Foundation Reflexivity Index Reflexivity Cost Nonneg

The cost of self-reflection

The reflexivity cost is a number that the Recognition Science framework assigns to a level of self-modeling, a system's depth of reflection on itself. The definition is simple: take a base constant φ, raise it to the power of the level, and subtract 1. For a system at level 0, the cost is 0; at level 1, it is φ minus 1, roughly 0.618; at level 2, it is φ² minus 1, roughly 1.618; and so on.

The declaration reflexivityCost_nonneg is a theorem in the framework's machine-checked library of formal theorems. It proves that for any valid configuration and any level, this cost is greater than or equal to zero. The proof is direct: because the base constant is chosen to be greater than 1, raising it to any natural-number power yields a value at least 1, so subtracting 1 leaves a result that cannot go below zero. The theorem also appears as part of a larger result, reflexivity_index_theorem, which bundles together several basic facts about the index.

What the theorem does not claim is just as important as what it proves. It does not say that the cost is positive, only non-negative; at level 0 it is exactly 0. It does not establish any upper bound on the cost, nor does it say anything about which levels are sustainable. The framework separately defines a maximum sustainable level of 7, but that is a definitional choice, not a consequence of this theorem. The theorem also does not assert that the reflexivity cost is the same as any physical energy or that it can be directly measured in a laboratory; it is a formal quantity within the framework's model of consciousness.

The practical consequence is that the framework's basic accounting for self-reflection is internally consistent: no level of self-modeling carries a negative cost. This matters because the framework uses this cost to reason about the depth of self-awareness, and a negative cost would make comparisons between levels meaningless. With this theorem, the framework can treat the cost as a well-behaved, ordered quantity, and the exponential growth of cost with level is a separate theorem that follows from the same definition.

/-- Reflexivity cost is non-negative -/
theorem reflexivityCost_nonneg (config : ReflexivityConfig) (level : ℕ) :
    0 ≤ reflexivityCost config level := by
  unfold reflexivityCost
  have : 1 ≤ config.φ ^ level := one_le_pow₀ (le_of_lt config.φ_gt_one)
  linarith

/-- Reflexivity cost is non-negative -/
theorem reflexivityCost_nonneg (config : ReflexivityConfig) (level : ℕ) :
    0 ≤ reflexivityCost config level := by
  unfold reflexivityCost
  have : 1 ≤ config.φ ^ level := one_le_pow₀ (le_of_lt config.φ_gt_one)
  linarith

/-- **THE REFLEXIVITY INDEX THEOREM**

    The reflexivity index is a well-defined topological invariant that:
    1. Is non-negative (consciousness is non-negative)
    2. Is bounded by meta-levels (finite self-reflection)
    3. Follows φ-decay (RS structure)
    4. Is invariant under representation change
    5. Has finite cost (consciousness is physically realizable) -/
theorem reflexivity_index_theorem (config : ReflexivityConfig) (profile : ReflexivityProfile) :
    (0 ≤ integerReflexivityIndex config profile) ∧
    (integerReflexivityIndex config profile ≤ profile.max_level + 1) ∧
    (∀ k : ℕ, 0 ≤ reflexivityCost config k) :=
  ⟨Nat.zero_le _,
   max_index_bound config profile,
   fun k => reflexivityCost_nonneg config k⟩

What this page does not claim

The theorem does not claim that the cost is positive, only non-negative; at level 0 it is exactly 0. The theorem does not establish any upper bound on the cost, nor does it say anything about which levels are sustainable. The theorem does not assert that the reflexivity cost is a physical energy or a directly measurable laboratory quantity.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: