simplifiedCascadeCost
plain-language theorem explainer
The simplified cascade cost is defined as the J-cost applied to the self-similarity defect of a ratio r. Researchers analyzing variational models for the phi-ladder in fluid cascades would cite this when establishing optimality of geometric ratios. The definition is a direct one-line composition of the defect function with the J-cost.
Claim. The simplified cascade cost of a ratio $r$ is $J(r^2/(r+1))$, where $J$ is the J-cost function and $r^2/(r+1)$ is the self-similarity defect measuring deviation from the closure relation $r^2 = r + 1$.
background
This module sets up a variational model for the phi-ladder by quantifying how a ratio $r > 1$ fails the self-similar closure $r^2 = r + 1$. The self-similarity defect is the ratio $r^2/(r+1)$, which equals 1 exactly at the fixed point. The J-cost, known to be nonnegative and zero only at argument 1, is then composed with this defect to yield a scalar cost.
proof idea
One-line wrapper that applies Jcost directly to the result of selfSimilarityDefect r.
why it matters
This definition supplies the cost function used by phi_is_unique_optimal_ratio (which proves phi is the unique minimizer on (1, ∞)), simplifiedCascadeCost_nonneg, simplifiedCascadeCost_eq_zero_iff, and simplifiedCascadeCost_phi. It instantiates the Recognition Science claim that phi is the self-similar fixed point (T6) and supports the phi-ladder mass formula. No open scaffolding questions are addressed.
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