hadamardDiv in IndisputableMonolith.Cost.Ndim.Core

(1) In plain English, hadamardDiv defines componentwise division on n-dimensional vectors: given two vectors x and y (each a function from Fin n to ℝ), it returns a new vector whose i-th entry is exactly x_i divided by y_i. This is the vector analogue of ordinary division, applied independently to each coordinate.

(2) In Recognition Science this matters because the N-dimensional cost JcostN is built by lifting the scalar J-cost through a weighted log-aggregate. Componentwise division (via hadamardDiv) lets the framework handle quotients of positive coordinates while preserving the reciprocal symmetry J(r) = J(1/r) that is forced by the Law of Logic. The theorem dot_log_hadamardDiv shows the log-aggregate of a quotient equals the difference of the separate aggregates, which is the algebraic step underlying reciprocity in higher dimensions.

(3) The formal statement is:

noncomputable def hadamardDiv {n : ℕ} (x y : Vec n) : Vec n := fun i => x i / y i

• {n : ℕ} is the dimension.
• Vec n is the type Fin n → ℝ (coordinate functions).
• The body fun i => x i / y i applies ordinary real division to each component.
• noncomputable indicates the definition uses real division, which is not executable in Lean.

(4) Visible dependencies in the supplied source: it is defined after Vec, hadamardMul and hadamardInv; it is used directly by dot_log_hadamardDiv (which proves the log-aggregate identity under positivity assumptions) and by JcostN_reciprocal (which lifts scalar reciprocity to the vector setting). No separate certificate or axiom is attached to the definition itself.

(5) The declaration does not prove any property; it is a pure definition. All theorems that use it (e.g., the log-aggregate identities and reciprocity) are proved separately in the same module. It supplies no uniqueness result, no connection to the scalar J-cost beyond the surrounding definitions, and no link to the forcing chain or physical constants.