Cost Real Trace Root Mul Dalembert Diff Sq Trace

The difference-square identity

The classical d'Alembert equation, named for Jean le Rond d'Alembert's 1747 work on vibrating strings, is the functional law g(xy) + g(x/y) = 2 g(x) g(y). It governs functions that behave like cosines under multiplication: the cosine addition formula is the standard example. In the Recognition Science framework, the same law appears as a recognition cost, a discrete record of how much it costs to recognize one event given another, and the framework's library of machine-checked formal theorems proves algebraic consequences of it.

The specific declaration mulDAlembert_diff_sq_trace establishes a difference-square identity. If a function T satisfies the d'Alembert law in doubled-trace form, T(xy) + T(x/y) = T(x) T(y), and is normalized so that T(1) = 2, then for any nonzero x and y the square of the difference between T(xy) and T(x/y) equals (T(x)² − 4)(T(y)² − 4). In symbols: (T(xy) − T(x/y))² = (T(x)² − 4)(T(y)² − 4). The identity is proved in the machine-checked library from the law and the normalization alone, with no further assumptions about the function's regularity or behavior.

The result matters because it connects the d'Alembert law to the trace form that underlies the framework's cost function. The quantity T(x)² − 4 is the discriminant-like term that appears in the principal root of X² − T(x)X + 1 = 0, the object the framework uses to model recognition costs. The identity shows that the difference between two composite costs factors exactly into the product of the individual discriminants, a structural fact that holds for every function satisfying the law and normalization, not just for the specific cosine-like solution.

What the declaration does not claim is equally precise. It does not assert that any particular function satisfies the d'Alembert law; it takes the law as a hypothesis. It does not identify the function T with any physical quantity, nor does it derive the framework's central cost function J(x) = (x + 1/x)/2 − 1. The identity is pure algebra: a conditional statement about all functions that happen to satisfy the stated hypotheses. It is a lemma in the framework's development, not a physical prediction.

In Recognition Science, the identity is a stepping stone toward the uniqueness theorem for the cost function, but the step itself is neutral. A reader can verify the algebra directly: expand the square, apply the d'Alembert law, and the factorization falls out. The declaration's value is that the machine-checked library has done that verification once, for all functions at once, and recorded the result as a reusable theorem.

/-- **Difference square in doubled-trace units** (`T 1 = 2`, reduce via `g = T/2`). -/
theorem mulDAlembert_diff_sq_trace {T : ℝ → ℝ}
    (hd : ∀ x y, x ≠ 0 → y ≠ 0 → T (x * y) + T (x / y) = T x * T y)
    (h2 : T 1 = 2) :
    ∀ x y, x ≠ 0 → y ≠ 0 →
      (T (x * y) - T (x / y)) ^ 2
        = (T x ^ 2 - 4) * (T y ^ 2 - 4) := by
  intro x y hx hy
  set g := fun z => T z / 2
  have hg :
      ∀ u v, u ≠ 0 → v ≠ 0 → g (u * v) + g (u / v) = 2 * g u * g v := by
    intro u v hu hv
    simp only [g]
    have := hd u v hu hv
    field_simp
    linarith
  have hg1 : g 1 = 1 := by simp [g, h2]
  have hsq := mulDAlembert_diff_sq hg hg1 x y hx hy
  simp only [g] at hsq
  have hl : (T (x * y) / 2 - T (x / y) / 2) ^ 2
      = (T (x * y) - T (x / y)) ^ 2 / 4 := by ring
  have hr : 4 * ((T x / 2) ^ 2 - 1) * ((T y / 2) ^ 2 - 1)
      = (T x ^ 2 - 4) * (T y ^ 2 - 4) / 4 := by ring
  rw [hl, hr] at hsq
  have h4 : (4 : ℝ) ≠ 0 := by norm_num
  field_simp at hsq
  exact hsq

What this page does not claim

The declaration does not prove that any specific function satisfies the d'Alembert law. It does not identify T with a physical quantity or derive the framework's cost function J. It does not assert uniqueness of the solution; that requires additional hypotheses beyond the law and normalization.

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/RealTraceRoot.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: