Foundation Reflexivity Index Phi Layer Strength Decreasing

The strength decay law

The golden ratio φ, approximately 1.618, appears throughout mathematics as the solution to r² = r + 1. Its powers form a geometric sequence: φ¹ ≈ 1.618, φ² ≈ 2.618, φ³ ≈ 4.236, and so on. Because φ is greater than 1, its positive powers grow without bound, while its negative powers shrink toward zero. The reciprocal sequence 1/φ, 1/φ², 1/φ³, ... decreases by a constant factor at each step.

In the Recognition Science framework, this decay becomes a structural law. The framework models a conscious system as having layers of self-modeling, where level 0 is the base experience and each higher level is a model of the level below. The strength of a layer is defined as φ raised to the negative of its level number: layer k has strength φ^(−k). So layer 0 has strength 1, layer 1 has strength 1/φ ≈ 0.618, layer 2 has strength 1/φ² ≈ 0.382, and so on.

The machine-checked theorem phiLayerStrength_decreasing proves that this sequence is strictly decreasing: for any two levels k₁ < k₂, the strength at k₂ is strictly less than the strength at k₁. The proof is a direct application of the monotonicity of real exponentiation, requiring only the assumption that φ > 1. The theorem also establishes that all strengths are positive, since any positive base raised to any real power is positive.

This decay law has a concrete consequence within the framework: deeper layers of self-reflection are always weaker than shallower ones, and the weakening is exponential, not linear. The framework uses this to define a reflexivity index, a non-negative integer counting how many layers exceed a threshold strength. The default threshold is 1/φ, so only layers 0 and 1 qualify in the default configuration, giving a baseline index of 2 for minimally conscious systems. Higher indices require either a lower threshold or a different base φ.

What the theorem does not claim is just as important as what it proves. It does not assert that any particular conscious system actually has these layer strengths; that is a modeling choice, not a derivation. It does not say that φ is the only possible base, since the theorem holds for any real φ > 1. And it does not claim that the reflexivity index measures anything about real brains; the framework's connection to neural integration measures such as Φ from integrated information theory remains a testable prediction, not a theorem.

/-- φ-layer strength decreases with level -/
theorem phiLayerStrength_decreasing (φ : ℝ) (hφ : 1 < φ) (k₁ k₂ : ℕ) (hk : k₁ < k₂) :
    phiLayerStrength φ k₂ < phiLayerStrength φ k₁ := by
  unfold phiLayerStrength
  apply Real.rpow_lt_rpow_of_exponent_lt hφ
  simp only [neg_lt_neg_iff, Nat.cast_lt]
  exact hk

/-- **PhiLayerStrength**: Expected strength at each level follows φ-decay.

    In a "natural" cognitive system, the strength at level k is
    approximately φ^(-k). This is the RS prediction. -/
noncomputable def phiLayerStrength (φ : ℝ) (level : ℕ) : ℝ :=
  φ ^ (-(level : ℝ))

/-- φ-layer strength is positive -/
theorem phiLayerStrength_pos (φ : ℝ) (hφ : 0 < φ) (level : ℕ) :
    0 < phiLayerStrength φ level :=
  Real.rpow_pos_of_pos hφ _

What this page does not claim

The theorem does not claim that any real conscious system actually has these layer strengths. The theorem does not claim that φ is the only possible base for the decay law. The theorem does not claim that the reflexivity index measures anything about real brains.

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.

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