phiHierarchy
plain-language theorem explainer
The φ-hierarchy is the geometric sequence K(ℓ) = K₀ φ^ℓ. Researchers studying J-cost minimization on cache hierarchies cite it as the canonical self-similar solution under the Fibonacci partition constraint. The definition is supplied directly as a noncomputable geometric progression with common ratio phi.
Claim. The φ-hierarchy is the map given by $K(ℓ) = K_0 φ^ℓ$ for real $K_0$ and natural number $ℓ$.
background
The module proves J-cost gradient descent on sequences of positive reals (cache capacities) converges to the Fibonacci/φ partition. Total J-cost is the sum of pairwise ratio costs J(K(ℓ+1)/K(ℓ)). The minimum-cost self-similar solution under the growth constraint is the unique φ-geometric sequence. This definition supplies the explicit form K(ℓ) = K₀ φ^ℓ. It extends the upstream spectral-peak structure where characteristic frequencies satisfy ω_k = ω_0 φ^k.
proof idea
Direct definition as the geometric progression K₀ multiplied by phi raised to the power ℓ. No lemmas or tactics are applied inside the definition body; the object is primitive and immediately unfolded in all downstream results.
why it matters
This definition anchors the exponential-growth claims that follow. It is invoked by cumulative_growth_lower_bound to bound total complexity by K₀ φ^N and by phi_hierarchy_exponential_growth to obtain exact equality at level N. It realizes the self-similar fixed point (T6) and supplies the phi-ladder required for the eight-tick octave and D = 3 structure in the forcing chain.
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