Paper XVIII - Horizons and Saturation from Finite Information Flow

DOI: 10.5281/zenodo.18009234 Horizons and Saturation from Finite Information Flow Paper XVIII of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Contents 1 Introduction 1 2 Finite information capacity and records 1 3 Influence flow and saturation 2 4 Horizons as saturation interfaces 2 5 Entropy as inaccessible records 2 6 Emergent thermality from information scaling 3 7 Discussion and conclusion 3 orizons are traditionally treated as geometric or causal boundaries in spacetime. In this paper we derive horizons as saturation phenomena in ordered dynamics with finite information capacity and bounded influence propagation. A horizon forms when the local rate of record production exceeds the capacity for influence to export information outward in operational time. Entropy is identified with the number of stored records inaccessible to distant observers, and horizon area arises as the effective interface between saturated and unsaturated regions. We further derive an emergent temperature scaling T∝1/M purely from information flow constraints, without invoking quantum field theory. No singularities or divergent fields are required: horizons reflect capacity limits, not geometric pathologies. 1 Introduction In earlier papers of the Ordered-Dynamics Reconstruction Program, spatial locality, gravity, gauge structure, quantum kinematics, and chronology protection were derived from bounded influence propagation and finite information capacity. Spacetime geometry plays no fundamental role in this framework. Locality, causal structure, and relativistic symmetry arise as large-scale, operational regularities of ordered dynamics. This raises a sharp question: what is a horizon when geometry itself is emergent? In this paper we show that horizons are information-saturation interfaces. They arise when finite information capacity and bounded influence prevent records generated in one region from being exported to another within operational time. Entropy counts inaccessible records; area laws arise from boundary bandwidth; thermality follows from information scaling alone. 2 Finite information capacity and records [Finite information capacity] Each update event on the interaction graph has finite capacity to store records. Arecord is any locally stored piece of information generated by an update: measurement outcomes, interaction histories, or internal state changes. Records persist unless erased or overwritten by subsequent dynamics. – 1 – Definition 1 (Accessible information).For an observer associated with a chain of updates, the accessible information acc consists of all records reachable via influence paths within operational time. Finite capacity implies that records may accumulate faster than they can be exported. When this imbalance persists, saturation becomes unavoidable. 3 Influence flow and saturation Let R(U) denote the rate of record production in a region U, and let F(∂U) denote the maximal rate at which influence can export information across the boundary of U. Definition 2 (Saturation condition).A region Uis saturated if R(U)> F(∂U) over operational timescales. [Horizon formation] If a region remains saturated over operational time, then distant observers lose access to an increasing fraction of records generated within the region. Proof. Bounded influence limits the rate at which records can be exported across ∂U. If record production exceeds export capacity, inaccessible records accumulate regardless of microscopic dynamics. Remark 1.This mechanism depends only on finite capacity and bounded influence. No assumptions about spacetime geometry, curvature, or collapse are required. 4 Horizons as saturation interfaces Definition 3 (Operational horizon).An operational horizon is the effective boundary separating saturated and unsaturated regions of the interaction graph. Operational horizons are observer-dependent in the same sense as Rindler horizons, but their existence is objective: they reflect limits on information flow, not coordinate artifacts. [Absence of singularities] Horizon formation does not require divergent fields or breakdown of dynamics. Saturation replaces singularity. 5 Entropy as inaccessible records Definition 4 (Horizon entropy).The entropy associated with a horizon is the number of records stored in the saturated region that are inaccessible to a given observer. [Area scaling] In statistically isotropic interaction graphs (as derived in earlier papers), the maximal information export rate across a boundary scales with the number of boundary links. Consequently, horizon entropy scales with boundary area rather than volume. Sketch. Influence export is mediated by boundary edges. In isotropic graphs, the number of such edges grows with boundary size, while interior volume grows faster. Record accumulation is therefore regulated by boundary area. Remark 2.This provides a non-geometric origin of holographic scaling. – 2 – 6 Emergent thermality from information scaling Energy in the saturation regime is identified with accumulated information content, which scales linearly with effective radius, E∝R. Boundary scaling yields S∝A∝R2, and therefore S∝E2. Temperature defined by 1 T:= dS dE then satisfies T∝1 E. Remark 3.Negative specific heat follows immediately. This reproduces the qualitative Hawking scaling without invoking quantum field theory or vacuum fluctuations. 7 Discussion and conclusion Horizons arise when finite information capacity and bounded influence prevent records from being exported. Entropy counts inaccessible records; temperature follows from scaling; singularities are replaced by saturation limits. This result completes the thermodynamic layer of the Ordered-Dynamics Reconstruction Program and prepares the ground for ultraviolet and cosmological extensions developed in subsequent papers. Conclusion. Horizons are not geometric obstructions but operational limits on information flow. They arise inevitably in ordered dynamics with finite capacity and bounded influence. – 3 –