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.
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.
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) fixes the maximal clock–system mutual information of a ℤ₂ parity sector to a quantum of 2 bits, saturated at the balanced state, and the vacuum asymmetry identity I_parity + 2S(A|B) = 0 shows that this information is exchanged between the two sides of a bipartition without being annihilated, the conserved stabilizer P_A ⊗ P_B acting as a membrane. Here we promote that single-channel membrane to the full conserved-charge content of a Standard-Model (SM) excitation. Each excitation is mirrored into a reservoir partner carrying the conjugate of every additive charge, and the membrane becomes the stabilizer group G = ⊗_c S_c with one generator per conserved label (parity, colour, weak isospin, hypercharge, spin, B−L). We show that (i) the membrane is preserved on every channel, abelian and non-abelian; (ii) the relational information per channel is the cutoff-free value 2 log₂ d_ρ fixed by the irrep dimension via Schur's lemma, recovering the I = 2 log₂ n law of the ℤₙ extension; and (iii) the exchange flows the parity weight bidirectionally, ⟨P_A⟩(θ) = ⟨P_A⟩₀ cos 2θ, carrying weight from the reservoir to spacetime and back through balance and sign reversal, while the stabilizer holds at every angle. All three statements are realized on IBM Kingston superconducting hardware in a single 13-circuit job: the abelian six-channel membrane stays at ⟨Z_A Z_B⟩ ≥ 0.88 across a full exchange period, and the SU(2) and SU(3) colour membranes hold at 0.94–0.97 in both directions. The construction demonstrates that the exchange-without-annihilation mechanism extends, intact and bidirectional, to the complete charge structure of the Standard Model.
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.
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.
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 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.