The Gauss-Bonnet Reformulation of g* = 4π — An Exact Equivalence, Its Physical Interpretation, and the Genuine Open Problem

Paper 45 of the Leveille Framework series. A systematic audit of the three independent proofs required to elevate the g* = 4π geometric nucleation barrier (established in Paper 44) from observation to derivation. Two routes are ruled out by exact calculation: the diffuse/sharp crossover condition yields ρc = 2√2·ln (3) = 3. 107, not 3. 000, and the Δ > 0 living requirement rules out the degenerate sharp-interface limit but does not uniquely select ε = 2/9. One exact identity — wₚlanar = √2·ln (3) for the planar double-well interface width — emerged as a genuine new result. The sole remaining open path is the Gauss-Bonnet topological reduction, connecting the Cahn-Hilliard saddle-point barrier to the Euler characteristic of the bounding surface. The Leveille Framework posits that exact zero tolerance is not physically realizable (Δ > 0) and derives consequences across domains.