The Parity-Entanglement Bound

We prove that any bipartite pure state satisfying a Z₂ parity constraint (P_C ⊗ P_S)|Ψ⟩ = +|Ψ⟩ with both sectors populated carries mutual information I(C;S) ≤ 2 bits, with equality if and only if the two Schmidt coefficients are equal (Parity-Entanglement Bound, PEB). Within the Page–Wootters relational-time framework, the PEB implies that the clock–system mutual information is at most 2 bits, achieving exactly 2 bits for the maximum-entropy ground state, independently of system size N, of which subsystem is the clock (clock-position-independence), and of how many qubits the clock contains (clock-size-independence). We further prove that among the uniform ground states of cyclic Z_n constraints, Z₂ uniquely minimises I(C;S) subject to I(C;S) > 0. N- and clock-independence are verified on IBM Kingston quantum hardware for N = 2, 3, 4, 5 qubits and all clock-qubit positions at N = 4, with all deviations below 0.15σ from the predicted 2 bits. The PEB answers why Moreva et al.'s photon experiment measured I ≈ 2 bits: this is not a feature of their setup but a consequence of the underlying Z₂ constraint.