The Parity-Entanglement Bound

In the Page–Wootters relational-time framework a ℤ₂ parity constraint (P_C ⊗ P_S)|Ψ⟩ = +|Ψ⟩ ties a clock C to a system S. We show that the constraint controls one specific term of the entanglement entropy — the sector (charge) entropy — which obeys H({p_e, p_o}) ≤ 1 bit with equality iff the two parity sectors are equally weighted, for every system size N, every clock size k, and every state in the constrained subspace. The familiar value I(C;S) = 2 bits is recovered exactly when the complementary configurational term vanishes, i.e. on the rank-2 family, to which the uniform ℤ₂ ground state belongs; there I = 2 bits independently of N, of k and of which qubits form the clock. We further prove an exact complementarity V² + P² = 1 between the clock–system fringe visibility and the sector predictability. On IBM quantum hardware we measure the quantity the theory actually predicts: the weight of the reduced clock state inside the rank-2 subspace. For an 8-dimensional (k = 3) clock register at N = 5 we find 0.920, against 0.250 expected if the register carried the naive 2k = 6 bits; the two surviving eigenvalues are 0.445 and 0.475. A depolarisation parameter calibrated from an independent stabilizer fidelity witness returns I = 1.998, 1.999, 1.997 bits for k = 1, 2, 3. A separable control state is included: a single-basis clock-marginal estimator returns 1.9991 bits on it, while the tomographic estimator used here returns 0.835 bits and V² + P² = 0.0005, so the protocol separates the constrained state from a classical mixture.