pairCost
plain-language theorem explainer
pairCost defines the J-cost of the ratio between consecutive levels in a hierarchy sequence K. Researchers modeling cache growth under J-cost minimization cite it to decompose total hierarchy costs into per-pair contributions. The definition is a direct one-line application of Jcost to K(ℓ+1)/K(ℓ).
Claim. The pair cost at level ℓ for hierarchy K is J(K(ℓ+1)/K(ℓ)), where J is the J-cost function induced by the multiplicative recognizer comparator.
background
The module proves J-cost gradient descent on cache hierarchies converges to the Fibonacci/phi partition. A hierarchy is a sequence of positive reals K : ℕ → ℝ giving cache-level capacities; total J-cost is the sum over ℓ of J(K(ℓ+1)/K(ℓ)). The J-cost itself is the cost of a recognition event, taken from ObserverForcing.cost as Jcost of the event state and from MultiplicativeRecognizerL4.cost as derivedCost of the comparator.
proof idea
one-line wrapper that applies Jcost to the ratio of consecutive hierarchy values.
why it matters
This definition is invoked by the theorem phiHierarchy_pairCost to show every adjacent pair in the phi-hierarchy carries cost J(phi). It supports the module claim that the phi-geometric sequence is the unique minimum-cost self-similar hierarchy under growth constraints, consistent with J-uniqueness (T5) and the forced phi fixed point (T6) in the Recognition Science chain.
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