This preprint develops the second black-hole module in the TEBAC 9D/9D+ program. Its scope is deliberately restricted to the static admissible branch imported from BH-I. The manuscript does not claim a general black-hole thermodynamics theory, a rotating-branch theory, a Hawking-radiation theorem, an entropy theorem, an information-transport theorem, or an interior-core theorem. Instead, it closes the BH-II layer in theorem-bearing form by organizing the near-horizon effective geometry, the admissible static variation package, the intrinsic first-order variation laws, the correction class, the renormalized horizon-energy class, and the first-law-compatible horizon-side relation on the imported static branch. More precisely, the paper defines the horizon-effective geometry package and the admissible static variation family, proves that the surface-gravity variation and horizon-area variation are well defined intrinsically, constructs the first-law-compatible correction class and the renormalized horizon-energy class, proves the near-horizon effective geometry closure theorem, and proves the first-law-compatible correction-structure theorem together with the BH-III export-readiness corollary. In addition to the theorem-bearing layer, the preprint includes an explicitly marked illustrative physical interpretation of the correction term on the canonical internal branch \ (K₅=S¹ T⁴\). There the correction side is interpreted through a frozen Yukawa-type horizon profile, a lowest Kaluza--Klein scalar induced from the \ (S¹\) sector, and a compactification modulus response field, clarifying how the correction term can act either as positive horizon-energy dressing or as an energy-lowering contribution depending on the effective coupling signs. The paper should be read as a static-branch BH-II module inside the larger TEBAC 9D/9D+ black-hole program and as a noncircular export layer for the later BH-III entropy package.
Tosho Lazarov Karadzhov (Thu,) studied this question.