The Primordial Algebra: The {-1, 0, 1} Generation and C-Closure Theorem

We establish that the set S = {−1, 0, +1} is the unique finite algebra satisfying the conditions of identity, reproduction, and cancellation. Beginning from three primordial states; Creation, Destruction, and Potential. We demonstrate that demanding closure, totality, and stability forces exactly one algebraic structure. This structure generates all subsequent number systems through iteration and extension, terminating uniquely at the field of complex numbers \( \mathbb{C} \) . We prove that Euler’s Identity, \( e^{i\pi} + 1 = 0 \) serves as the formal termination certificate of this extension sequence, resolving entirely to the elements of the primordial alphabet. The central result: S = {−1, 0, +1} is the unique algebra that is complete before extension and generative after it.