«Instant Inversion Formula for Symmetric Elements in Clifford Algebras (R,C,H,O)

We present and prove a universal closed-form formula for the instantaneous computation of the inverse element for the symmetric hypercomplex numbers of the type S = a(1 + e₁ + … + eₙ). The formula, S⁻¹ = (1/(a(n+1)))(1 – e₁ – … – eₙ), is rigorously validated for the four normed division algebras (ℝ, ℂ, ℍ, 𝕆) and generalized to all Clifford algebras Cl(0, n-1) with anti-commuting imaginary units. This result bypasses the need for norm computation and conjugation, offering an O(1) algorithmic advantage and revealing a fundamental structural symmetry within these algebras.