Explain the Lean def `additiveQuadratic` in module `IndisputableMonolith.Cost.Ndim.Bridge`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

(1) What it says in plain English

additiveQuadratic computes the scaled sum of squared components of an $n$-dimensional vector $\varepsilon$. Mathematically, it defines the expression: $$ \text{additiveQuadratic}(\varepsilon) = \frac{1}{2} \sum_{i=0}^{n-1} \varepsilon_i^2 $$ It represents an isotropic, decoupled quadratic sum over $n$ independent dimensions.

(2) Why it matters in Recognition Science

In Recognition Science, the fundamental cost function $J(x)$ has a minimum at $x=1$ where it locally resembles a quadratic well. When extending RS cost structures to $N$-dimensional spaces, the framework must distinguish between strictly independent dimensions and coupled dimensions. additiveQuadratic serves as the foundational MODEL for the independent, non-interacting limit. This baseline is essential for analyzing geometric cost projections. For instance, the module goes on to prove via multiplicative_le_additive_of_sqNorm_le_one that multiplicative projections onto normalized vectors are bounded by this additive capacity, leaving a non-negative compensatoryQuadratic residual.

(3) How to read the formal statement

• noncomputable def: It is a mathematical definition over the real numbers (ℝ). Because exact real arithmetic involves limits, Lean cannot execute it as a standard computational algorithm.
• {n : ℕ}: An implicit parameter specifying the integer number of dimensions. Lean automatically infers this from the vector provided.
• (ε : Vec n): The input argument, an $n$-dimensional vector (representing perturbations, errors, or displacements).
• : ℝ :=: The declaration returns a single real number.
• (1 / 2 : ℝ) * ∑ i : Fin n, (ε i) ^ 2: The literal formalization of $\frac{1}{2} \sum_i \varepsilon_i^2$, using Lean's Finset.sum over the finite index set Fin n.

(4) Visible dependencies and downstream usage

Within the supplied source, additiveQuadratic depends on Vec n, a type defined in the imported (but unseen) IndisputableMonolith.Cost.Ndim.Core module. It acts as a direct building block for several proved theorems:

• compensatoryQuadratic: Defines the residual between the additive and multiplicative costs.
• additive_decomposition: A THEOREM verifying the trivial algebraic identity that additive cost equals the sum of the multiplicative and compensatory parts.

(5) What this declaration does not prove

This declaration is purely a definitional MODEL. It does not prove that nature fundamentally uses this specific additive cost, nor does it establish the rigorous Taylor-expansion linkage connecting the unique, parameter-free $J(x)$ cost function to this generic multi-dimensional quadratic approximation. Finally, while it deals with $n$-dimensional vectors, it has no bearing on proving why spatial dimensions $D=3$; that is handled upstream via the S¹ cohomology axioms in the RS forcing chain.