Explain the Lean theorem `zero_cost_iff_dot_zero` in module `IndisputableMonolith.Cost.Ndim.Neutrality`. 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.

(1) Plain English

The theorem zero_cost_iff_dot_zero states that, for any dimension n and any two n-vectors α and x, the N-dimensional J-cost JcostN(α, x) equals zero if and only if the weighted sum of the logarithms of the components of x (with weights from α) is exactly zero. In other words, the total recognition cost vanishes precisely when the ledger is balanced in logarithmic scale.

(2) Role in Recognition Science

In RS the J-cost is the unique reciprocal-symmetric cost function forced by the Law of Logic (see the forcing chain in the primer). Zero cost is the neutral point of the ledger. This theorem therefore supplies the precise algebraic condition under which a multi-dimensional recognition event incurs no net cost, linking the cost functional directly to the aggregate-neutrality surface. It is used to prove the companion statement zero_cost_iff_aggregate_one, which equates zero cost with the aggregate equaling one.

(3) Reading the Formal Statement

 theorem zero_cost_iff_dot_zero {n : ℕ} (α x : Vec n) :
     JcostN α x = 0 ↔ dot α (logVec x) = 0 :=
   JcostN_eq_zero_iff α x

• {n : ℕ} fixes the dimension.
• α x : Vec n are the weight vector and the state vector.
• JcostN α x = 0 is the left-hand side (zero total cost).
• dot α (logVec x) = 0 is the right-hand side (weighted log-sum vanishes).
• The proof is a direct appeal to the auxiliary lemma JcostN_eq_zero_iff (imported from Core).

(4) Visible Dependencies in the Supplied Source

The module IndisputableMonolith.Cost.Ndim.Neutrality imports IndisputableMonolith.Cost.Ndim.Core and immediately defines three related theorems:

• aggregate_eq_one_iff proves aggregate α x = 1 ↔ dot α (logVec x) = 0.
• zero_cost_iff_dot_zero is the statement above.
• zero_cost_iff_aggregate_one chains the two biconditionals to obtain JcostN α x = 0 ↔ aggregate α x = 1.

No sorry appears in this module; the proofs are short constructor applications.

(5) What the Declaration Does Not Prove

It does not define JcostN, Vec, dot, logVec or aggregate (those live in the imported Core module). It does not prove the uniqueness of the J-cost functional itself, nor any physical interpretation of the neutrality surface, nor any link to the forcing chain or constants derived elsewhere. Those connections are established in other modules (e.g., IndisputableMonolith.Foundation.RecognitionForcing and IndisputableMonolith.Cost.FunctionalEquation).