Cost Ndim Bridge Multiplicative Le Additive Of Sq Norm Le One

The quadratic bridge

In ordinary linear algebra, the Cauchy-Schwarz inequality bounds a dot product by the product of the two vectors' lengths. Squaring both sides turns that bound into a statement about quadratic forms: the square of a dot product is at most the product of the two squared norms. The Recognition Science declaration multiplicative_le_additive_of_sqNorm_le_one is a specialized, machine-checked instance of this classical fact, written in the framework's own notation for recognition costs.

The framework models recognition events as entries in a ledger, a discrete record of events, and assigns each event a cost. Two natural quadratic approximations appear in its n-dimensional bridge: an additive cost, half the sum of squared errors, and a multiplicative cost, half the square of a weighted sum of those errors. The declaration proves that whenever the squared norm of the weight vector is at most one, the multiplicative cost is bounded above by the additive cost. In plain terms, a weighted average of errors, squared, cannot exceed the plain sum of squared errors, as long as the weights are not too large.

The proof is a direct consequence of Cauchy-Schwarz. The library also records two companion facts: the additive cost decomposes exactly into the multiplicative cost plus a residual term, and that residual term is nonnegative under the same normalized-weight condition. Together these three theorems form the quadratic bridge between the additive and multiplicative pictures, a bridge that holds for any finite dimension n.

What the declaration does not claim is just as important. It does not say the multiplicative cost is always smaller, only that it is no larger under the stated norm condition. It does not assert that the bound is tight, nor does it identify when equality holds. It says nothing about the framework's deeper forcing chain or about physical constants; it is a standalone inequality about quadratic forms, useful because it is checked and available for later steps.

/-- 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

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

What this page does not claim

The declaration does not establish equality conditions for the bound. It does not claim the multiplicative cost is always smaller, only no larger under the norm condition. It makes no statement about the framework's physical constants or forcing chain.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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