Explanation of multiplicativeQuadratic in IndisputableMonolith.Cost.Ndim.Bridge

(1) What the declaration says in plain English

The definition introduces a real-valued function that takes a dimension n, two vectors α and ε of that dimension, and returns half the square of their dot product. It is a quadratic form that captures a multiplicative interaction between the vectors rather than summing independent squared components.

(2) Why it matters in Recognition Science

This declaration belongs to the additive-multiplicative quadratic bridge in the N-dimensional cost module. It supplies one side of the decomposition that relates additive quadratic approximations (sum of squared entries) to multiplicative ones (square of the weighted dot product). The bridge supports modeling recognition costs in vector spaces, where the compensatory term quantifies the difference; related results in the module bound the multiplicative form by the additive form when the weight vector satisfies ‖α‖² ≤ 1.

(3) How to read the formal statement

noncomputable def multiplicativeQuadratic {n : ℕ} (α ε : Vec n) : ℝ :=
  (1 / 2 : ℝ) * (dot α ε) ^ 2

• noncomputable indicates the definition relies on real arithmetic that Lean does not treat as executable.
• {n : ℕ} is an implicit parameter (curly braces).
• Vec n is the vector type imported from Core.
• dot α ε is the dot-product operation on those vectors.
• The body is ordinary real arithmetic: scale the squared dot product by 1/2.

(4) Visible dependencies or certificates in the supplied source

The module imports IndisputableMonolith.Cost.Ndim.Core (supplying Vec and dot). The same file defines the companion functions additiveQuadratic and compensatoryQuadratic. Key certificates include:

• additive_decomposition proving additiveQuadratic ε = multiplicativeQuadratic α ε + compensatoryQuadratic α ε.
• dot_sq_le_sqNorm_mul (Cauchy-Schwarz form).
• multiplicative_le_additive_of_sqNorm_le_one and compensatory_nonneg_of_sqNorm_le_one giving the norm-1 bound. No sorry appears in the module.

(5) What this declaration does not prove

It is a local definitional bridge only; it does not derive the one-dimensional J-cost functional equation, connect to φ or any RS constant, establish uniqueness of the cost law, or participate in the forcing chain. The supplied source contains no theorems linking this quadratic form to physical predictions or the broader Recognition Science ledger.