Foundation Inequalities J Formula Min At One

The minimum of J

The recognition cost function, written J(x), measures the price of recognition in a discrete ledger of events. The declaration J_formula_min_at_one establishes that when x equals 1, this cost is exactly zero. It is a simple arithmetic fact: (1 + 1/1) / 2 - 1 = 0. The theorem does not merely assert that zero is a possible value; it identifies the specific input that produces it.

The broader context comes from the AM-GM inequality, a classical result stating that for any positive number x, the sum x + 1/x is at least 2. The framework's library proves this inequality and then derives from it that J(x) is never negative. The minimum at x = 1 is the sharp edge of that inequality: the cost cannot go below zero, and it touches zero only at that single point. Away from x = 1, for any other positive value, the cost is strictly positive.

In Recognition Science, this minimum has a structural meaning. The cost function J is forced by five plain conditions, and its uniqueness is a proved theorem in the framework's machine-checked library. The fact that J bottoms out at x = 1 means that recognition at unity, recognizing something as exactly itself, carries no cost. Every other recognition event, comparing something to something it is not, incurs a positive price. The golden ratio φ appears later in the framework as the unique self-similar scaling, and its cost is (√5 - 2)/2, a positive number consistent with this minimum.

What the declaration does not claim is equally precise. It does not say that x = 1 is the only point where J is defined; J is defined for all positive x. It does not claim that the minimum is unique in any broader sense, only that the value at x = 1 is zero and all other values are larger. The theorem is a local fact about a specific formula, not a statement about the origin of that formula or its philosophical implications.

/-- 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 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

/-- The AM-GM inequality for x and 1/x: for all x > 0, x + 1/x ≥ 2.

    This is the fundamental inequality that forces J-cost ≥ 0.

    **Proof**: Use Mathlib's `add_div_two_ge_sqrt_mul_self_of_sq_le_sq` or direct algebra. -/
theorem am_gm_reciprocal {x : ℝ} (hx : x > 0) : x + 1/x ≥ 2 := by
  have h1 : x * (1/x) = 1 := by field_simp
  have h2 : (x - 1/x)^2 ≥ 0 := sq_nonneg _
  -- (x - 1/x)² = x² - 2 + 1/x²
  -- So x² + 1/x² ≥ 2
  -- We need: x + 1/x ≥ 2
  -- Use: (x + 1/x)² = x² + 2 + 1/x² ≥ 4, so x + 1/x ≥ 2 (since both positive)
  have hx_inv_pos : 1/x > 0 := by positivity
  have h_sum_pos : x + 1/x > 0 := by linarith
  -- Alternative: direct Mathlib lemma
  have h3 : x + 1/x = x + x⁻¹ := by rw [one_div]
  rw [h3]
  -- Use add_inv_le_iff or similar
  nlinarith [sq_nonneg (x - 1), sq_nonneg (x - x⁻¹), sq_nonneg x, sq_nonneg x⁻¹,
             mul_pos hx hx_inv_pos]

/-- J-cost is strictly positive away from x = 1. -/
theorem J_formula_pos {x : ℝ} (hx : x > 0) (hne : x ≠ 1) : (x + 1/x) / 2 - 1 > 0 := by
  have h := am_gm_reciprocal_strict hx hne
  linarith

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

What this page does not claim

The declaration does not prove that x = 1 is the only point where J is defined; J is defined for all positive x. The theorem does not explain why the cost function has this form; it only establishes a property of the given formula. This result does not claim that the minimum is unique in any broader sense beyond the value at x = 1 being zero and all other values being larger.

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

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