Cost Ndim Bridge Compensatory Nonneg Of Sq Norm Le One

The compensatory term

In cost analysis, one often approximates a complicated cost by a quadratic form. The Recognition Science framework's cost library defines two such quadratic approximations. The first, the additive quadratic cost, is half the sum of the squares of the components of an error vector. The second, the multiplicative quadratic cost, is half the square of a weighted sum of those same components, using a weight vector. These two approximations differ by a residual term, called the compensatory quadratic term, which is simply the additive cost minus the multiplicative cost.

The declaration compensatory_nonneg_of_sqNorm_le_one establishes a condition under which this residual term is never negative. The condition is that the squared norm of the weight vector is at most one, meaning the weights are normalized. Under that condition, the theorem proves that the additive quadratic cost is always at least as large as the multiplicative quadratic cost. This is a direct consequence of the Cauchy-Schwarz inequality, which bounds the square of a dot product by the product of the squared norms. The proof is carried out in the framework's machine-checked library of formal theorems, so the inequality is verified by the kernel.

What this theorem does not claim is equally important. It does not say that the compensatory term is always nonnegative for any weight vector; the normalization condition is essential. Without it, the multiplicative cost can exceed the additive cost. It also does not claim that the compensatory term is small, only that it is nonnegative. The theorem is a structural fact about the relationship between two quadratic approximations, not a statement about any particular physical cost or about the value of the compensatory term itself.

/-- Under normalized weights (`‖α‖² ≤ 1`), the compensatory term is nonnegative. -/
theorem compensatory_nonneg_of_sqNorm_le_one {n : ℕ}
    (α ε : Vec n) (hα : dot α α ≤ 1) :
    0 ≤ compensatoryQuadratic α ε := by
  unfold compensatoryQuadratic
  have hle := multiplicative_le_additive_of_sqNorm_le_one α ε hα
  linarith

theorem additive_decomposition {n : ℕ} (α ε : Vec n) :
    additiveQuadratic ε
      = multiplicativeQuadratic α ε + compensatoryQuadratic α ε := by
  unfold compensatoryQuadratic
  ring

What this page does not claim

The theorem does not claim the compensatory term is nonnegative without the normalization condition. The theorem does not claim the compensatory term is small or has a specific value. The theorem does not make any statement about physical costs or empirical measurements.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/Ndim/Bridge.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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