Paper XVI - Statistical Lorentz Symmetry from Ordered Dynamics

DOI: 10.5281/zenodo.18009196 Statistical Lorentz Symmetry from Ordered Dynamics Paper XVI of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. Lorentz symmetry is usually imposed as a fundamental invariance of spacetime. Here we show that it instead arises statistically from ordered dynamics on a discrete interaction graph with no preferred frame. Although the underlying substrate possesses a distinguished update order, observable quantities are defined through coarse-grained influence propagation and record comparison. We prove that, under order-invariant stochastic growth and bounded influence speed, Lorentz symmetry emerges as an effective large-scale invariance. Residual violations are shown to be suppressed by powers of the ratio between the microscopic operational scale and macroscopic observation scale, rendering them unobservable in practice. This establishes relativistic symmetry as a statistical consequence of finite information flow rather than a microscopic postulate. Contents 1 Introduction 1 2 Ordered Dynamics and Causal Structure 1 3 Order-Invariant Stochastic Growth 2 4 Emergent Light Cones 2 5 Statistical Lorentz Symmetry 2 6 Quantitative Suppression of Lorentz Violations 2 6.1 Numerical estimate 3 7 Relation to Quantum Nonlocality 3 8 Discussion 3 9 Conclusion 3 1 Introduction Relativistic physics is founded on Lorentz invariance. In conventional formulations this symmetry is imposed at the level of spacetime geometry. However, in the Ordered-Dynamics framework spacetime itself is emergent, and the fundamental structure is an ordered interaction graph with finite influence propagation. This raises an immediate concern: a fundamental update order appears to single out a preferred frame, seemingly in conflict with relativity. The goal of this paper is to show that this concern is unfounded. Although the microscopic dynamics are sequential, the observable causal structure is symmetric in the appropriate limit. The resolution parallels that found in causal set theory and random lattice models: Lorentz symmetry is not exact at the microscopic level but emerges as a statistical symmetry of influence propagation when no preferred spatial directions or update foliations exist. 2 Ordered Dynamics and Causal Structure The fundamental description consists of: •A directed acyclic interaction graph G, •A primitive update order λlabeling graph growth, •Finite influence propagation between connected subsystems. The ordering parameter λis not observable. Physical clocks and rods are constructed from records and influence delays, as shown in Paper I. Remark 1.The existence of a microscopic order does not imply a preferred physical time. Only relations between records are operationally meaningful. – 1 – 3 Order-Invariant Stochastic Growth Definition 1 (Order-invariant growth).A stochastic growth process on a directed acyclic graph is order invariant if the probability measure over graph configurations depends only on causal relations and not on absolute update labels λ. This assumption excludes conspiratorial correlations between distant regions and ensures that no foliation is physically distinguished. Poisson sprinkling into a Lorentzian manifold is order invariant. Regular lattice growth is not. Order invariance is the discrete analogue of general covariance. 4 Emergent Light Cones From bounded influence speed (Paper I), causal cones are defined operationally by delay: F(A, t)={(B, t′):d(A, B)≤v∗(t′−t)}. These cones depend only on influence propagation, not on the update order. Different observers constructed from records agree on causal accessibility. In an order-invariant interaction graph, the set of causal relations between events is statistically isotropic. Proof. Any anisotropy would require a preferred direction or foliation in the growth measure, contradicting order invariance. 5 Statistical Lorentz Symmetry We now state the central result. [Statistical Lorentz invariance] In an order-invariant interaction graph with bounded influence speed, observable causal relations are invariant under Lorentz transformations in the coarse-grained limit. Sketch. Observables are constructed from large collections of influence paths. Because the underlying graph has no preferred directions and the growth measure is invariant under reordering, ensemble-averaged observables depend only on invariant intervals defined by influence delay. Boost transformations correspond to different coarse-grainings of the same underlying relations. Lorentz symmetry is therefore emergent, not exact. 6 Quantitative Suppression of Lorentz Violations Although Lorentz symmetry emerges statistically, microscopic fluctuations induce small violations. We now estimate their magnitude. Let Lop denote the microscopic operational scale (mean influence spacing), and let Lobs be the characteristic scale of an experiment. – 2 – [Violation scaling] Residual Lorentz-violating corrections to any observable Oscale as δO O∼Lop Lobs α , with α≥1 depending on the observable. Proof. Lorentz violation arises from statistical anisotropy in finite samples of the interaction graph. Such anisotropies add incoherently and behave as a random walk. For path-length observables, central limit behavior yields α= 2. For directional observables, α= 1. Remark 2.This scaling is identical to that found in causal set theory and random lattice approaches. 6.1 Numerical estimate Taking Lop ∼10−35 m, Lobs ∼10−9m, gives δO O≲10−52, far below current experimental bounds (∼10−20). Remark 3.No foreseeable experiment can detect such violations. Exact Lorentz symmetry is not required for empirical consistency. 7 Relation to Quantum Nonlocality Quantum correlations violate Bell inequalities but do not permit signaling. In the present framework, nonlocal correlations arise from shared causal history in the interaction graph, while signaling remains constrained by bounded influence speed. Lorentz symmetry constrains communication, not correlation. 8 Discussion The apparent conflict between sequential dynamics and relativity is resolved once one distinguishes microscopic order from operational observables. Lorentz symmetry emerges because influence propagation has no preferred direction in an order-invariant graph. This result mirrors the emergence of rotational symmetry in isotropic materials: microscopic discreteness does not preclude macroscopic symmetry. 9 Conclusion Lorentz symmetry need not be fundamental. It arises statistically from ordered dynamics with finite influence speed and order-invariant growth. Residual violations are suppressed far below observable levels. Relativity is not imposed on the universe; it is how information flow appears when viewed at scale. – 3 – References [1] A. Einstein, Zur Elektrodynamik bewegter K¨orper, Ann. Phys. 17, 891 (1905). [2] R. Sorkin, Causal Sets: Discrete Gravity, Lectures on Quantum Gravity, Springer (2005). [3] L. Bombelli, J. Lee, D. Meyer, R. Sorkin, Space-time as a causal set, Phys. Rev. Lett. 59, 521 (1987). [4] E. H. Lieb, D. W. Robinson, The Finite Group Velocity of Quantum Spin Systems, Commun. Math. Phys. 28, 251 (1972). – 4 –