Black Holes in TEBAC 9D/9D+: BH-I Static Admissible Branches, Horizon Consistency, and Export Readiness

This preprint develops the first black-hole module in the TEBAC 9D/9D+ program. Its scope is deliberately limited to the static nonrotating branch and is framed as a theorem-bearing intermediate closure result rather than a full black-hole theory. The paper defines the canonical black-hole branch datum, the static first branch, and the admissible horizon class, and proves the first near-horizon closure package for this setting. More precisely, the manuscript establishes: (i) off-horizon induced regularity for the static defect metric, (ii) a near-horizon normal form and a well-defined surface gravity for the admissible static branch, (iii) defect-bulk projection compatibility in the horizon collar, (iv) a frozen horizon spectral package with self-adjointness, compact resolvent, and heat-trace admissibility, (v) descent of the relevant T2 admissibility data to the black-hole horizon sector, (vi) a near-horizon effective closure theorem, (vii) a horizon-consistency theorem for the frozen TEBAC package, and (viii) export readiness of the resulting BH-I package for downstream modules. The results are intentionally restricted to the static admissible branch. Rotating branches, entropy theorems, information-transport statements, and interior/singularity reinterpretation are not claimed in this paper and are deferred to later modules. In this precise sense, the present manuscript positions BH-I as the static admissible-branch foundation for subsequent black-hole work in TEBAC 9D/9D+.