Explain the Lean theorem `JcostN_eq_Jcost_aggregate` in module `IndisputableMonolith.Cost.Ndim.Core`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

The Lean theorem JcostN_eq_Jcost_aggregate formally connects multi-dimensional cost to the fundamental scalar cost.

1. Plain English Translation

The declaration states that computing the $n$-dimensional recognition cost of a coordinate vector $x$ given a weight vector $\alpha$ is mathematically identical to first compressing those coordinates into a single scalar value—the logarithmic aggregate—and then applying the standard 1D scalar cost function. The aggregate is mathematically equivalent to the weighted geometric product $\prod x_i^{\alpha_i}$.

2. Significance in Recognition Science

In RS, all physics derives from a single invariant functional equation for cost. When the framework scales from single scalar recognition events to multi-component ($n$-dimensional) systems, it cannot introduce arbitrary new cost functions without breaking the framework. This theorem provides a structural MODEL ensuring that higher-dimensional cost is strictly tethered to the fundamental 1-dimensional scalar Jcost. By routing through a logarithmic aggregate, RS perfectly preserves the reciprocal symmetries in multi-component spaces without ad-hoc parameterization.

3. Reading the Formal Statement

@[simp] theorem JcostN_eq_Jcost_aggregate {n : ℕ} (α x : Vec n) :
    JcostN α x = Jcost (aggregate α x) := by
  rfl

• @[simp]: A tag telling Lean's automated simplifier to replace the left side with the right side whenever it is encountered.
• {n : ℕ} (α x : Vec n): The theorem applies to any dimension $n$ (a natural number), taking two $n$-dimensional vectors of real numbers: $\alpha$ (the weights) and $x$ (the values).
• JcostN α x = Jcost (aggregate α x): The core equality. It states that the multidimensional cost function JcostN evaluates exactly to the 1D Jcost applied to the aggregate of $\alpha$ and $x$.
• := by rfl: The proof. rfl stands for reflexivity, meaning the left and right sides are equivalent simply by unfolding their definitions.

4. Visible Dependencies

Within the supplied slice, the theorem directly depends on:

• aggregate: Defined as $\exp(\sum \alpha_i \ln x_i)$.
• JcostN: The multi-component cost, defined via JlogN and logVec.
• Vec: The $n$-dimensional real vector type defined as Fin n → ℝ.

The proof is closed, containing zero sorry gaps and invoking no external axioms.

5. What This Declaration Does Not Prove

• Why the scalar Jcost is unique: This theorem assumes the scalar Jcost function. The proof that J(x) = (x + x⁻¹)/2 - 1 is the unique reciprocal-symmetric cost is a THEOREM (washburn_uniqueness_aczel or T5), which the primer notes is established in Cost.FunctionalEquation (not in this specific slice).
• The physical meaning of $\alpha$: It does not assign a physical interpretation (like mass or scaling dimensions) to the weight vector $\alpha$; it merely provides the mathematical architecture for multi-dimensional aggregation.
• Multi-dimensional reciprocity: While it establishes the identity bridge, properties like reciprocity under inversion and non-negativity are proved by downstream theorems in the same module (e.g., JcostN_reciprocal).