The Parity–Entanglement Bound (PEB) fixes a ceiling: any ℤ₂-constrained bipartite pure state with both sectors populated carries I(C;S) ≤ 2 bits. Whether the 2-bit value is also a floor — a genuine quantum of information — has remained implicit in the program's language. Here we settle it, by proof and by hardware, in both directions at once. There is no unconditional floor: I takes every value in (0,2] on the constrained family, and we measure states down to I = 0.394 bits. There are, however, exactly two conditional floors, each activated by a different operational requirement. Floor I (locality): every local k=1 autoencoder of the balanced state has reconstruction fidelity exactly 1/4 — a flat floor, proved in three lines and measured flat at 0.250 ± 0.009 across five distinct local encoders on ibm_fez, while a single cut-crossing gate restores the device ceiling (0.973 vs. the k=2 control's 0.975: global compression is free). Floor II (clock function): the rank-2 rigidity bound I ≥ 2 − 2log₂(1+ε), with ε the tick overlap, holds at every point of a measured Schmidt sweep (8/8, forbidden region empty), with the two independently measured curves I = 2h(b) and ε = |2b−1| tracking theory to ≤ 0.03 bits and ≤ 0.007 respectively. The two floors meet the PEB ceiling at a single point: ε = 0 ⇔ b = 1/2 ⇔ I = 2. The 2-bit quantum is therefore not the floor of the constraint but the pinch of the unconditional ceiling with the conditional floors — information below 2 bits is always paid for, exactly, in tick overlap, and the shared bits can be globally compressed for free but locally not at all. Statements are labelled [P] proved, [K] known, [N] measured.
The Parity–Entanglement Bound (PEB) fixes a ceiling: any ℤ₂-constrained bipartite pure state with both sectors populated carries I(C;S) ≤ 2 bits. Whether the 2-bit value is also a floor — a genuine quantum of information — has remained implicit in the program's language. Here we settle it, by proof and by hardware, in both directions at once. There is no unconditional floor: I takes every value in (0,2] on the constrained family, and we measure states down to I = 0.394 bits. There are, however, exactly two conditional floors, each activated by a different operational requirement. Floor I (locality): every local k=1 autoencoder of the balanced state has reconstruction fidelity exactly 1/4 — a flat floor, proved in three lines and measured flat at 0.250 ± 0.009 across five distinct local encoders on ibm_fez, while a single cut-crossing gate restores the device ceiling (0.973 vs. the k=2 control's 0.975: global compression is free). Floor II (clock function): the rank-2 rigidity bound I ≥ 2 − 2log₂(1+ε), with ε the tick overlap, holds at every point of a measured Schmidt sweep (8/8, forbidden region empty), with the two independently measured curves I = 2h(b) and ε = |2b−1| tracking theory to ≤ 0.03 bits and ≤ 0.007 respectively. The two floors meet the PEB ceiling at a single point: ε = 0 ⇔ b = 1/2 ⇔ I = 2. The 2-bit quantum is therefore not the floor of the constraint but the pinch of the unconditional ceiling with the conditional floors — information below 2 bits is always paid for, exactly, in tick overlap, and the shared bits can be globally compressed for free but locally not at all. Statements are labelled [P] proved, [K] known, [N] measured.
We prove that any bipartite pure state satisfying a Z2 parity constraint (PC ⊗PS)|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 and bipartition choice (clock-independence). We further prove that among the uniform ground states of cyclic Zn constraints, Z2 uniquely minimises I(C; S) subject to I(C; S) > 0. N-independence andbclock-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 [2] measured I ≈ 2 bits: this is not a feature of their setup but a consequence of the underlying Z2 constraint.
The Parity-Entanglement Bound (PEB) fixes the clock-system mutual information of a ℤ₂ constrained bipartite pure state at exactly two bits. We show that this invariant functions as the zero point of a metric geometry built from information. Six results are given. (i) The deviation Π = 1 - 1/2 I is the unique local-unitary (LU) invariant of the minimal state (pure, ℤ₂, bipartite); relaxing each hypothesis adds a definite number of further axes, which we count numerically (mixedness +1, ℤn+(n-2), tripartite +2; a continuous gauge group opens a continuous axis), (ii) The correlation quantity d = -ln κ is non-negative, symmetric, and vanishes in the coincidence limit; in the gapped regime, at separations above a floor r*(m), it obeys the triangle inequality and is a metric. (iii) The validity window is mapped: the floor r*(m) grows with the gap and the window closes entirely once the correlation length falls below the lattice scale (m ≥ 1.7 in 1D, ≥ 2 in 2D, lattice units); in the opposite, gapless limit the inequality survives with growing slack (0 violations atm m = 0.02) and an asymptotic argument establishes it analytically in the logarithmic regime. Non-Gaussianity per se does not damage the inequality: on a φ⁴ chain the observed violations are reproduced quantitatively by a free field of matched effective gap. (iv) Local noise acts on d as an exactly additive per-region dilation, d → d + δA + δB with δ ≥ 0; pure dephasing leaves d exactly invariant, and local noise can only increase the triangle slack, so the metric is noise-robust by construction. (v) The PEB value equals a combinatorial crossing number of stabiliser generators, hence is independent of system size, clock size, and clock position; it is formally analogous to a topological entanglement entropy. (vi) Following the Dirac-equation construction [6], the two-dimensionality dim ℋC = 2 underlies a distance/time split. The construction is verified by exact and Monte-Carlo simulation (geodesic locking 60/60 across five seeds; zero triangle violations above the floor in more than 1.6 × 10⁴ triples spanning gapped, near-critical, interacting, and noisy regimes) and, as a proof of concept, by a state-preparation study on quantum hardware (IBM FEZ). Formal correspondences to gravitational behaviour geodesic bending in a non-uniform gap and a volume-to-area transition in the crossing count are recorded in Sec. no gravitational derivation is claimed. Statements are labelled [P] proved, [K] known-with-citation, [N] numerical, or [0] open.
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.
The Parity-Entanglement Bound (PEB) and its Zn / non-abelian extensions fix the clock-system mutual information I(C;S) of a symmetry-constrained pure state to an absolute, calculable value: 2.000 bits for a uniform Z2 constraint, 2 log2(n) for Zn, and representation-dependent values (e.g., 3.84 bits for S3). These invariants are independent of qubit number, clock size, clock position, and physical platform. Here we convert these mathematical results into a platform-agnostic, rank-agnostic benchmarking protocol for quantum processors. A single experimental run provides three-axis error diagnosis - passive decoherence, active gate noise, and logical survivability - all referenced to the absolute invariant, without any external baseline or classical simulation. The same protocol also enables lightweight entanglement certification in quantum networks (single-qubit, two-basis measurement). The method is demonstrated on IBM Kingston hardware for N = 2-5 qubits; all measured values lie within 0.15 sigma of the target, and a Schmidt-balance sweep follows the theoretical curve with R^2 = 0.998. The PEB thus supplies a universal ruler for calibration, error diagnosis, and cross-platform comparison - a tool that assists quantum computation without being a fault-tolerance code.
The parity–entanglement bound (PEB) fixes the relational information of a Zₙ-constrained uniform state at I(A:B) = 2 log₂ n bits for every bipartition and every system size. The topological entanglement entropy of a Zₙ gauge theory is γ = log n, likewise independent of the size and shape of the region. We show that these are one constant: I_PEB = 2γ, exactly, for every n, and that the agreement is not numerology but the same superselection structure read twice — the cut crosses n charge sectors, and both quantities are the logarithm of that count. Extending to a finite group G, the programme's decomposition gives I = 2 log₂|G| − ΔI_NC with ΔI_NC = 2 Σ_ρ P_ρ log₂ d_ρ, so the non-abelian censorship of the programme is exactly the shortfall of the bound below 2γ, and it vanishes precisely on the abelian groups. Both statements are verified exactly, by singular values for n = 2–5 at every cut and size and by representation theory for nine groups [N]. Both sides are then measured on IBM Heron processors, with nothing fitted [N]. Abelian: nine independent cuts across three sizes give a weighted mean of 1.999 bits with a spread of 0.004 bits, encoded Z₃ and Z₄ registers give 3.199 and 3.978 bits against 3.170 and 4.000, and the reading is unchanged at 27 qubits and depth 107. Non-abelian: on the S₃-constrained state the two informations are read from one tomography and differ by 1.329 ± 0.016 bits against the predicted ΔI_NC = 1.3333, a 0.3σ agreement and 83σ from zero, while the same analysis on an abelian control returns −0.008 ± 0.004; the censored value agrees with the one prepared by a different route on a third processor in earlier work. The practical content is that the programme's two-bit constant is a quantity other fields have already realised, without reference to this programme, as topological order on three platforms and as a dense-coding capacity on a fourth [K]; what is new is that one constant, read three ways across seven platforms, agrees throughout, and that the censorship term acquires a standard identity as the edge-mode contribution. Statements are labelled [P] proved, [K] known, [N] numerical or measured.
The Parity-Entanglement Bound (PEB) fixes a minimal relational-information quantum of exactly two bits for a bipartite pure state under a Z2 constraint. We show that this quantum is best understood as a single local-unitary equivalence class of states with fixed mutual information I = 2, and that three features of the framework are faces of this one object: (i) non-locality -- the class is invariant under all local unitaries, so no local operation can change it; (ii) balance without mirror symmetry -- the class contains highly asymmetric representatives whose two sides carry equal entropy yet differ in structure, so global balance needs no symmetric copy; and (iii) sector exchange -- an internal rotation moves weight between the two Z2 parity sectors at conserved total while the state stays in the class. We give the exchange dynamics as a continuity equation, confirm by simulation (including a realistic noise model) that the local/non-local distinction survives realistic decoherence, and address anticipated objections. The significance is structural: this same two-bit class is the quantum from which the framework derives a single time dimension, the invariant speed c, and the Dirac structure. The statement is platform-independent and testable on any entanglement-capable hardware. This work is part of a wider framework built on information conservation (Delta I = 0) and the minimal two-bit quantum.