theorem activation_threshold_with_weights
{N : ℕ} (e : WeightedHierarchicalCESEconomy N)
(hN : 0 < N) :
1 < (weightedCycleProductBeta e) ^ ((1 : ℝ) / ↑N)
↔ 1 < weightedCycleProductBeta e := by
unfold weightedCycleProductBeta
exact activation_threshold_iff_product
(cycleProductBeta_pos e.toHierarchicalCESEconomy) hN## Activation with Weights and IRS