P43 V1.2 The Origin of G_eff(ρ) = G_N exp(-ρ/ρ_c): Rigorous Derivation from DBI Brane Dynamics and Gauss-Codazzi Projection

"Erratum minor" v1.2 (April 3, 2026): Enhanced exposition and rigor over v1.0 (no changes to core physics): 1. DBI kinetic squaring: Explicitly shown ∂ρ κ²₄ ∝ [∂ρ ϕ]² emerges from nonlinear DBI structure (not ad hoc) 2. Randall-Sundrum ansatz: Justified why metric takes form Eq. (2) and how modulus field ϕ represents brane position 3. ODE origin: Explained why dϕ/dρ = −1/(2ρ_c)ϕ in overdamped limit and factor 1/(2ρ_c) physical meaning (characteristic damping rate) 4. Exponent factor of 2: Clarified how 1/(2ρ_c) in ODE → 1/ρ_c in G_eff via squaring mechanism [ϕ]² 5. P39 reconciliation: Explicit statement that P39 (full Gauss-Codazzi) and P43 (DBI IR limit) are complementary, both give same result 6. NEW Section…