Explanation of additiveQuadratic in IndisputableMonolith.Cost.Ndim.Bridge

(1) Plain English

The declaration additiveQuadratic defines a function that takes an n-dimensional vector ε and returns half the sum of the squares of its components. In plain terms, for small deviations or errors ε₁, ε₂, ..., εₙ it computes (1/2) × (ε₁² + ε₂² + ... + εₙ²). This is the standard quadratic additive approximation.

(2) Why it matters in Recognition Science

This definition is central to the additive-multiplicative quadratic bridge in higher-dimensional cost modeling within Recognition Science. It provides the additive side of the approximation used to relate different quadratic forms of recognition costs, enabling decomposition and bounds that connect to the broader cost structure (J-cost symmetry and ledger forcing) in the framework.

(3) How to read the formal statement

The Lean code is:

noncomputable def additiveQuadratic {n : ℕ} (ε : Vec n) : ℝ :=
  (1 / 2 : ℝ) * ∑ i : Fin n, (ε i) ^ 2

• noncomputable signals that the definition involves real-number operations not guaranteed to be executable.
• {n : ℕ} is an implicit parameter for the dimension.
• ε : Vec n is a vector whose components are accessed by index (Vec is imported from the Core module).
• The body is exactly the mathematical expression ½ Σᵢ (εᵢ)², using Lean's Fin-indexed sum.

(4) Visible dependencies or certificates in the supplied source

The module imports IndisputableMonolith.Cost.Ndim.Core (supplying Vec and dot). Visible theorems that reference or build on this definition include additive_decomposition, dot_sq_le_sqNorm_mul, multiplicative_le_additive_of_sqNorm_le_one, and compensatory_nonneg_of_sqNorm_le_one. No sorry or axioms appear in this module.

(5) What this declaration does not prove

Being a definition, it introduces the term without proving properties. All relational results (decomposition, bounds, non-negativity) are proved in the subsequent theorems listed above. It does not establish any connection to the full Recognition Science forcing chain, physical constants, or empirical predictions.