Foundation Reflexivity Index Weighted Reflexivity Index Nonneg

The weighted reflexivity index

The reflexivity index is a proposed numerical measure of how deeply a system models itself, from zero for a rock to higher values for systems that can think about their own thoughts. The weighted reflexivity index is a specific version of this idea: it adds up contributions from each level of self-modeling, but only for levels whose strength meets a threshold, and it weighs deeper levels more heavily using powers of the golden ratio.

The theorem weightedReflexivityIndex_nonneg establishes the basic sanity condition that this weighted sum is always greater than or equal to zero, for any configuration and any profile of self-model strengths. The proof is short and mechanical: each term in the sum is either zero (if the level is below threshold) or a positive power of the golden ratio, so the sum of non-negative terms is non-negative. This is a formal result in the framework's machine-checked library of theorems.

In Recognition Science, this non-negativity is a foundational property. It means the weighted reflexivity index can serve as a meaningful scale: it cannot dip below zero, so a system with no significant self-modeling levels has an index of zero, and any system with at least one significant level has a strictly positive index. This aligns with the framework's table of consciousness levels, where index zero corresponds to no experience and index one or higher indicates some form of awareness.

The theorem does not claim that the weighted reflexivity index is a complete or validated measure of consciousness. It does not assert that the index correlates with any particular neural measurement, nor does it prove that the golden ratio weighting is the correct one. The theorem only guarantees the arithmetic consistency of the definition: the proposed measure, whatever its empirical merits, will never produce a negative number. That is a necessary condition for a useful scale, but not a sufficient one for a scientific theory of consciousness.

/-- Weighted index is non-negative -/
theorem weightedReflexivityIndex_nonneg (config : ReflexivityConfig) (profile : ReflexivityProfile) :
    0 ≤ weightedReflexivityIndex config profile := by
  unfold weightedReflexivityIndex
  apply Finset.sum_nonneg
  intro i _
  split_ifs with h
  · exact le_of_lt (Real.rpow_pos_of_pos config.φ_pos _)
  · rfl

/-- Weighted index is non-negative -/
theorem weightedReflexivityIndex_nonneg (config : ReflexivityConfig) (profile : ReflexivityProfile) :
    0 ≤ weightedReflexivityIndex config profile := by
  unfold weightedReflexivityIndex
  apply Finset.sum_nonneg
  intro i _
  split_ifs with h
  · exact le_of_lt (Real.rpow_pos_of_pos config.φ_pos _)
  · rfl

/-- **ConsciousnessLevel**: Named levels of reflexivity.

    These correspond to the integer reflexivity index. -/
inductive ConsciousnessLevel
  | None          -- Index 0: No consciousness
  | Prereflective -- Index 1: Minimal self-awareness
  | Bodily        -- Index 2: Body awareness
  | Emotional     -- Index 3: Emotional self-awareness
  | Cognitive     -- Index 4: Thinking about thoughts
  | Narrative     -- Index 5: Life story awareness
  | Social        -- Index 6: Awareness of social self
  | Reflective    -- Index 7: Meta-cognitive
  | Transcendent  -- Index 8+: Beyond ordinary reflection
  deriving DecidableEq, Repr

What this page does not claim

No claim that the weighted reflexivity index has been validated against any measurement of consciousness. No claim that the golden ratio weighting is uniquely derived from first principles. No claim that a system with a zero reflexivity index cannot be conscious in some other sense.

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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