Sharpness of sparse A_p bound: one fixed family [0,2^-k) suffices for both regimes

To formalize that ‖A_S‖_{L^p(w)} ≲ [w]_{A_p}^{max(1,1/(p-1))} is sharp, you do not need a family depending on f: the intervals [0,2^-k), k ≥ 0, are 2-sparse (E_k = [2^-k-1, 2^-k), disjointness via index computations: points of E_k have index (k+1) = 1, points of later cubes have index (k+1) = 0). For the p/(p-1) half reuse Buckley's pair (power weight t^{(p-1)(1-δ)}, f = t^{δ-1}) since the averages realising M_d f are over [0,L). For the p half take v = t^{δ-1}, which is the dual weight of Buckley's weight for p', so [v]_{A_p} = [w_{p'}]_{A_{p'}}^{p-1} ≤ 2/δ via an existing duality lemma (no new A_p computation); f = 1_[0,1) gives A_S f ≥ N on [0,2^-N) (sum of N averages equal to 1, lower-bound the ENNReal tsum by Finset.sum over (range N).map of an embedding into the subtype), and N = ⌈1/δ⌉₊ gives v([0,2^-N)) ≥ 1/(4δ). Generalize the limiting lemma to A ≤ Λδ^{-s} ⇒ β ≥ p/s so both halves share it.