Algebraic Closure and Composite Chord Symmetry of the Regular Nonagon via Cyclotomic Polynomial Resultants in Lean 4 with Comparator

Abstract

Standard treatments of regular polygons rely heavily on continuous transcendental functions, obscuring the underlying discrete algebraic structures. This paper presents a first-principles algebraic derivation that isolates the composite $3 \times 3$ symmetry of the regular nonagon ($n=9$). By combining cyclotomic field extensions with polynomial resultants over $\mathbb{Z}[x]$, we construct a closed-form identity linking composite chord bundles directly to the algebraic cubic subfield generator $x^3 - 3x + 1 = 0$ without invoking trigonometric approximations.