Explain the Lean theorem `JcostN_eq_cosh_logsum` 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.

Explanation of JcostN_eq_cosh_logsum

1. What the declaration says in plain English The theorem establishes that the $N$-dimensional positive-coordinate reciprocal cost, when evaluated with a weight vector $\alpha$ and a state vector $x$, is mathematically identical to the hyperbolic cosine of the weighted sum of the natural logarithms of $x$, minus one. In standard mathematical notation: $$J_N(\alpha, x) = \cosh\left( \sum_{i=1}^n \alpha_i \ln(x_i) \right) - 1$$

2. Why it matters in Recognition Science In RS, every physical structure is downstream of a unique, reciprocal-symmetric cost function $J(r) = J(r^{-1})$. To apply this foundational Law of Logic to multi-component systems (like multidimensional phase spaces or spacetime coordinates), the cost must aggregate individual variations without breaking the reciprocal symmetry. JcostN_eq_cosh_logsum proves that by taking a weighted sum in logarithmic coordinates, the multi-dimensional cost lifts smoothly to a $\cosh$ potential. This ensures that $N$-dimensional state spaces inherit the fundamental reciprocal symmetry natively.

3. How to read the formal statement

• {n : ℕ}: The theorem is structurally true for any natural number of dimensions $N$.
• (α x : Vec n): It takes two $N$-dimensional vectors: $\alpha$ (typically configuration weights) and $x$ (coordinates or ratios).
• dot α (logVec x): logVec takes the natural logarithm of each component $x_i$, and dot computes the inner product (weighted sum) of these logarithms with $\alpha$.
• Real.cosh (...) - 1: The aggregated scalar cost evaluates directly to the $\cosh$ of this dot product, shifted by $-1$ so the identity cost minimum remains zero.

4. Visible dependencies in the supplied source The proof simpa [JcostN, JlogN] using (Jcost_exp_cosh (dot α (logVec x))) shows exactly how it is derived:

• It expands the multi-dimensional cost definition JcostN, which passes the logarithmic vector to JlogN.
• It relies on dot and logVec to structure the sum.
• It invokes an upstream foundational identity Jcost_exp_cosh (imported from the base IndisputableMonolith.Cost module), which proves the scalar identity $J(e^t) = \cosh(t) - 1$.

5. What this declaration does not prove

• The physical dimensionality of space: This theorem is purely a structural identity valid for any dimension $n$. It does not restrict spacetime to 3 spatial dimensions; that physical constraint is forced separately via external math axioms (e.g., topological linking in alexander_duality_circle_linking).
• Input positivity: It assumes the components of $x$ are evaluated correctly by Lean's Real.log. It does not independently prove that coordinates must be strictly positive (though the lower-bound behavior of the cost itself is established in adjacent theorems like JcostN_nonneg).