Foundation Jcost Geometry Simultaneous Differs From Sequential

The geometric mean and the arithmetic mean

The geometric mean of two positive numbers is the square root of their product, while the arithmetic mean is their sum divided by two. For example, the geometric mean of 2 and 8 is 4, and the arithmetic mean is 5. A classical theorem states that if the two numbers are not equal, these two means are always different. In Recognition Science, this theorem is named simultaneous_differs_from_sequential and it underpins a distinction between two ways of reducing a cost, a quantity that measures how far a ratio is from 1.

The framework models a system as a ledger, a discrete record of events, where each event compares a value with a neighbor. The cost of a comparison is J(x) = (x + 1/x)/2 - 1, which is zero when the ratio x equals 1 and positive otherwise. When a value has two neighbors, the total cost is the sum of the costs of each comparison. The framework proves that this total cost is minimized when the value equals the geometric mean of its two neighbors, not the arithmetic mean.

Because the geometric mean and the arithmetic mean differ whenever the two neighbors differ, the two ways of lowering cost are not the same. Adjusting a value to the geometric mean of its neighbors is a simultaneous adjustment, while moving toward the arithmetic mean would be a sequential adjustment. The theorem simultaneous_differs_from_sequential states this difference formally: for any two positive unequal numbers, the geometric mean is not the arithmetic mean. This is a proved theorem in the framework's machine-checked library of formal theorems.

This distinction matters because it shows that the order of operations changes the outcome. In a system where values adjust to reduce cost, doing all adjustments at once versus one after another leads to different results. The theorem is a precise statement of that difference, and it is a building block for understanding how the framework's cost geometry behaves.

/-- **Key structural fact**: Sequential single-bond descent (take v = n₁, then v = n₂,
    etc.) converges toward the arithmetic mean, while simultaneous descent converges
    to the geometric mean. The two differ for distinct neighbors. -/
theorem simultaneous_differs_from_sequential {n₁ n₂ : ℝ}
    (hn₁ : 0 < n₁) (hn₂ : 0 < n₂) (hne : n₁ ≠ n₂) :
    Real.sqrt (n₁ * n₂) ≠ arithmeticMean n₁ n₂ :=
  geometric_ne_arithmetic hn₁ hn₂ hne

What this page does not claim

This does not claim that the geometric mean is always the optimal value for more than two neighbors. This does not claim that the arithmetic mean is a cost-minimizing value under any condition. This does not claim that the theorem establishes anything about the physical interpretation of the cost function.

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

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