Explanation of aggregate_eq_one_iff

(1) Plain English

The theorem asserts that, for vectors α and x in any finite dimension n, the scalar aggregate α x equals exactly 1 if and only if the weighted sum dot α (logVec x) equals zero. In other words, the aggregate condition holds precisely when the log-weighted inner product vanishes.

(2) Why it matters in Recognition Science

This equivalence sits at the neutrality surface of the N-dimensional ledger. It directly links the aggregate function to the vanishing of the weighted log sum, which is the condition for zero J-cost in the multi-dimensional extension. The same module uses it to prove zero_cost_iff_aggregate_one, connecting ledger neutrality to the cost-minimizing property that forces recognition structure.

(3) How to read the formal statement

The declaration reads:

theorem aggregate_eq_one_iff {n : ℕ} (α x : Vec n) :
    aggregate α x = 1 ↔ dot α (logVec x) = 0

• {n : ℕ} is an implicit dimension parameter.
• α x : Vec n are two n-dimensional vectors.
• The left side is the aggregate scalar; the right side is the dot product of α with the component-wise logarithm of x.
• The proof unfolds aggregate, rewrites the equality as an exponential equality, and applies injectivity of the real exponential.

(4) Visible dependencies or certificates in the supplied source

The theorem appears verbatim in IndisputableMonolith.Cost.Ndim.Neutrality. It is immediately reused by the sibling declarations zero_cost_iff_dot_zero and zero_cost_iff_aggregate_one within the same module. No further certificates or axioms are attached inside the supplied slice.

(5) What this declaration does not prove

It does not supply the definition of aggregate, Vec, dot, or logVec (those reside in the imported Core module). It does not address the uniqueness of the J-cost functional, the forcing chain from distinction to physics, or any empirical calibration. Those topics appear only in other supplied modules such as IndisputableMonolith.Foundation.RealityFromDistinction and IndisputableMonolith.Cost.Ndim.Neutrality itself.