Cost Ndim Bridge Compensatory Quadratic

The residual term

The declaration compensatoryQuadratic defines a residual term in a cost function, the framework's measure of recognition effort. For a vector of errors ε and a vector of weights α, the additive quadratic cost is half the sum of squared errors, ½Σεᵢ². The multiplicative quadratic cost is half the square of the weighted sum, ½(α·ε)². The compensatory term is their difference: additive minus multiplicative.

The definition establishes an identity: the additive cost always equals the multiplicative cost plus the compensatory term. This is a theorem in the machine-checked library of formal theorems, proved by simple algebra. The decomposition matters because the two quadratic forms behave differently. The additive form treats each error independently; the multiplicative form couples them through the weights. Their difference measures how much the coupling changes the cost.

When the weights are normalized so that α·α ≤ 1, the compensatory term is nonnegative. This follows from the Cauchy-Schwarz inequality: the square of the dot product cannot exceed the product of the squared norms. With normalized weights, the multiplicative cost is bounded above by the additive cost, so their difference is never negative. The theorem states this as α·α ≤ 1 implies 0 ≤ compensatoryQuadratic α ε.

The declaration does not claim that the compensatory term is zero, nor that the multiplicative form is the true cost. It defines a residual and proves its basic properties. The nonnegativity result holds only under the normalization condition; without it, the term can be negative. The framework does not use this term to derive any physical constant or to force a dimension; it is a technical lemma in the cost decomposition.

In plain terms, the declaration gives a way to split a quadratic cost into a coupled part and a leftover. The leftover is always a nonnegative correction when weights are normalized. This is a building block for later work, not a standalone physical prediction.

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

/-- 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

/-- If `‖α‖² ≤ 1`, multiplicative quadratic cost is bounded by additive quadratic cost. -/
theorem multiplicative_le_additive_of_sqNorm_le_one {n : ℕ}
    (α ε : Vec n) (hα : dot α α ≤ 1) :
    multiplicativeQuadratic α ε ≤ additiveQuadratic ε := by
  have hsq : (dot α ε) ^ 2 ≤ ∑ i : Fin n, (ε i) ^ 2 := by
    have hcs : (dot α ε) ^ 2 ≤ (dot α α) * (∑ i : Fin n, (ε i) ^ 2) :=
      dot_sq_le_sqNorm_mul α ε
    have hsum_nonneg : 0 ≤ ∑ i : Fin n, (ε i) ^ 2 := by
      exact Finset.sum_nonneg (fun i _ => sq_nonneg (ε i))
    have hmul : (dot α α) * (∑ i : Fin n, (ε i) ^ 2) ≤ 1 * (∑ i : Fin n, (ε i) ^ 2) :=
      mul_le_mul_of_nonneg_right hα hsum_nonneg
    exact le_trans hcs (by simpa using hmul)
  have hhalf : (0 : ℝ) ≤ 1 / 2 := by norm_num
  have hscaled := mul_le_mul_of_nonneg_left hsq hhalf
  simpa [multiplicativeQuadratic, additiveQuadratic, one_mul] using hscaled

What this page does not claim

The compensatory term is zero in general. The multiplicative quadratic form is the true recognition cost. The declaration derives any physical constant or dimension.

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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