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 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. The lattice modular Hamiltonian converges to slope 2π; analogue black-hole data agree with the thermal reading.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. Version 5 extends this to 3+1: every quadratic write on the pair preserves the membrane, the writes form Sp(4,ℝ) ≅ Spin(3,2), and inside it the beam splitters and squeezers close on the Lorentz algebra, with the programme's write exactly one boost. The quadratures carry the Majorana spinor of SL(2,ℂ), the remaining generators form a four-vector whose invariant norm is the zero-point energy, and the Page–Wootters constraint of the pair is the rotation J₃ = (n_a − n_b)/2, which commutes with the write.Hardware. On two IBM Heron processors (ibm_marrakesh, ibm_fez) 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 1/2, with opposite phases for the two sectors and agreement between processors 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 an injected coherent error was recovered with gain 1.00 ± 0.04 while the geometric slope was unchanged. 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 are stated as seven protocols with predictions and refutation criteria for cQED, trapped-ion, photonic, Rydberg and BEC platforms. All code, the IBM results in JSON, and a numpy-only reanalysis are deposited.
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 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. The lattice modular Hamiltonian converges to slope 2π; analogue black-hole data agree with the thermal reading.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. Version 5 extends this to 3+1: every quadratic write on the pair preserves the membrane, the writes form Sp(4,ℝ) ≅ Spin(3,2), and inside it the beam splitters and squeezers close on the Lorentz algebra, with the programme's write exactly one boost. The quadratures carry the Majorana spinor of SL(2,ℂ), the remaining generators form a four-vector whose invariant norm is the zero-point energy, and the Page–Wootters constraint of the pair is the rotation J₃ = (n_a − n_b)/2, which commutes with the write.Hardware. On two IBM Heron processors (ibm_marrakesh, ibm_fez) 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 1/2, with opposite phases for the two sectors and agreement between processors 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 an injected coherent error was recovered with gain 1.00 ± 0.04 while the geometric slope was unchanged. 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 are stated as seven protocols with predictions and refutation criteria for cQED, trapped-ion, photonic, Rydberg and BEC platforms. All code, the IBM results in JSON, and a numpy-only reanalysis are deposited.
The parity–entanglement programme carries a single sector-exchange generator, X_A X_B, whose flow moves the vacuum bias while the membrane Z_A Z_B is conserved. Extending it from one link to a network of N excitations, and from ℤ₂ to an arbitrary abelian group, we obtain four results, all exact. (i) Every link generator X_i X_j commutes with the membrane — which is the single global check Z₁···Z_N, not one check per link — and all link generators commute with one another, so the network flow factorises identically and the state space of the network is a torus (ℤ₂)^(N−1) rather than a many-body problem [P]. (ii) The links are not independent: X_a X_v · X_v X_c = X_a X_c, so a cycle carries no new generator and the independent content of any network is a spanning tree, the same size-independence that fixes the crossing number at two bits [P]. (iii) On the saturated state every link acts as a pure phase, |⟨Ψ|e^(iθX_i X_j)|Ψ⟩| = 1 to twelve figures: at the bound the network is frozen, so the vacuum's strict even bias is not a blemish but the condition for a link to move at all [P/N]. (iv) The sector exchange and the product update are two different channels — the first conserves the membrane and moves the Schmidt spectrum, the second conserves the spectrum and is blind to the membrane — so one writes on the record and the other only carries it, and a link writes on exactly those cuts it crosses [P/N]; a write is moreover exactly internal to its link, conserving S₂ of the link with the rest of the network to 1.2×10⁻¹⁵ and leaving every outside expectation untouched. (v) Two links sharing a vertex are not additive, and the failure has a closed form with no free parameter, ⟨Z_v⟩(θ_A,θ_B) = cos2θ_A cos2θ_B ⟨Z_v⟩ − sin2θ_A sin2θ_B ⟨Z_v X_a X_c⟩, whose cross term is carried by the link closing the triangle — so (ii) and (iv) are one operator identity [P]. Disjoint links are additive to 1.9×10⁻¹⁶ at every N tested, and the closed form is exact to 10⁻¹⁶ for N = 3…6 over θ = 0.02…0.90 [N]. The triangle law is a law of the write channel, and the interaction is measured on ibm_marrakesh through the parity of the law in θ_A: the cross term is odd and the rest even, while any depolarising decay multiplies both, so the ratio of the antisymmetric to the symmetric combination cancels the decay identically. The measurement returns the interaction at 40σ against its absence and constant across six (θ_A,θ_B) points to within 12%, with an overall amplitude 36% high whose source is excluded by a dedicated run: a circuit-depth null and two gate-sign nulls, each exact for ideal circuits, return 0.6σ, 1.0σ and 0.1σ. The construction is carried out for a general ℤ_n: the membrane, the torus factorisation and the closing-link identity AB = X_a X_c† hold for every n, while the two-term closure is a ℤ₂ property, and we identify the reason — it requires the shift operator to be a real symmetric involution. For a non-abelian group the link generators cease to commute (‖[A,B]‖ = 1.0 in the regular representation of S₃) and the factorisation, hence the law, fails; the loop closure survives as a group identity. Statements are labelled [P] proved, [K] known, [N] numerical, [F] formal.
The parity–entanglement programme takes as primitive a stabiliser membrane: one conserved global check and a family of commuting two-local generators that write on it. A companion paper showed that this structure is the attractor of a survival dynamics defined on a population of candidate checks, and left as open the question of a selection dynamics defined intrinsically on the vacuum. We supply one. The reservoir is given three rules and nothing else: it attempts every operator at once, so that attempts which fail to commute cancel and contribute nothing; it is hot and dense and lowers its energy per degree of freedom, which for a horizon record means E/S = k_BT ∝ 1/ξ and so grows the record without bound; and what it retains must remain writable. No parity, no X, no Z, no membrane and no lattice enters the rules. Run blind, the three produce the structure. Cancellation alone does not: it accumulates constraints until the state is unique, reaching k = N in 100% of 1200 runs at N = 4, 6, 8, a frozen point with no record and no motion. Adding the cooling drive stops it exactly at one constraint — k = 1 in 100% of 1500 runs at N = 6, 8, 10 — because the energy per surviving cell, k/(N−k), is minimal there among structures that record anything at all. Requiring the result to remain writable then selects the weight: a constraint of weight N built from a single Pauli letter commutes with every two-local generator of the other two letters, 60/60 at N = 6 and 112/112 at N = 8, which no other weight achieves. What emerges is an [N, N−1] code with a global check and two-local commuting writes, which is the membrane. The remaining structure follows from results already proved: the independent content is a spanning tree of N−1 generators, and among candidate graphs the sparsest connected one wins because the crossing number is capped at two bits while every link costs a refresh — 92.8 for a spanning tree against 43.5 for a square lattice and 11.6 for the complete graph at N = 16. Statements are labelled [P] proved, [K] known, [N] numerical, [O] open.
Every entanglement event in the universe writes into the same vacuum record, and the number of such events is enormous while the record is one object. Three questions follow: how simultaneous writes avoid disturbing one another, how the record stays globally consistent without a shared clock, and how the balance that defines it is maintained while every local fluctuation moves. We show that the parity–entanglement register answers all three structurally rather than dynamically, and we verify each on an exact 4×4 lattice of 16 cells. Non-interference is commutativity: the link generators are X-type Pauli strings, so every pair commutes exactly, joint and sequential evolution agree to 0.00 and reversing the order changes the state by 1.8×10⁻¹⁶ [P/N]. A write is confined to its own link, conserving the link's entanglement with the rest of the lattice and leaving every outside expectation unchanged, so an event at one place moves that place's row of the emergent distance matrix and nothing else [P/N]. Consistency needs no communication: the membrane is a single global check Z₁···Z_N which every write preserves, holding at +1.000 through fifteen consecutive events [P/N]. And the balance is not maintained at all — it is global purity. For any partition of the whole into any number of parts, Σᵢ[½I(Pᵢ;rest) + S(Pᵢ|rest)] = 0 identically, because S(P) = S(rest) for every part of a pure state; the sum is exactly zero for five different partitions before and after twenty-five events, and |S(A) − S(B)| vanishes across all 696 cuts [P/N]. The two terms of the balance are not two quantities meeting at zero but one quantity counted twice with opposite signs, which is why they move together and the sum cannot drift. Breaking it requires discarding information: tracing out one cell makes the global state mixed and the identity fails immediately. Counting the real register, the horizon holds 4.71×10¹²² bits with as many independent write channels, and the Margolus–Levitin bound puts the writes performed since t=0 at 2.6×10¹²¹, so the record is a few per cent full. Statements are labelled [P] proved, [K] known, [N] numerical.
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.
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.
For a region of a free-fermion ground state the entanglement Hamiltonian K_A = log[(1 - C_A) C_A^-1] splits into a part proportional to the identity and a trace-free remainder. The identity part shifts every level rigidly, so every observable built from level differences is blind to it. This paper is about that part. It is not a bulk quantity. The block carries one ladder per entangling cut, the trace is the difference of the two cut-phase charges, and on the exactly dimerised chain with a single cut carrying the whole charge it has the closed form 2 artanh(v / sqrt(v^2 + t^2)), derived in one line and measured to 10^-11. In exact arithmetic the trace is a definite number: zero to thirty-three digits when the two cut phases agree, and 0.786153199019314523 when they do not. The apparent divergence reported in earlier work was double-precision truncation of a finite sum, and is retired here. Along the two lines through the Rice-Mele plane where the half-line entanglement Hamiltonian is known in closed form, it is proportional to the Bisognano-Wichmann deformation T = (1/2){X, H} with no additive constant, and the ladder spacing and phase are fixed by the literature. Between those lines the commuting operator survives, measured at 10^-13 against controls that fail at 1.5 and 29, but the proportionality does not: the level map acquires an intercept b that is zero to 3 x 10^-9 on both lines and of order 10^-1 in the interior, and is identical whether the region has one entangling cut or two. That intercept is the measured failure of exact Bisognano-Wichmann proportionality. At first order in the staggering v the intercept is derived rather than fitted. The symbol's determinant is det(z - P) = z(z - 1) independently of momentum, so the factorisation is canonical everywhere; its logarithm is affine in an involution R = H/E with R^2 = I; the abelian Szego-type boundary functional is identically even in v and therefore cannot carry an odd intercept; and the first-order piece follows from the chiral zero mode of the dimerised chain, whose amplitudes are (-r)^j C(2j,j) 4^-j in closed form. The result is b/v = 2(E' - K')/(1 - r^2) + K', K' = K(1 - r^2), E' = E(1 - r^2), r = t1/t2, whose expansion in the modulus u is, in exact rationals, h(u)/(pi/8) = 1 - (5/8) u^2 - (9/64) u^4 - (65/1024) u^6 - (595/16384) u^8 - ... So pi/8, -5/8 and -9/64 are consequences rather than recognitions, and the natural elliptic candidate [9E - K]/16 - pi/8 is excluded by 7.7 percent on the first coefficient it does not get to choose. The closed form agrees with direct measurement extrapolated to v -> 0 by two to three orders better than the fitted series it replaces; a previously reported u^6 coefficient of -0.0648 is corrected here to -65/1024 and identified as truncation. The trace-free half is treated in parallel. A Clausius relation with a single temperature holds exactly when that half is proportional to the region's own Hamiltonian. At a ground state it never is, and the measured temperature is zero, which is correct rather than a failure. In a thermal state the same instrument returns 1/beta to five significant figures, and the boundary of the region where it does so collapses onto two dimensionless ratios. Two further results are included. The offset vanishes when the two cuts sit at the same phase in the unit cell, without requiring chirality, time reversal, inversion or cleanliness, and a mirror-symmetrised disordered chain gives a larger offset than its unsymmetrised control, which rules reflection out as the mechanism. And in two dimensions the offset is exactly additive along the boundary for a transverse coupling proportional to the identity in sublattice space, for a reason short enough to state, with a derived crossing past which additivity fails and a sigma_z coupling that breaks it smoothly. Every claim carries one of four grades: [T] proven here, [V] computationally verified with stated controls and a stated failure mode, [C] imported from the cited literature, [H] interpretation. The two exact lines, the commuting deformation, the boundary-locality of the entanglement spectrum, and the first law of entanglement entropy are not this work's and are used as given. Sixteen retired claims, including two withdrawn from earlier versions of this manuscript, are listed with the measurement that killed each. Files in this deposit: the_offset_belongs_to_the_boundary_v8_submission.pdf — 35 pages, gate output verbatim, script listings supplied separately. the_offset_belongs_to_the_boundary_v8_complete.pdf — 59 pages, identical body with all scripts printed inline. offset_v8_scripts.zip — 23 files. Every number quoted in the paper is printed by one of them; the two gate files reproduce their stored output byte for byte. KEYWORDS entanglement Hamiltonian; entanglement spectrum; modular Hamiltonian; Bisognano-Wichmann; Rice-Mele model; SSH model; free fermions; boundary charge; cut-phase charge; corner transfer matrix; Toeplitz operators; Wiener-Hopf factorisation; Szego-Widom asymptotics; algebraic Heun operator; complete elliptic integrals; entanglement first law; area law; Landau-Widom counting function ADDITIONAL NOTES Supersedes earlier versions of this manuscript. Two claims present in versions 1 to 3 are withdrawn and recorded in the kill list: that the incommensurate offset has no value, and that the offset is one quantum. Both were double-precision truncation of a finite sum. The u^6 coefficient of the shape function reported in the companion results ledger is likewise superseded by the derived value. Computational verification, gating and drafting assistance by an AI collaborator (Claude, Anthropic). Every gate is independently re-runnable from the ancillary scripts.