Foundation Dalembert Counterexamples Fquad On Exp

The quadratic cost counterexample

The d'Alembert equation is a classical functional equation studied since Jean le Rond d'Alembert's work in the 1740s. It asks for functions H satisfying H(t+u) + H(t-u) = 2 H(t) H(u), a relation that forces a distinctive shape, often involving exponential or trigonometric functions. The equation appears across mathematics, from probability to harmonic analysis, and in the Recognition Science framework it is the target structure that the cost function's log-lift should satisfy.

The machine-checked library of formal theorems in the framework contains a counterexample module. It defines Fquad(x) as (log x)^2 / 2, a quadratic log-cost. The declaration Fquad_on_exp proves that Fquad(e^t) equals t^2 / 2, which is the log-lift written in the variable t. This is a simple lemma, but it sets up the key fact: the shifted log-lift Hquad(t) = t^2 / 2 + 1 does not satisfy the d'Alembert equation. The theorem Hquad_not_dAlembert states this failure explicitly.

Why does this matter? The framework's central result, the forced form J(x) = (x + 1/x)/2 - 1, depends on a chain of hypotheses. One weak hypothesis is the mere existence of some combiner P such that F(xy) + F(x/y) = P(F(x), F(y)). The counterexample shows that this weak hypothesis alone is not enough. The quadratic cost Fquad satisfies that weak consistency condition with the additive combiner Padd(u,v) = 2u + 2v, and it also satisfies the symmetry Fquad(x) = Fquad(1/x) and the calibration condition on its log-lift. Yet its log-lift fails the d'Alembert equation.

In Recognition Science, this is a structural obstruction, not a failure. The framework proves that any theorem forcing the d'Alembert form from the weak hypothesis must add at least one further nondegeneracy axiom. The counterexample therefore sharpens the boundary of what the framework's theorems require. It does not weaken the main forcing theorem, which uses stronger hypotheses; it clarifies why those stronger hypotheses are necessary.

The consequence for a reader is a precise picture of the framework's logical structure. The quadratic cost is a genuine solution to the weak consistency condition, so the weak condition alone cannot select the framework's unique cost function. The framework's own theorems must and do assume more. This counterexample is the reason the framework's claims are stated with the exact hypotheses they need, and it is a reminder that a uniqueness theorem is only as strong as its assumptions.

lemma Fquad_on_exp (t : ℝ) : Fquad (Real.exp t) = Gquad t := by
  -- log(exp t) = t
  simp [Fquad, Cost.F_ofLog, Gquad]

theorem Hquad_not_dAlembert :
    ¬ (Hquad 0 = 1 ∧ ∀ t u : ℝ, Hquad (t + u) + Hquad (t - u) = 2 * Hquad t * Hquad u) := by
  intro h
  have h0 : Hquad 0 = 1 := h.1
  have hdA := h.2
  -- Evaluate the d'Alembert identity at t = 1, u = 1.
  have h11 := hdA 1 1
  -- Compute both sides explicitly; they disagree (4 ≠ 9/2).
  have hL : Hquad (1 + 1) + Hquad (1 - 1) = 4 := by
    calc
      Hquad (1 + 1) + Hquad (1 - 1)
          = ((1 + 1) ^ 2 / 2 + 1) + ((1 - 1) ^ 2 / 2 + 1) := by
              simp [Hquad_simp]
      _ = 4 := by
            norm_num
  have hR : 2 * Hquad 1 * Hquad 1 = (9 : ℝ) / 2 := by
    simp [Hquad_simp]
    ring
  -- Contradiction
  have : (4 : ℝ) = (9 : ℝ) / 2 := by
    calc (4 : ℝ) = Hquad (1 + 1) + Hquad (1 - 1) := by simpa using hL.symm
      _ = 2 * Hquad 1 * Hquad 1 := h11
      _ = (9 : ℝ) / 2 := hR
  norm_num at this

lemma Fquad_consistency :
    ∀ x y : ℝ, 0 < x → 0 < y →
      Fquad (x * y) + Fquad (x / y) = Padd (Fquad x) (Fquad y) := by
  intro x y hx hy
  -- Work in log-coordinates: let t = log x, u = log y
  have hx0 : x ≠ 0 := hx.ne'
  have hy0 : y ≠ 0 := hy.ne'
  have hlog_mul : Real.log (x * y) = Real.log x + Real.log y := by
    simpa using Real.log_mul hx.ne' hy.ne'
  have hlog_div : Real.log (x / y) = Real.log x - Real.log y := by
    simpa [div_eq_mul_inv, Real.log_mul, Real.log_inv, hy0] using Real.log_div hx.ne' hy.ne'
  -- Now compute
  simp [Fquad, Cost.F_ofLog, Gquad, Padd, hlog_mul, hlog_div]
  ring

theorem Hquad_not_dAlembert :
    ¬ (Hquad 0 = 1 ∧ ∀ t u : ℝ, Hquad (t + u) + Hquad (t - u) = 2 * Hquad t * Hquad u) := by
  intro h
  have h0 : Hquad 0 = 1 := h.1
  have hdA := h.2
  -- Evaluate the d'Alembert identity at t = 1, u = 1.
  have h11 := hdA 1 1
  -- Compute both sides explicitly; they disagree (4 ≠ 9/2).
  have hL : Hquad (1 + 1) + Hquad (1 - 1) = 4 := by
    calc
      Hquad (1 + 1) + Hquad (1 - 1)
          = ((1 + 1) ^ 2 / 2 + 1) + ((1 - 1) ^ 2 / 2 + 1) := by
              simp [Hquad_simp]
      _ = 4 := by
            norm_num
  have hR : 2 * Hquad 1 * Hquad 1 = (9 : ℝ) / 2 := by
    simp [Hquad_simp]
    ring
  -- Contradiction
  have : (4 : ℝ) = (9 : ℝ) / 2 := by
    calc (4 : ℝ) = Hquad (1 + 1) + Hquad (1 - 1) := by simpa using hL.symm
      _ = 2 * Hquad 1 * Hquad 1 := h11
      _ = (9 : ℝ) / 2 := hR
  norm_num at this

What this page does not claim

The counterexample does not disprove the framework's main forcing theorem, which uses stronger hypotheses. The quadratic cost Fquad is not a valid cost function in the framework, only a counterexample to a weak hypothesis. This answer does not claim the d'Alembert equation is the only structure the framework derives.

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

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