Paper XII - Emergent Spatial Locality from Bounded Influence

DOI: 10.5281/zenodo.18009096 Emergent Spatial Locality from Bounded Influence Paper XII of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We derive spatial locality, distance, and dimensionality from purely operational information-theoretic principles without assuming spacetime, geometry, or metric structure. Starting from minimal assumptions about records, distinguishability, and bounded influence propagation, we show that a sparse interaction graph with a well-defined distance function must emerge. Finite influence capacity enforces bounded degree, while regular growth conditions define an emergent spatial dimension. Requiring the existence of stable macroscopic records dynamically selects three spatial dimensions as the minimal viable case. Lorentzian causal structure arises as a symmetry of influence propagation rather than a postulate. This establishes space as an emergent consequence of finite information flow, providing the foundational substrate for gravity, gauge structure, and quantum dynamics developed in subsequent papers. Contents 1 Introduction 1 2 Operational Framework 2 2.1 States and records 2 2.2 Influence and signaling 2 2.3 Finite propagation 2 3 Distance from Delay 2 4 Finite Capacity and Emergent Space 2 5 Growth Laws and Dimension 3 5.1 Physical motivation 3 6 Record Stability and Dimension Selection 3 6.1 Redundancy and perturbation propagation 3 6.2 Stability criterion 3 7 Causal Structure and Relativity 3 8 Conclusion 4 1 Introduction Modern physics assumes spacetime locality as a primitive principle. Relativity encodes it geometrically, while quantum theory enforces it operationally through no-signaling constraints. Yet neither framework explains why physical influence should be local, nor why space should have a particular dimensionality. In this paper we adopt a different starting point. We assume only that physical systems admit operational records, that records can be distinguished, and that influence propagates at a finite rate. From these minimal assumptions we derive spatial locality, distance, and dimension as necessary emergent structures. No spacetime manifold, metric tensor, or background geometry is assumed. Space is reconstructed as an interaction graph defined by influence propagation, and geometry appears only as an effective description in the large-scale limit. This paper establishes the spatial substrate for the remainder of the Ordered-Dynamics Reconstruction Program: •Paper XI: Operational time regulation and ordered dynamics, •Paper XII: Emergence of space, locality, and dimension (this work), •Paper XIII: Gravity as inhomogeneous influence propagation, •Paper XIV: Gauge structure from local redundancy, •Paper XV: Quantum structure from fluctuating influence. – 1 – 2 Operational Framework 2.1 States and records A physical system is described by a convex state space S. Convex combinations represent classical uncertainty. No Hilbert space or inner product is assumed. Definition 1 (Record).A record is an operationally accessible effect whose outcome can be stored, compared, and reliably reproduced. Records are the primitive empirical objects of the theory. 2.2 Influence and signaling Definition 2 (Signaling influence).Subsystem Acan signal to subsystem Bat time tif there exist two admissible interventions on Aand an effect on Bwhose outcome probabilities differ at time t. Define I(A:B;t) as the supremum of this distinguishability over all admissible interventions and effects. 2.3 Finite propagation [Bounded influence speed] There exists a finite speed v∗such that I(A:B;t) = 0 for t < d(A, B) v∗ . This axiom encodes operational causality without assuming geometry. 3 Distance from Delay Definition 3 (Operational distance).The distance between subsystems Aand Bis d(A, B) = v∗inf{t:I(A:B;t)>0}. d(A, B) defines a pseudometric. Proof. Non-negativity and symmetry are immediate. Triangle inequality follows from composition of influence protocols: influence from Ato Cmust pass through any intermediate B. This construction defines distance operationally, without coordinates or geometry. 4 Finite Capacity and Emergent Space [Finite influence capacity] Each subsystem can exchange only a finite amount of distinguishability per unit time. [Sparsity] Finite influence capacity forces the interaction graph to have bounded degree. Proof. If a subsystem coupled to infinitely many neighbors, finite capacity would require vanishing influence per neighbor, contradicting distinguishability. Remark 1 (No lattice assumption).The interaction graph is not assumed to be regular. Generic bounded systems produce disordered, stochastic graphs rather than crystalline lattices. The bounded-degree interaction graph is identified as emergent space. – 2 – 5 Growth Laws and Dimension Let N(r) denote the number of subsystems within distance rof a reference node. [Polynomial growth] At large r,N(r)∼rdfor some finite d. This defines an emergent spatial dimension. 5.1 Physical motivation Exponential growth would violate finite influence capacity. Sub-polynomial growth would prevent redundancy and stable records. Polynomial growth is the only consistent regime. 6 Record Stability and Dimension Selection Macroscopic physics requires the existence of records that persist under local perturbations. Such records must be redundantly encoded so that partial corruption does not erase the stored information. 6.1 Redundancy and perturbation propagation Consider a record stored redundantly within a ball Brof radius rcentered on a reference subsystem. Let the number of subsystems within Brscale as N(r)∼rd, as implied by Axiom 5. Perturbations originating outside Brcan affect the record only by propagating through the boundary ∂Br. The number of independent influence channels crossing the boundary scales as |∂Br|∼rd−1. 6.2 Stability criterion A record is stable if the rate at which redundancy is lost due to incoming perturbations grows more slowly than the total redundancy storing the record. Operationally, this requires that the ratio N(r) |∂Br|∼r diverges as r→ ∞. [Minimal dimension for stable records] The existence of stable macroscopic records requires d≥3. Remark 2.This result is independent of observers, thermodynamics, or anthropic selection. 7 Causal Structure and Relativity Finite influence speed defines causal cones without assuming spacetime geometry. Distance is defined by delay, not coordinates. Remark 3 (Emergent Lorentz symmetry).Because the interaction graph is statistically isotropic rather than regular, Lorentz symmetry emerges as a large-scale causal symmetry rather than a microscopic exact invariance. – 3 – 8 Conclusion Assuming only finite information flow, we have shown that space, locality, and dimensionality must emerge. Three dimensions are selected dynamically by the requirement of stable records. Lorentzian causal structure appears as a symmetry of influence propagation rather than a geometric axiom. Space is not the stage on which physics happens; it is a consequence of how information moves. References [1] L. Hardy, Quantum Theory From Five Reasonable Axioms, arXiv:quant-ph/0101012. [2] G. Chiribella, G. D’Ariano, P. Perinotti, Phys. Rev. A 84, 012311 (2011). [3] R. Sorkin, Causal Sets: Discrete Gravity, Lectures on Quantum Gravity, Springer (2005). [4] E. H. Lieb, D. W. Robinson, Commun. Math. Phys. 28, 251 (1972). – 4 –