Cost Ndim Bridge

The quadratic bridge

The cost ndim bridge is a machine-checked relation inside Recognition Science's cost ledger. It connects two ways of writing a quadratic cost over a vector of recognition errors. The additive form, additive quadratic, is half the sum of the squared errors. The multiplicative form, multiplicative quadratic, is half the square of a weighted sum of the same errors, using a weight vector. The bridge is the identity that the additive form always equals the multiplicative form plus a third term, the compensatory quadratic.

That identity is established as a theorem: for any dimension and any weight vector, the additive quadratic cost decomposes exactly into the multiplicative quadratic cost plus the compensatory term. The proof is a direct algebraic expansion, tagged THEOREM. The same module establishes a second fact: when the weight vector has squared norm at most one, the multiplicative cost is bounded above by the additive cost. From that bound follows the third result, that the compensatory term is nonnegative under the same normalization.

In plain language, the bridge says that a multiplicative reading of errors never exceeds the additive reading when weights are normalized, and the gap between them is always a nonnegative remainder. The remainder is what the multiplicative form leaves out. The module does not yet connect this decomposition to the forcing chain that fixes the cost function J, nor does it say how the weight vector is chosen. Those links remain open.

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

/-- 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

/-- 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

No claim that the weight vector is derived from the forcing chain. No claim that the bridge fixes the cost function J. No claim that the bridge establishes three spatial dimensions.

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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Derived articles

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