Alfic Algebra I : A Unit-Relative Arithmetic System

We introduce Alfic Algebra, a number system in which the additive identity is 1 rather than 0, defined by the transformation x = x − 1. This shift reflects a foundational principle: that every quantity carries an intrinsic unit derived from its own structure, and that imposing a universal zero-based unit introduces artificial infinities that are artifacts of measurement mismatch rather than features of the underlying reality. We present six core theorems with full proofs and demonstrate that Alfic Algebra forms an Abelian group under Alfic addition with identity element e = 1. A key result is that the Alfic squaring of 2 yields 3, which serves as the structural seed for a companion ternary geometry system (Nonic Geometry). Unlike structural unification approaches such as Alpay Algebra, Alfic Algebra addresses a more fundamental question: what is the natural unit from which arithmetic itself should be built? The answer may directly affect how algebraic structures can and should be unified.