theorem cross_level_amplification_with_weights
{N : ℕ} (e : WeightedHierarchicalCESEconomy N)
(hP : 1 < weightedCycleProductBeta e) :
∃ n : Fin N, 1 < e.beta n := by
unfold weightedCycleProductBeta cycleProductBeta at hP
exact cross_level_amplification _
(fun n => e.hbeta n) hP## Activation with Weights and IRS