Foundation Reflexivity Index Reflexivity Index Theorem

The reflexivity index theorem

The reflexivity index is a proposed number that tries to capture how deeply a system models itself. The idea is simple: a rock models nothing, a mouse may model its body, and a person can model their own thoughts. The index assigns a non-negative integer to each level of this self-modeling, with zero meaning no self-model and larger numbers meaning deeper reflection.

In a machine-checked library of formal theorems, the declaration reflexivity_index_theorem establishes three basic facts about this index. First, the index is never negative. Second, it cannot exceed the maximum level of self-modeling considered, plus one. Third, the cost associated with each level of reflection is always non-negative. These are the theorem's only claims. It proves these bounds for any configuration of the model and any profile of strengths, meaning the result is fully general within the framework's definitions.

The theorem does not claim that the reflexivity index measures actual consciousness in any biological or psychological sense. It does not prove that a higher index corresponds to greater subjective experience, nor does it validate the proposed mapping of index values to states like flow states or meditation. The index is a formal construction, and the theorem is a statement about that construction's internal consistency, not about the world.

In Recognition Science, this theorem is a foundational check. It shows that the proposed index is well-behaved: it stays within expected bounds and never produces negative costs. This is a necessary first step if the index is ever to be used in further derivations, but it is far from sufficient. The leap from a formal index to a claim about real consciousness remains entirely open.

/-- **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⟩

/-- **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 reflexivity index corresponds to actual consciousness. The theorem does not validate the mapping of index values to specific mental states like flow or meditation. The theorem does not prove that a higher reflexivity index is better or more evolved.

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

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