Foundation Inequalities Am Gm Reciprocal Strict

The strict inequality

The arithmetic mean of a positive number and its reciprocal is always at least 1. For any x > 0, the average of x and 1/x is (x + 1/x)/2, and this average is never less than 1. Equivalently, x + 1/x is always at least 2. This is a classical result, a direct application of the AM-GM inequality, which states that the arithmetic mean of non-negative numbers is at least their geometric mean. For two numbers, the geometric mean of x and 1/x is 1, so their arithmetic mean is at least 1.

The strict version sharpens this. If x is not equal to 1, then the average is strictly greater than 1, meaning x + 1/x > 2. The only case where equality holds is x = 1, where the sum is exactly 2. This is not a new mathematical discovery; it is a precise, machine-checked formalization of a centuries-old fact, recorded in the framework's library of formal theorems.

In Recognition Science, this inequality is load-bearing. The framework's central object is the cost function, a measure of the forced expense of a recognition event. The cost is defined as J(x) = (x + 1/x)/2 - 1. From the non-strict inequality, it follows immediately that J(x) is always non-negative. The strict version adds the key detail: the cost is zero only when x = 1, and is strictly positive for any other positive x. This gives the cost function its unique minimum and its interpretation as a measure of deviation from the identity.

The theorem am_gm_reciprocal_strict establishes that for all x > 0 with x ≠ 1, x + 1/x > 2. Its proof in the library is short: it combines the non-strict inequality with the equality condition. The framework's library proves this with no gaps, relying only on the standard axioms of its ambient logic. This single inequality is the foundation for the framework's claim that cost is never negative and is zero only at the point of perfect agreement.

What this declaration does not claim is more modest than the framework's larger ambitions. It does not, by itself, prove that the cost function J is the only possible cost function, nor does it derive the golden ratio or any physical constant. It is a lemma, a single brick in a larger edifice. The theorem proves a property of a specific, chosen formula; it does not prove that this formula is forced by any deeper principle. That forcing is the work of other, more substantial theorems in the framework's library.

/-- Strengthened AM-GM: x + 1/x > 2 when x ≠ 1. -/
theorem am_gm_reciprocal_strict {x : ℝ} (hx : x > 0) (hne : x ≠ 1) : x + 1/x > 2 := by
  have h := am_gm_reciprocal hx
  have hne' : ¬(x + 1/x = 2) := by
    intro heq
    exact hne ((am_gm_reciprocal_eq hx).mp heq)
  exact lt_of_le_of_ne h (Ne.symm hne')

/-- 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 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

What this page does not claim

The theorem does not prove that the cost function J is the only possible cost function. The theorem does not derive the golden ratio or any physical constant. The theorem does not establish the framework's larger claims about space or time.

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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