Plain English (1): compensatoryQuadratic computes the difference between the additive quadratic form (½ Σ εᵢ²) and the multiplicative quadratic form (½ (Σ αᵢ εᵢ)²) for vectors α and ε of length n. It isolates the residual term that compensates between the two approximations.
Role in Recognition Science (2): The definition supplies the bridge term in the N-dimensional cost module, enabling decomposition of quadratic costs into additive and multiplicative parts. This supports analysis of recognition costs under normalized weights, as shown by the non-negativity result when ‖α‖² ≤ 1.
Reading the formal statement (3): The declaration is
noncomputable def compensatoryQuadratic {n : ℕ} (α ε : Vec n) : ℝ := additiveQuadratic ε - multiplicativeQuadratic α ε
It is parameterized by dimension n and two vectors, returning a real; the body is a direct subtraction of the two sibling definitions in the same module.
Dependencies and certificates (4): Depends on the sibling definitions additiveQuadratic and multiplicativeQuadratic. The supplied source contains the decomposition certificate additive_decomposition (proved by ring after unfolding), the Cauchy-Schwarz bound dot_sq_le_sqNorm_mul, the comparison multiplicative_le_additive_of_sqNorm_le_one, and the non-negativity certificate compensatory_nonneg_of_sqNorm_le_one.
What it does not prove (5): The declaration itself establishes only the syntactic difference; it does not prove uniqueness of the bridge, any link to the J-cost functional equation, physical constants, or forcing-chain results.