Foundation Reflexivity Index Reflexivity Cost Exponential

The cost of deeper reflection

The reflexivity index is a proposed measure of how deeply a system models itself, from zero for a rock to higher integers for self-aware minds. The framework's library, a machine-checked collection of formal theorems, defines a specific cost for each level of this index: the cost at level k is φ^k − 1, where φ is the golden ratio, about 1.618. This is a definitional choice, a model, not a derived fact.

What the library proves about this cost is a simple monotonicity statement: if one level is deeper than another, its cost is strictly larger. The theorem reflexivityCost_exponential states that for any two levels k₁ and k₂ with k₁ < k₂, the cost at k₁ is strictly less than the cost at k₂. Because the base φ is greater than 1, the cost grows exponentially with level: each step up in self-reflection is more expensive than the one before it.

This is a theorem about the framework's own model, not a claim about any actual mind. It says nothing about whether the reflexivity index is a correct measure of consciousness, whether the golden ratio is the right base, or whether the cost function has any physical meaning. The theorem holds for any configuration where the base is greater than 1; the specific choice of φ is a separate, unproved assumption of the model.

The practical consequence within the framework is that the cost of self-reflection is not linear: moving from level 3 to 4 costs more than moving from 2 to 3. This supports the model's picture of why higher-order reflection is rare or effortful, but the framework does not claim this matches any measured property of minds. The theorem is a formal fact about a chosen definition, not an empirical finding.

/-- The cost of level k grows exponentially with k -/
theorem reflexivityCost_exponential (config : ReflexivityConfig) (k₁ k₂ : ℕ) (hk : k₁ < k₂) :
    reflexivityCost config k₁ < reflexivityCost config k₂ := by
  unfold reflexivityCost
  have h : config.φ ^ k₁ < config.φ ^ k₂ := pow_lt_pow_right₀ config.φ_gt_one hk
  linarith

/-- **ReflexivityCost**: The J-cost of maintaining a given reflexivity level.

    Higher reflexivity requires more "energy" (has higher J-cost).
    This explains cognitive limits on self-reflection.
    Uses ℕ-power: `config.φ ^ level - 1`. -/
noncomputable def reflexivityCost (config : ReflexivityConfig) (level : ℕ) : ℝ :=
  config.φ ^ level - 1

What this page does not claim

The reflexivity index is a validated measure of consciousness in any real system. The golden ratio base φ is derived from first principles rather than chosen. The exponential growth of cost has any observed counterpart in psychology or neuroscience.

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

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