1. What the declaration says in plain English
JlogN defines the $N$-dimensional reciprocal cost evaluated in logarithmic coordinates. Given an $n$-dimensional vector of weights $\alpha$ and an $n$-dimensional vector of log-coordinates $t$, it computes their weighted sum $\alpha \cdot t$, exponentiates the result to return to positive coordinates, and feeds it into the fundamental scalar recognition cost function $J$.
2. Why it matters in Recognition Science
Recognition Science derives physics from the unique reciprocal-symmetric scalar cost function $J(x) = (x + x^{-1})/2 - 1$. To evaluate the cost of multi-component systems, the framework must lift this scalar kernel to $N$ dimensions. By aggregating components via a weighted sum in log-space and applying $J$, JlogN creates a multi-dimensional cost that preserves the central $r \leftrightarrow 1/r$ reciprocal symmetry.
3. How to read the formal statement
noncomputable def JlogN {n : ℕ} (α t : Vec n) : ℝ :=
Jcost (Real.exp (dot α t))
noncomputable def: A definition in Lean involving real numbers (mathematically rigorous, but not algorithmically executable in finite steps).{n : ℕ}: An implicit natural number specifying the dimensionality of the vectors.(α t : Vec n): Two $n$-dimensional vectors. $\alpha$ holds the coordinate weights, and $t$ holds the logarithmic values.: ℝ: The function returns a real number.Jcost (...): The foundational scalar Recognition cost function.Real.exp (dot α t): The exponential of the weighted dot product of $\alpha$ and $t$.
4. Visible dependencies and certificates
JlogN relies directly on dot for the weighted sum and the external Jcost.
It serves as the underlying engine for the positive-coordinate multi-dimensional cost JcostN, which is defined simply as JlogN α (logVec x).
A key structural consequence proved in this module is JlogN_eq_cosh_sub_one, which establishes that JlogN α t is exactly equal to $\cosh(\alpha \cdot t) - 1$, tightly linking the log-coordinate cost to hyperbolic geometry.
5. What this declaration does not prove
JlogN is a MODEL (a definitional choice) for aggregating multi-component costs. While the scalar function $J(x)$ is forced as the unique reciprocal-symmetric cost (by t5_holds / washburn_uniqueness_aczel, outside this module slice), this module does not prove that JlogN is the unique way to lift $J$ to $N$ dimensions. It merely defines a consistent lifting strategy and proves that this specific strategy successfully preserves reciprocity under componentwise inversion (via JcostN_reciprocal).