Holographic Bit–Mode Balance Regularization of Wheeler–DeWitt Cosmology: Determinant Running and Inflationary Closure

We present a sharpened Holographic Bit-Mode Balance (HBMB) quantum-cosmology framework in which the Wheeler-DeWitt problem is first deparametrized with a scalar internal clock and only then projected onto a holographically accessible sector of superspace. This avoids applying Feshbach-type machinery directly to the raw Hamiltonian constraint. The omitted high-l sector induces an accessible-sector self-energy. We show, however, that the overlap scaling must be interpreted carefully: in an explicit S^2 boundary-strip model the raw geometric strip overlap behaves as |V_l^strip|^2 proportional to lambda_l^(-1/2), while the stronger scaling relevant for the HBMB kernel emerges only after the strip state is decomposed into local tangential channels. Tangential isotropy then implies that the retained coarse mode couples only to the single uniform channel combination, so the local patch amplitude acquires the 1/sqrt(N_t) normalization algebraically and the channel matrix is rank one, where N_t denotes the number of local tangential strip channels at fixed l. The exactly telescoping kernel proportional to (2l + 1)/[l^2 (l + 1)^2] is therefore interpreted as an exactly summable representative of the correct K_l ~ l^(-3) asymptotic universality class, not as a unique microscopic kernel. We also perform the omitted-sector determinant expansion explicitly. The raw cutoff sum contains not only area-like terms but also an interface contribution proportional to L ln L, where L is the accessible multipole threshold; after subtracting the bulk and local code-boundary counterterms the renormalized residual omitted-sector determinant is Gamma_Q^ren(L) = Gamma_0 - (2/3) ln L - (31/360) L^(-2) + O(L^(-4)). From this logarithmic coefficient we partially fix the inflationary plateau parameters. Under a minimal unit-response closure the derived residual running coefficient b = 2/3 implies alpha = 1/(2b) = 3/4, while the end-of-inflation matching condition yields an asymptotic shift Delta_asy = sqrt(3)/2 approximately 0.866 and an exact matched value Delta_match approximately 0.819 once the subleading determinant term is retained. Scalar and tensor perturbations are then formulated on this background, and reheating is represented by an HBMB-specific tail-decoupling source term. The first-principles core established here consists of the deparametrized projection, the asymptotic tail scaling, the explicit determinant expansion, the partial fixing of alpha (the inflationary plateau coefficient, not the fine-structure constant) and Delta (the matching shift parameter), and the resulting plateau inflationary closure. The microscopic code-averaging dynamics beyond leading order and the reheating efficiency remain open.