Cost Ndim Dalembert Jcost N D Alembert

The multidimensional identity

The recognition cost, a discrete record of how expensive it is to recognize one state from another, obeys a striking identity in any number of dimensions. For two vectors x and y with all positive entries, the cost of their componentwise product plus the cost of their componentwise quotient equals twice the cost of x plus twice the cost of y plus twice the product of the two costs. In symbols: J(x·y) + J(x/y) = 2J(x) + 2J(y) + 2J(x)J(y), where the dot and slash mean multiplication and division entry by entry. This is the multidimensional d'Alembert identity, a direct analogue of the classical d'Alembert equation that characterizes quadratic forms.

The identity is a theorem in the framework's machine-checked library of formal theorems. It holds for every dimension n, for every positive vector, and for every choice of the cost's internal parameter α. The proof is short: the identity follows from the one-dimensional cost formula, applied to each coordinate separately, then summed. Because the quotient term is always nonnegative, the identity immediately yields an inequality: the cost of the product is at most twice the cost of x plus twice the cost of y plus twice their product. That inequality is the lemma JcostN_submult, and it is the form the framework uses when it needs to bound the cost of a combined recognition event.

What the declaration does not claim is just as precise. It does not assert that the identity characterizes the cost function; it merely states a consequence of the already-established one-dimensional form. It does not claim that the product cost equals the bound; the inequality can be strict. And it says nothing about the physical meaning of the vectors, only about the algebraic structure of the cost. The identity is a tool, not a new postulate.

theorem JcostN_dAlembert {n : ℕ} (α x y : Vec n)
    (hx : ∀ i, 0 < x i) (hy : ∀ i, 0 < y i) :
    JcostN α (hadamardMul x y) + JcostN α (hadamardDiv x y)
      = 2 * JcostN α x + 2 * JcostN α y + 2 * JcostN α x * JcostN α y := by
  let u : ℝ := Real.exp (dot α (logVec x))
  let v : ℝ := Real.exp (dot α (logVec y))
  have hu : 0 < u := by
    dsimp [u]
    exact Real.exp_pos _
  have hv : 0 < v := by
    dsimp [v]
    exact Real.exp_pos _
  have hmul :
      JcostN α (hadamardMul x y) = Jcost (u * v) := by
    calc
      JcostN α (hadamardMul x y)
          = Jcost (Real.exp (dot α (logVec (hadamardMul x y)))) := by
              simp [JcostN, JlogN]
      _ = Jcost (Real.exp (dot α (logVec x) + dot α (logVec y))) := by
            rw [dot_log_hadamardMul α x y hx hy]
      _ = Jcost (u * v) := by
            simp [u, v, Real.exp_add]
  have hdiv :
      JcostN α (hadamardDiv x y) = Jcost (u / v) := by
    calc
      JcostN α (hadamardDiv x y)
          = Jcost (Real.exp (dot α (logVec (hadamardDiv x y)))) := by
              simp [JcostN, JlogN]
      _ = Jcost (Real.exp (dot α (logVec x) - dot α (logVec y))) := by
            rw [dot_log_hadamardDiv α x y hx hy]
      _ = Jcost (u / v) := by
            simp [u, v, Real.exp_sub]
  have hbase := dalembert_identity (x := u) (y := v) hu hv
  calc
    JcostN α (hadamardMul x y) + JcostN α (hadamardDiv x y)
        = Jcost (u * v) + Jcost (u / v) := by rw [hmul, hdiv]
    _ = 2 * Jcost u + 2 * Jcost v + 2 * Jcost u * Jcost v := hbase
    _ = 2 * JcostN α x + 2 * JcostN α y + 2 * JcostN α x * JcostN α y := by
          simp [u, v, JcostN, JlogN]

lemma JcostN_submult {n : ℕ} (α x y : Vec n)
    (hx : ∀ i, 0 < x i) (hy : ∀ i, 0 < y i) :
    JcostN α (hadamardMul x y)
      ≤ 2 * JcostN α x + 2 * JcostN α y + 2 * JcostN α x * JcostN α y := by
  have h := JcostN_dAlembert α x y hx hy
  have hnonneg : 0 ≤ JcostN α (hadamardDiv x y) := JcostN_nonneg α (hadamardDiv x y)
  linarith

What this page does not claim

The identity does not characterize the cost function; it is a consequence of the one-dimensional form. The inequality JcostN_submult does not assert equality; the quotient term can make the bound strict. No physical meaning is assigned to the vectors; the theorem is purely algebraic.

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/DAlembert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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