A Structurally Symmetric Reformulation of the Classical Legendre Relation

We present a structurally symmetric reformulation of the classical Legendre relation for complete elliptic integrals, specialized at the lemniscate modulus k = 1/\sqrt{2}. The result is a dual-hypergeometric representation of \pi expressed as a rational function of two convergent {}_2F_1 series evaluated at the elementary point z = 1/2. The formula does not achieve the convergence speed of Ramanujan–Sato series, but offers complete structural transparency: it requires no large modular invariants or obscure integer constants and exhibits a natural parameter-sign symmetry between the two companion series. We provide rigorous truncation error bounds with explicit constants, prove the formula from first principles, and contextualize it within the classical theory of elliptic integrals. All numerical computations are reproducible using the algorithm provided in the Appendix.