phiHierarchy_unique
plain-language theorem explainer
Any positive constant-ratio sequence obeying the Fibonacci recurrence must have ratio exactly the golden ratio phi. Researchers modeling minimal J-cost cache hierarchies or self-similar structures under recognition costs cite this uniqueness result. The proof is a one-line application of the forcing lemma that any Fibonacci partition forces the phi ratio.
Claim. Let $K : ℕ → ℝ$ be a sequence of positive reals satisfying the Fibonacci recurrence and having constant ratio $r > 0$. Then $r = ϕ$, where $ϕ$ is the golden ratio.
background
The module proves J-cost gradient descent on cache hierarchies necessarily converges to the Fibonacci/phi partition. A hierarchy is a sequence of positive reals $K$ representing cache-level capacities; total J-cost is the sum of pairwise ratio costs $∑ J(K(ℓ+1)/K(ℓ))$. Under the Fibonacci partition constraint forced by J-symmetry at optimal boundaries, the minimum-cost self-similar solution is the unique phi-geometric sequence. Upstream, the constant $K$ is defined as $ϕ^{1/2}$ and the cost of a recognition event is its J-cost; the magnitude-of-mismatch structure forces symmetry on carriers.
proof idea
The proof is a one-line wrapper that applies the lemma fibonacci_partition_forces_phi to the hypotheses on positivity, Fibonacci recurrence, and constant ratio.
why it matters
This theorem establishes the uniqueness of the phi ratio for any constant-ratio Fibonacci hierarchy, closing the loop that J-cost descent forces convergence to the phi-ladder. It realizes the self-similar fixed point (T6) in the forcing chain and supports the module claim that the phi-hierarchy is the unique constant-ratio hierarchy satisfying Fibonacci. No downstream uses are recorded yet.
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