Full Action: The Gauss–Bonnet Topological Sector and the Two-Channel Structure of the Holographic Paradigm

Attention: This version is preliminary and contains errors that will be corrected in subsequent publications. A full action S_FQHP = S_grad + ζ(ε)·S_D + w·S_GB is proposed—the fundamental functional of the fractal quantum holographic paradigm (FQHP), unifying the gravitational, barrier, and topological sectors. It is shown that the historically first (3+1)-dimensional version of the action possessed six structural deficits: absence of a hidden sector, absence of a carrier of the N^+ ↔ N^- channel, topological inertness of the Gauss–Bonnet term, an incorrect carrier of the scalar sector, an unsplit coupling to curvature, and absence of scale dependence. The minimal extension eliminating all six deficits is a five-dimensional shell: in 5D the Gauss–Bonnet term becomes the senior Lovelock term with second-order equations of motion without ghosts. After compactification of the fifth dimension, a topological trace remains—the transition weight w = e^(−8π·κ_top) ≈ 3.13 × 10^−8 at κ_top = 11/16. The barrier channel is realized by the hyperfield Φ = (φ + iχ)/√2 on a fractal manifold of dimension D_1 = 2 + α_1, α_1 = Ω_b = 0.0493, with the Caputo fractional covariant derivative of order 1 + α_1 = 1.0493 and running coefficient ζ(ε) = (ℓ_P/ε)^α_1. An effective expression is obtained for the coupling constant κ_D = c_η·η_total·v²/2, free from the Planck factor. The topological sector is shown to have a two-level structure: dynamics in 5D (Lovelock tensor) and a topological weight w in the effective theory on the barrier. The work distinguishes between the topological constant κ_top = 11/16 and the static conformal constant ξ_conf = 5/16, related by the matching rule κ_top + ξ_conf = 1, and introduces the dynamic conformal coupling ξ_dyn(η). An attribution table separates standard components of the action from original constructions of the present work. Limiting cases and numerical consequences amenable to testing are presented, including comparison with the isotropic distance measurement DESI DR1. Keywords: fractal quantum holographic paradigm, full action, Gauss–Bonnet term, Lovelock theory, five-dimensional shell, topological transition weight, conformal coupling, Caputo fractional derivative, informaton, fractal barrier, Kaluza–Klein compactification, Scherk–Schwarz twist.