Asymptotic Solution of Fermats Last Theorem Using Trigonometric Analysis

This manuscript develops an asymptotic solution of the trigonometric system induced by Fermat's Last Theorem 1 . A hypothetical positive solution of x n + y n = z n determines an acute triangle and an associated right triangle, yielding tan 2 θ = (sin α/sin β) n . Fixed nonoscillatory roots collapse to zero, while regular interior accumulation for general fixed frequencies is confined to the level set sin (at)/sin (bt) = 1. For normalized Fermat angles, the unit-frequency region b ≤ 1 is geometrically impossible. On the ordered acute branch, the fast and slow regimes and the boundary layer are classified using a uniform Taylor remainder and the joint cluster set. The global multiplicity of critical roots remains open. Lean 4 verifies the fixed-frequency results, root bounds, the nonempty admissible interval, the exact boundary cosine reduction, the unit-frequency obstruction, and endpoint limit transfers.