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.
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.
The Parity–Entanglement Bound (PEB) program takes the two-bit parity stabilizer—the membrane—as its primitive: prior work derives relational time, an emergent metric, and a geometric field law once the membrane is assumed. Here we ask where the membrane itself comes from, and answer with a selection principle. We define a resource-bounded survival dynamics on random local check structures whose only currency is relational information retained per unit resource under noise, with every rule fixed a priori and no tuned weight. Across independent random histories the dynamics converges, 12/12, to a single structure: a crossing Z-type generator (the membrane) and a crossing X-type logical (the exchange), mutually commuting, of minimal weight—exactly the stabilizer pair {ZᵢZ_N, XᵢX_N} of the [N,N−1,2] parity code, with e_{C|S}=2 and I=2 bits. The membrane is thus not an assumption but an attractor: the most information-efficient survivor of a competition among structures. The convergence is stable under larger populations and longer runs, and identical under depolarizing, dephasing, and amplitude-damping retention—a property of the structure, not of the noise model. The commutativity that lets the exchange move sector weight without annihilating the membrane—"exchange without annihilation" in the vacuum-asymmetry analysis—emerges from selection rather than being imposed. We further map the framework's own viability: on the emergent-geometry side the metric axioms hold only on an interior window of the information axis, Π∈(0,Π_c), with a parameter-free coupling that diverges at both walls—a measured sensitivity diagram for the structure's constants. Statements are labelled [P] proved, [K] known, [N] numerical.
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 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 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.
Within the ΔI=0 program, where information is continually redistributed between spacetime and the vacuum reservoir, one question is central: what mechanism lets information pass between the two sides, in both directions, without the two cancelling each other to nothing? We answer it, kinematically, for the Z2 parity sector of a free scalar field's vacuum. Recasting the sector in the language of negative conditional entropy, we prove a structural identity — the positive mutual information and the negative conditional entropy cancel exactly, I_parity + 2 S(A|B)_parity = 0, for every value of the parity weight, with no even/odd symmetry required. We then exhibit the mechanism: entanglement is a bidirectional gate. The shared Z2 parity bit is exchanged across the cut with no local operation crossing it — the membrane is the conserved stabilizer P_A ⊗ P_B — so weight flows between the sectors while the stabilizer holds and the sum stays zero: exchange without annihilation, a border crossing without contact. The vacuum is moreover always even-biased, ⟨P_A⟩ > 0, so its conditional entropy never reaches the maximally-negative value −1; the residual 1 − h(|α|²) is an order parameter of the asymmetry — there is no perfect mirror. Scope: parity sector only, vacuum state (not energy); the identity holds for any parity-even theory. The reading of the two sides as spacetime and reservoir, and of the exchange as a temporal update, is a working hypothesis, not used in any proof.
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.