Foundation Rhat Fixed Point Faster Contraction Faster Thinking

Faster contraction, faster thinking

In mathematics, a contraction is a function that brings points closer together with each application. If you start with any positive number and repeatedly apply a contraction that pulls toward 1, the values get closer to 1 at a rate governed by a number called the contraction rate, which lies between 0 and 1. A smaller rate means each step shrinks the distance more aggressively, so the sequence reaches its target faster. This is a standard fact in analysis, and it underlies many iterative methods used in numerical computation.

The Recognition Science framework applies this idea to its model of intelligence. The framework defines a recognition event as a discrete record of a comparison, and it uses a fixed cost function J(x) = (x + 1/x)/2 - 1 to measure the defect of a state x from the ideal state x = 1. The framework models thinking as a sequence of recognition steps that reduce this cost, and it treats the process as a contraction on the positive real line. The theorem faster_contraction_faster_thinking states that if one contraction rate is smaller than another, then after any positive number of steps, the error term for the faster contraction is no larger than the error for the slower one. In plain language: a smaller contraction rate means the state approaches the fixed point at least as quickly, and the framework interprets this as faster thinking.

The theorem is proved in the framework's machine-checked library of formal theorems. It relies on the monotonicity of powers: for positive bases less than 1, a smaller base raised to any positive power is less than or equal to a larger base raised to the same power, and multiplying by a positive error preserves the inequality. The proof is short and direct, with no gaps.

The framework does not claim that this theorem alone establishes any specific speed for human or artificial intelligence. It establishes a comparative statement: given two contractions, the one with the smaller rate converges no slower. The framework's further claim, that this rate corresponds to thinking speed, is an interpretation built on the model, not a consequence of the theorem itself. The theorem also does not say that a smaller rate always produces a strictly smaller error after every step; it only guarantees the inequality holds, with equality possible in degenerate cases.

/-- The convergence rate determines thinking speed:
    smaller contraction rate = faster convergence = faster thinking. -/
theorem faster_contraction_faster_thinking (c₁ c₂ : ℝ)
    (h₁ : 0 < c₁) (h₂ : 0 < c₂)
    (h_faster : c₁ < c₂) (error : ℝ) (he : 0 < error) (n : ℕ) :
    c₁ ^ n * error ≤ c₂ ^ n * error := by
  apply mul_le_mul_of_nonneg_right
  · exact pow_le_pow_left₀ (le_of_lt h₁) (le_of_lt h_faster) n
  · exact le_of_lt he

/-- The convergence rate determines thinking speed:
    smaller contraction rate = faster convergence = faster thinking. -/
theorem faster_contraction_faster_thinking (c₁ c₂ : ℝ)
    (h₁ : 0 < c₁) (h₂ : 0 < c₂)
    (h_faster : c₁ < c₂) (error : ℝ) (he : 0 < error) (n : ℕ) :
    c₁ ^ n * error ≤ c₂ ^ n * error := by
  apply mul_le_mul_of_nonneg_right
  · exact pow_le_pow_left₀ (le_of_lt h₁) (le_of_lt h_faster) n
  · exact le_of_lt he

What this page does not claim

The theorem does not prove that any particular cognitive process has a specific speed. The theorem does not claim that a smaller contraction rate always gives a strictly smaller error after every step. The framework's interpretation of contraction rate as thinking speed is a modeling choice, not a proved fact.

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

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