Foundation Inequalities J Cost Phi

The cost at the golden ratio

The golden ratio φ is the number that satisfies φ² = φ + 1, approximately 1.618. It has been studied for millennia, appearing in the geometry of the regular pentagon and in the Fibonacci sequence, where successive ratios approach it. One of its classical identities is that φ plus its reciprocal equals the square root of 5, a fact that follows directly from the defining quadratic equation.

In Recognition Science, the framework models a recognition event, a discrete act of matching one thing to another, as carrying a forced cost, a price that cannot be avoided. The framework's cost function is J(x) = (x + 1/x)/2 - 1, which is proved to be non-negative and to reach its minimum of zero only at x = 1. The declaration J_cost_phi establishes that when this cost function is evaluated at the golden ratio, the result takes the simple closed form (√5 - 2)/2, approximately 0.118.

This result is a direct consequence of the golden ratio's own properties. Since φ + 1/φ = √5, substituting into the cost formula gives (√5)/2 - 1, which simplifies to (√5 - 2)/2. The theorem is proved in the framework's machine-checked library of formal theorems, meaning the derivation is verified step by step from the definitions. It does not introduce new physics; it is a calculation about a specific number within an already-established framework.

The value is not a new fundamental constant. It is a derived property of the golden ratio under a particular cost model. The theorem does not claim that the golden ratio minimizes cost, nor that this cost function is the only possible one. It simply states a clean algebraic fact: the cost at φ equals (√5 - 2)/2.

/-- J-cost is non-negative: J(x) = (x + 1/x)/2 - 1 ≥ 0 for x > 0.

    This follows directly from AM-GM. -/
theorem J_formula_nonneg {x : ℝ} (hx : x > 0) : (x + 1/x) / 2 - 1 ≥ 0 := by
  have h := am_gm_reciprocal hx
  linarith

/-- J-cost achieves minimum 0 at x = 1. -/
theorem J_formula_min_at_one : (1 + 1/(1 : ℝ)) / 2 - 1 = 0 := by norm_num

/-- J-cost of φ -/
theorem J_cost_phi : (φ + 1/φ) / 2 - 1 = (Real.sqrt 5 - 2) / 2 := by
  rw [phi_plus_inv]
  ring

/-- φ + 1/φ = √5 -/
theorem phi_plus_inv : φ + 1/φ = Real.sqrt 5 := by
  unfold φ Constants.phi
  have hroot_pos : (0 : ℝ) < 5 := by norm_num
  have hroot_ne : Real.sqrt 5 + 1 ≠ 0 := by
    have := Real.sqrt_nonneg 5
    linarith
  field_simp
  ring_nf
  rw [Real.sq_sqrt (le_of_lt hroot_pos)]
  ring

What this page does not claim

The golden ratio minimizes the cost function. The cost function J(x) is the only possible cost function. The value (√5 - 2)/2 is a fundamental constant with independent physical meaning.

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