Explanation of the theorem JcostN_eq_Jcost_aggregate in IndisputableMonolith.Cost.Ndim.Core
(1) In plain English, the theorem states that for any dimension n and vectors α (weights) and x (positive values), the n-dimensional cost JcostN(α, x) equals the scalar Jcost applied to the aggregate of α and x. This aggregate is the exponential of the weighted sum of logarithms of the components of x.
(2) In Recognition Science, this equivalence ensures that the cost defined by the Law of Logic extends uniformly to multi-dimensional cases. It allows all scalar properties of Jcost (non-negativity, symmetry under inversion) to carry over directly, supporting parameter-free derivations in higher-dimensional recognition structures.
(3) The formal statement reads: For all natural numbers n and vectors α, x : Vec n, JcostN α x = Jcost (aggregate α x). Here Vec n is the type of functions from Fin n to reals. The proof is by reflexivity (rfl), showing the equality holds by definition after unfolding JcostN as JlogN α (logVec x) and aggregate as exp(dot α (logVec x)).
(4) Visible dependencies include the definitions of aggregate, JcostN, JlogN, dot, logVec, and hadamard operations in the same module. It is marked as a simplification rule. Related theorems in the source are aggregate_pos, which proves the aggregate is positive, and JcostN_nonneg, which applies this equality to show non-negativity. The module imports IndisputableMonolith.Cost for the base Jcost.
(5) This declaration does not prove the uniqueness of the J-cost functional or its forcing from the Law of Logic. It also does not address physical applications, empirical predictions, or connections to other RS structures like spacetime emergence.