The Membrane Is Fermion Parity: Hawking Temperature, Majorana Two Bits, and Relational Time as an Enclosed Area

The parity–entanglement programme rests on one object: a conserved global parity check (the membrane), commuting two-local writes, and one local order parameter ⟨P⟩. We show that this object is fermion parity and read three pieces of known physics off it.Hawking temperature. The Unruh–Hawking two-mode squeezed state is a membrane state with ⟨P⟩ = tanh(βω/2), so the modular scale λ = 2 artanh⟨P⟩ equals βω. The condition that a clock exists, ⟨P⟩ < 1, is the condition that the state is separating. With a positive-energy null translation, Borchers' theorem fixes the modular period to 2π, giving T_H = 1/8πM and S = A/4. Numerically, the lattice modular Hamiltonian converges to slope 2π. Analogue black-hole data agree with the thermal reading (χ² = 9.5/5).Membrane and geometry. Under Jordan–Wigner the membrane is (−1)^F, so its conservation is the fermion-parity superselection rule. Writes plus order-parameter terms are free fermions only on a path. The two bits are the Majorana entanglement of the Kitaev fixed point. In the three-dimensional fermionic completion the doubling is exactly twofold: two Majorana tastes forming one Dirac fermion. The emergent distance follows the coupling graph of the vacuum, and the write tree is a gauge fixing.Kinematics. Two-mode squeezing is a boost: predictability is 1/γ, visibility is v/c, and visibilities compose by the relativistic addition law. A sequence of writes is a polygon in velocity space: it returns the state iff the polygon closes, an unclosed one multiplies the energy by γ, and a closed one leaves a rotation equal to half the enclosed area. In the compact sector this is the Berry phase. The modular scale and the rapidity obey the Kramers–Wannier relation, self-dual at P = V = 1/√2.Hardware (versions 3–4). The compact-sector statements were tested on two IBM Heron processors (ibm_marrakesh, ibm_fez) on a membrane-encoded qubit. An ancilla interferometer measured the phase of a single parity-sector state after a closed loop of writes: |θ|/Ω = 0.4945 ± 0.0030 against the predicted 1/2 (1.8σ). The two sectors acquire opposite phases, and the processors agree to 0.7σ. The phase does not depend on the number of writes along a fixed path, a retraced loop leaves only a coherent offset, and a one-parameter coherent-error model reproduces the only failure of the slope. A deliberately injected coherent error was recovered with gain 1.00 ± 0.04 while the geometric slope was unchanged. The membrane interferometer thus separates control error from geometric phase. Entanglement across every cut of the Kitaev fixed point was certified for N = 4, 8 and 12. The measurements confirm the standard Berry phase within the programme's reading; its interpretation as relational time remains formal.Call for verification. The bosonic, critical and analogue statements cannot be tested on a qubit register. They are stated as seven protocols, each with a prediction and a refutation criterion, for cQED, trapped-ion, photonic, Rydberg and BEC platforms. Groups able to perform them are invited to contact the author. All code, the IBM results in JSON, and a numpy-only reanalysis script are deposited.