Foundation Inequalities J Formula Nonneg

The cost stays nonnegative

The arithmetic mean of a positive number and its reciprocal is always at least 2. For any x > 0, the average (x + 1/x)/2 is never smaller than 2; it equals 2 only when x = 1. This is a classical fact, the AM-GM inequality applied to x and 1/x, and it has a one-line proof.

In Recognition Science, the recognition cost (the price a ledger pays to record an event) is defined as J(x) = (x + 1/x)/2 - 1. Subtracting 1 shifts the AM-GM bound so that the cost is always at least 0. The declaration J_formula_nonneg in the framework's machine-checked library of formal theorems proves exactly this: for every positive x, J(x) is nonnegative. The same library proves that the cost reaches 0 only at x = 1, and that it is strictly positive for every other positive x.

This is a small but load-bearing result. It guarantees that the cost function never produces a negative value, which would be meaningless in a ledger that records recognition events. The minimum at x = 1 means that when the two quantities being compared are equal, the cost of recognizing their sameness is zero; any mismatch costs something positive. The theorem is a direct consequence of AM-GM, not an independent assumption.

In Recognition Science, this nonnegativity is the foundation for later results. The golden ratio φ appears in the same file, with proofs that φ² = φ + 1 and that 1/φ = φ - 1. The cost of recognizing φ itself is computed as (√5 - 2)/2, a positive number. These facts are part of the forcing chain that leads to the framework's larger claims about scaling and structure.

What J_formula_nonneg does not claim is broader than what it proves. It says nothing about why the cost takes this particular form; that uniqueness is a separate theorem. It does not assert that the ledger is physically real, nor that recognition events exist in nature. It is purely a mathematical statement about a defined function, and its proof relies only on standard arithmetic and the AM-GM inequality.

/-- 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

/-- Equality in AM-GM holds iff x = 1. -/
theorem am_gm_reciprocal_eq {x : ℝ} (hx : x > 0) : x + 1/x = 2 ↔ x = 1 := by
  constructor
  · intro h
    have h1 : (x - 1)^2 = x^2 - 2*x + 1 := by ring
    have h2 : x^2 + 1 = 2*x := by
      have hx_ne : x ≠ 0 := ne_of_gt hx
      field_simp at h
      linarith
    have h3 : (x - 1)^2 = 0 := by nlinarith [sq_nonneg x]
    have h4 : x - 1 = 0 := by
      rwa [sq_eq_zero_iff] at h3
    linarith
  · intro h
    rw [h]
    norm_num

/-- 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 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

/-- φ² = φ + 1 (the defining equation) -/
theorem phi_sq : φ^2 = φ + 1 := Constants.phi_sq_eq

/-- 1/φ = φ - 1 (the golden ratio property) -/
theorem phi_inv : 1 / φ = φ - 1 := by
  have hsq : φ ^ 2 = φ + 1 := Constants.phi_sq_eq
  have hpos : 0 < φ := Constants.phi_pos
  have hne : φ ≠ 0 := hpos.ne'
  have hmul : φ * (φ - 1) = 1 := by
    calc
      φ * (φ - 1) = φ ^ 2 - φ := by ring
      _ = (φ + 1) - φ := by simp [hsq]
      _ = 1 := by ring
  have hdiv : φ - 1 = 1 / φ := by
    apply (eq_div_iff hne).2
    simpa [mul_comm, mul_left_comm, mul_assoc] using hmul
  exact hdiv.symm

What this page does not claim

The declaration does not prove that the cost function is unique or that it must take this form. The declaration does not assert that recognition events or ledgers exist in physical reality. The declaration does not derive any physical constant or empirical prediction.

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