phiHierarchy_ratio
plain-language theorem explainer
The sequence K(ℓ) = K₀ φ^ℓ satisfies K(ℓ+1) = φ K(ℓ) for all natural numbers ℓ. Researchers minimizing J-cost on cache hierarchies cite this to confirm self-similar growth under the Recognition Composition Law. The proof is a direct algebraic verification obtained by unfolding the definition and applying ring simplification.
Claim. Let $K : ℕ → ℝ$ be defined by $K(ℓ) := K_0 φ^ℓ$. Then $K(ℓ+1) = φ · K(ℓ)$ holds for every $ℓ ∈ ℕ$.
background
A hierarchy is a sequence of positive reals K : ℕ → ℝ representing cache-level capacities. The constant_ratio property requires K(ℓ+1) = r K(ℓ) for all ℓ. The φ-hierarchy is the geometric sequence K(ℓ) = K₀ φ^ℓ. This module proves that J-cost gradient descent on hierarchies necessarily converges to the Fibonacci/φ partition, with the φ-geometric sequence as the unique self-similar minimum-cost solution under J-symmetry at optimal boundaries.
proof idea
The proof is a one-line wrapper that applies the definition of phiHierarchy and simplifies the resulting equality using ring tactics.
why it matters
This result establishes the constant-ratio property for the φ-hierarchy, which is required to invoke the key lemma that Fibonacci recurrence plus constant positive ratio forces r² = r + 1. It supports the argument that the φ-partition is the unique global minimum under J-symmetry constraints, tying directly into the self-similar fixed point forced at T6 of the Unified Forcing Chain.
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