The declaration zero_cost_iff_aggregate_one states that for any dimension n and vectors α, x, the N-dimensional cost vanishes exactly when the aggregate equals one.
(1) In plain English: JcostN(α, x) = 0 if and only if aggregate(α, x) = 1. This equates zero multi-dimensional recognition cost with the ledger neutrality condition where the weighted product (via exp of the log-dot-product) balances to unity.
(2) In Recognition Science this identifies the zero-cost surface of the ledger: equilibrium configurations incur no net cost precisely when the aggregate is one, formalizing neutrality in higher-dimensional recognition.
(3) The formal statement reads: theorem zero_cost_iff_aggregate_one {n : ℕ} (α x : Vec n) : JcostN α x = 0 ↔ aggregate α x = 1. The proof constructs the biconditional by composing two prior equivalences in the same module.
(4) Visible dependencies in the supplied source are aggregate_eq_one_iff (which equates aggregate = 1 with dot-product of logs = 0) and zero_cost_iff_dot_zero (which equates zero cost with the same dot-product condition). The module imports Core but supplies no further certificates or axioms here.
(5) This declaration does not prove existence of any zero-cost vectors, uniqueness of the cost functional, or any link to the forcing chain, constants, or spacetime emergence.