The Emergent Priority of the Multiplicative Lattice: A Rigorous Proof that the Additive Group Emerges from the Divisibility Poset

Mathematics has historically assumed addition to be the primary arithmetic operation, with multiplication defined as repeated addition. This paper overturns that foundational order using an emergence framework. We prove two theorems. The Emergence Existence Theorem demonstrates that the subtractive poset (additive group) \( A = (N, ≤) \) strictly emerges from the divisibility poset (multiplicative lattice) \( D = (N^+, ) \) through a purely algebraic construction---the exponential cloud isomorphism, Grothendieck group completion, and a canonical summation projection---without presupposing real numbers or any additive structure. The Emergence Asymmetry Theorem demonstrates that the reverse is impossible: no emergence functor can produce \( D \) from \( A \), because any order-preserving image of a total order remains a total order, while \( D \) is not. Consequently, the ontological order is definitively established: \( D ≥ A \), and the multiplicative identity \( 1 \) strictly precedes the additive identity \( 0 \). This result constitutes a refoundation of arithmetic, replacing the Peano-ZFC succession with a ``relation precedes existence'' bedrock in which the multiplicative lattice is the primitive structure.