Paper CLXXXIII: Many-Body Dynamical Localization as Balanced SSFP in Momentum The Gudermannian Function as Brane-Bulk Bridge and G₂ Permutohedron Structure

We show that many-body dynamical localization (MBDL) in the kicked Lieb-Liniger model [Yang et al., PRL 136, 123402 (2026)] is the exact many-body analog of balanced steady-state free precession (bSSFP) in magnetic resonance imaging, with the Gudermannian function gd(x) = arctan(sinh(x)) serving as the canonical mathematical bridge between the flat-space (brane) and hyperbolic-space (AdS₅ bulk) descriptions. Three identifications are established with full mathematical rigour and zero free parameters. (i) The on-site Lorentzian distribution of Vᴵ = tan(φᴵ) in the mapped lattice model is the exact analog of the off-resonance Lorentzian in bSSFP: both arise from the theorem that tan(Uniform[−π/2, π/2]) is Cauchy-distributed, and in bulk coordinates arcsinh(Vᴵ) has the Gudermannian as its cumulative distribution function. (ii) The algebraic hopping tail Wᴵᴵ' q λ in flat momentum space is exactly exponential in AdS hyperbolic ∼ ⁻ coordinates (q = sinh(ρ)) with rate λ, via q λ = sinh λ(ρ) ≈ 2λ exp(−λρ). (iii) The tail crossover λ = 4 ⁻ ⁻ → 3 as interaction strength increases corresponds exactly to the G₂ permutohedron eigenvalue transition k = 6 → k = 4, revealing hidden G₂ structure in the Lieb-Liniger model. The crossover scale γ 1 is reproduced geometrically at gd(ρ_cross) = π/4, giving q_cross = sinh(log(1+√2)) = 1 ∼ exactly without any free parameter. Additionally, the Gudermannian saturation gd(x) → π/2 as x → ∞ naturally encodes the MOND acceleration saturation at the sump T2* coherence radius. Three falsifiable predictions are stated. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.