Sharp embezzlement laws for quantum chains and finite Bell tests

For the finite open critical XX and Ising chains, we prove a sharp entanglement-embezzlement law from their exact discrete Schmidt spectra, including all degeneracies. For a maximally entangled target of fixed dimension d, the optimal squared local-unitary error is E_d = (ln d)^2/(4S) + O_d(S^-2), where S is the half-chain entanglement entropy. A counting criterion gives a Gaussian crossover for growing d and the same sharp law for the XXZ corner-transfer-matrix spectrum as Δ decreases to 1. We compare the optimal loss in a fixed Bell game with 243 question pairs to the Bell-pair embezzlement error with explicit constants, and prove the loss floor (3−2√2)/243 for maximally entangled states of every dimension. With a shared nine-dimensional maximally entangled register, the open XX ground state at N = 2^19 beats this floor in the original game. At N = 2^84 the chain alone suffices. A tagged test gives an exact tilted CHSH separation for the four-spin XX ground state, with classical play in the original game branch. For the infinite critical XX ground state, the half-chain factors are type III_1 and the game value is one without attainment.