Cost Cauchy Auxiliary Phi At Zero

The auxiliary function

In the study of functional equations, a d'Alembert solution is a continuous function H that satisfies H(t+u) + H(t−u) = 2·H(t)·H(u) for all real t and u, with H(0) = 1. Such functions appear in the classification of solutions to the cosine and hyperbolic cosine equations. For the branch where H takes values at least 1, the auxiliary function φ(t) = H(t) + √(H(t)² − 1) is defined. The theorem phi_at_zero establishes that this auxiliary function equals 1 at the origin: φ(0) = 1, given that H(0) = 1. This is a direct consequence of the definition and the assumption on H, and it is proved in the machine-checked library of formal theorems.

The statement is a structural lemma, not a classification result. It does not assert that H is a hyperbolic cosine, nor that φ is exponential. Those claims require additional hypotheses: the multiplicativity of φ, its continuity, and the bridge to the exponential function. The theorem phi_at_zero only pins down the value at a single point. Its role is to serve as a base case in a larger argument that classifies the d'Alembert solutions, a classification that remains conditional on those further assumptions.

In Recognition Science, this lemma belongs to the toolkit that analyzes the cost function J(x) = (x + 1/x)/2 − 1. The framework models recognition events as a ledger, a discrete record of events, and derives J as the forced cost of recognition. The function H = 1 + J is a d'Alembert solution that achieves values greater than 1, so it falls into the branch where the auxiliary function φ is used. The theorem phi_at_zero provides the initial condition φ(0) = 1 that the classification strategy requires. It is a small but necessary step in the chain that leads to the exponential form of φ and hence to the hyperbolic cosine form of H.

What the theorem does not claim is as important as what it proves. It does not prove the full classification of d'Alembert solutions; that remains conditional on the multiplicative property and the exponential bridge. It does not involve the golden ratio, the forcing chain, or any specific value of the cost function beyond the starting point. The theorem is a single, verified fact about a helper function, not a claim about the structure of recognition itself.

/-- 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]

/-- The full Aczél classification, conditional on the two bridge lemmas. -/
theorem aczel_classification_conditional
    (H : ℝ → ℝ)
    (h_one : H 0 = 1)
    (h_cont : Continuous H)
    (h_dA : ∀ t u, H (t + u) + H (t - u) = 2 * H t * H u)
    (h_phi_mult : H_PhiMultiplicative H)
    (h_cauchy : H_CauchyToExponential)
    (h_ge_one : ∀ t, 1 ≤ H t) :
    ∃ lam : ℝ, ∀ t, H t = Real.cosh (lam * t) := by
  have h_phi_cont : Continuous (phi H) := by
    unfold phi
    exact h_cont.add ((h_cont.pow 2).sub continuous_const).sqrt
  have h_phi_pos : ∀ t, 0 < phi H t := fun t => phi_pos H t (h_ge_one t)
  have h_phi_zero : phi H 0 = 1 := phi_at_zero H h_one
  have h_phi_cauchy : ∀ t u, phi H (t + u) = phi H t * phi H u :=
    fun t u => h_phi_mult t u (h_ge_one t) (h_ge_one u)
  obtain ⟨lam_, hlam⟩ := h_cauchy (phi H) h_phi_cont h_phi_pos h_phi_zero h_phi_cauchy
  refine ⟨lam_, fun t => ?_⟩
  have h_phi_exp : phi H t = Real.exp (lam_ * t) := hlam t
  have h_phi_neg : phi H (-t) = Real.exp (-(lam_ * t)) := by
    rw [hlam (-t)]; ring_nf
  have h_H_from_phi := H_from_phi H t (h_ge_one t)
  rw [h_phi_exp] at h_H_from_phi
  rw [Real.cosh_eq]
  convert h_H_from_phi using 1
  rw [exp_neg]

What this page does not claim

The full classification of d'Alembert solutions is proved. The value of the cost function at any point other than the origin is determined by this theorem. The golden ratio or the forcing chain is derived from phi_at_zero.

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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