Bianchi identity, a topological variational-silence theorem, Fredholm/Birman--Schwinger determinant criteria for defect fluctuations, determinant stability under trace-class perturbations, and a final closure-audit / acceptance-test layer. The paper distinguishes local defect sources, internal spectral sources, moduli contributions, projected bulk corrections, embedding corrections, and topological/invertible admissibility data. The topological sector is treated as a global consistency and admissibility filter, not as an automatic local stress-energy source unless an explicit local variational contribution is proved. The revised version also includes a methodological spectral-determinant bridge to 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 Einstein dynamics. This preprint does not claim that all effective gravitational constants, moduli, threshold terms, correction tensors, or low-energy observables have already been numerically extracted in final scheme-independent form. It is intended as a referee-readable, public-release gravity architecture module within the broader TEBAC 9D/9D+ research programme.