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.

Here is a breakdown of the theorem zero_cost_iff_dot_zero from the IndisputableMonolith.Cost.Ndim.Neutrality module.

1. Plain English Translation

The theorem states that for an N-dimensional state, the total generalized cost is exactly zero if and only if the weighted sum of the logarithms of the state's components is zero. In other words, logarithmic deviations from unity perfectly balance each other out across all dimensions, resulting in no net cost.

2. Why it Matters in Recognition Science

In Recognition Science, cost structures define stable configurations. A foundational principle is that self-recognition (or a perfectly balanced state) carries zero cost, formalized in 1D by theorems like global_minimum_is_self_recognition. This theorem generalizes that principle to N-dimensional configurations. By proving that zero cost is mathematically equivalent to the logarithmic dot product vanishing, it characterizes the "ledger neutrality surface." On this surface, an N-dimensional system is structurally balanced.

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

• {n : ℕ}: The dimension of the system is a natural number n.
• (α x : Vec n): The system takes two vectors of length n. α represents the weights, and x represents the values.
• JcostN α x = 0: The multi-dimensional cost evaluates to exactly zero.
• ↔: "If and only if" (logical equivalence).
• dot α (logVec x) = 0: The dot product of the weight vector α and the element-wise natural logarithm of the value vector x is zero.

4. Visible Dependencies and Certificates

In the supplied source, the theorem is proved by delegating to JcostN_eq_zero_iff α x. This indicates that the foundational proof logic resides in the imported IndisputableMonolith.Cost.Ndim.Core module.

Within its own module, this theorem serves as a critical lemma to prove zero_cost_iff_aggregate_one, which establishes that zero cost is equivalent to the system's geometric aggregate being exactly 1. This equivalence is logically bridged by aggregate_eq_one_iff.

5. What This Declaration Does Not Prove

• It does not define the algebraic form of JcostN, logVec, or dot, which are handled in the unseen Core module.
• It does not prove that any physical system dynamically targets this zero-cost surface, only identifying the algebraic condition for the surface itself.
• It does not describe the behavior of the system when JcostN > 0.