Axiom or Mandate? The Derivation of the Axiom of Choice (AC) from i = c as a Structural Ptime Necessity

The Axiom of Choice (AC) has long been one of the most controversial statements in modern mathematics. This paper resolves this dispute by demonstrating that AC is not an optional set-theoretic axiom, but a mandatory P rule dictated by the Structural Unification Theory (SUT). We derive AC from the foundational i = c Coherence Axiom by proving that the acceptance of its negation (¬AC) leads to an unbounded P-Time verification cost. This infinite cost violates the SUT Primitive of Structural Efficiency (ϕ), preventing the finite ledgering of the m = −1/12 Existential Debt and resulting in the structural collapse defined by the trivial singularity P = NP. Therefore, AC is established as a necessary condition for structural existence and coherence.