Cost Cauchy Auxiliary H Phi Multiplicative

The auxiliary function phi

In the classical theory of functional equations, a central question asks which continuous functions satisfy the d'Alembert equation H(t+u) + H(t-u) = 2H(t)H(u), with H(0) = 1. The standard solutions are the hyperbolic cosine, cosh(λt), and the ordinary cosine, cos(λt). A standard classification strategy, due to Aczél, splits into two branches depending on whether H ever exceeds 1. The Recognition Science framework's machine-checked library formalizes the first branch, the cosh case, which is the one relevant to its cost function.

The key move in that branch is to define an auxiliary function φ(t) = H(t) + √(H(t)² − 1). The declaration H_PhiMultiplicative states a conditional property: if H(t) ≥ 1 and H(u) ≥ 1, then φ(t+u) = φ(t)·φ(u). This is the multiplicative Cauchy equation, and it is the bridge that turns the d'Alembert equation into a simpler exponential form. The framework's library proves that φ(0) = 1 and that φ(t) > 0 whenever H(t) ≥ 1, and it shows that H can be recovered from φ via H(t) = (φ(t) + φ(t)⁻¹)/2.

What the declaration does not claim is that this multiplicative property actually holds for every d'Alembert solution. The formal statement is conditional: it assumes the property as a hypothesis rather than proving it from the d'Alembert equation alone. The library's own status notes mark the algebraic verification as blocked by the need to handle square-root branches carefully. Nor does the declaration assert that φ must be exponential. That step, the conclusion that continuous positive multiplicative functions take the form e^(λt), is a separate standard textbook result that the library records as a hypothesis, not a proved theorem.

In Recognition Science, this matters because the cost function J(x) = (x + 1/x)/2 − 1 grows unboundedly, so its associated H = 1 + J achieves values greater than 1. That places it in the cosh branch, where the auxiliary function and its multiplicative property are the natural tools. The conditional declaration is a precise statement of what the framework has established so far, and an honest inventory of what remains to be formalized.

/-- The key algebraic identity: if H satisfies d'Alembert and H(t), H(u) ≥ 1,
    then φ(t+u) = φ(t) · φ(u).

    PROOF STRATEGY: From d'Alembert, H(t+u) + H(t-u) = 2·H(t)·H(u).
    Combined with H(t+u) - H(t-u) involving the square-root terms,
    this forces the multiplicative relation on φ.

    STATUS: CONDITIONAL — proved assuming `H_PhiMultiplicative` below.
    The algebraic verification is a calculation; its formalization is
    blocked by the need to handle the square root branches carefully. -/
def H_PhiMultiplicative (H : ℝ → ℝ) : Prop :=
  ∀ t u, 1 ≤ H t → 1 ≤ H u → phi H (t + u) = phi H t * phi H u

/-- φ(0) = 1 when H(0) = 1. -/
theorem phi_at_zero (H : ℝ → ℝ) (h_one : H 0 = 1) : phi H 0 = 1 := by
  simp [phi, h_one]

/-- φ(t) > 0 when H(t) ≥ 1. -/
theorem phi_pos (H : ℝ → ℝ) (t : ℝ) (ht : 1 ≤ H t) : 0 < phi H t := by
  unfold phi
  have h_sq : 0 ≤ H t ^ 2 - 1 := by nlinarith
  have h_sqrt : 0 ≤ Real.sqrt (H t ^ 2 - 1) := Real.sqrt_nonneg _
  linarith

/-- H(t) can be recovered from φ: H(t) = (φ(t) + φ(t)⁻¹) / 2 when φ(t) > 0. -/
theorem H_from_phi (H : ℝ → ℝ) (t : ℝ) (ht : 1 ≤ H t) :
    H t = (phi H t + (phi H t)⁻¹) / 2 := by
  unfold phi
  set s := Real.sqrt (H t ^ 2 - 1)
  have hs_sq : s ^ 2 = H t ^ 2 - 1 := by
    exact Real.sq_sqrt (by nlinarith : 0 ≤ H t ^ 2 - 1)
  have hs_nonneg : 0 ≤ s := Real.sqrt_nonneg _
  have h_pos : 0 < H t + s := by linarith
  have h_inv : (H t + s)⁻¹ = H t - s := by
    have : (H t + s) * (H t - s) = 1 := by nlinarith [hs_sq]
    rw [eq_comm, inv_eq_of_mul_eq_one_right this]
  rw [h_inv]
  ring

What this page does not claim

H_PhiMultiplicative is not a proof that every d'Alembert solution satisfies the multiplicative property. The declaration does not assert that φ(t) = e^(λt) for some λ. The library does not claim to have formalized the full Aczél classification without additional hypotheses.

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

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