Paper XIX - Discrete Quantum Gravity from Non--Coarse--Grained Ordered Dynamics

DOI: 10.5281/zenodo.18009258 Discrete Quantum Gravity from Non–Coarse–Grained Ordered Dynamics Paper XIX 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 Discrete ordered substrate 1 3 Delay as a dynamical degree of freedom 2 4 Collective modes and gravitational excitations 2 5 Relation to emergent geometry 3 6 Quantization without the problem of time 3 7 Connections to other approaches 3 8 Discussion and outlook 3 revious papers showed that spacetime geometry and classical gravity emerge from coarse– graining a discrete ordered interaction graph with bounded influence propagation. In this paper we remove coarse–graining and examine the ultraviolet regime. We show that gravity persists as a genuine quantum phenomenon, but not as quantized geometry. Instead, gravitational degrees of freedom arise as collective excitations of dynamical influence–delay structure on the graph. This yields a discrete, background–independent quantum gravity framework without geometric singularities, background metrics, or a problem of time. 1 Introduction Most approaches to quantum gravity attempt to quantize spacetime geometry. Despite partial successes, this strategy encounters persistent difficulties: nonrenormalizability, background dependence, and the problem of time. In the ordered–dynamics reconstruction program, geometry is not fundamental. Locality, relativistic symmetry, and curvature arise from bounded influence propagation on a discrete, acyclic interaction graph. Paper II showed that gravity appears as an effective influence–delay field under coarse–graining, while Paper VIII showed that horizons, entropy, and thermality emerge from saturation of information flow. The goal of the present paper is to analyze the theory before coarse–graining. We show that the ultraviolet description already contains a quantum theory of gravity, realized as dynamical influence structure rather than quantized geometry. 2 Discrete ordered substrate [Ordered interaction graph] The fundamental structure is a locally finite directed acyclic graph whose nodes represent update events and whose edges represent channels of influence propagation. Each edge e∈carries a positive influence delay (e)>0. The ordering parameter specifies update precedence but carries no metric or geometric meaning. – 1 – Definition 1 (Operational time).The operational time between events xand yis defined by t(x, y) := min γ:x→yX e∈γ (e), where the minimum is taken over directed paths γfrom xto y. Operational time is emergent and observer–dependent; the ordering remains unobservable. 3 Delay as a dynamical degree of freedom [Local delay dynamics] Influence delays (e) evolve according to local update rules depending only on records in the causal neighborhood of e. This assumption enforces background independence: no ambient spacetime or metric structure is required to define dynamics. Remark 1.The delay field is not a matter field propagating in space. It is part of the causal structure from which space emerges. Treating it as a matter field would double–count degrees of freedom. Definition 2 (Delay fluctuation).A delay fluctuation is a localized deviation δ(e) from the statistical mean of on a subset of edges. Remark 2 (Entanglement–geometry relation).The influence delay (eAB) between two nodes Aand Bis inversely related to their shared mutual information I(A:B). High entanglement corresponds to strong correlation conductance, facilitating influence propagation and effectively shortening operational distance. Conversely, weakly correlated nodes exhibit larger delays. In the continuum limit, this relation reproduces area–law scaling of entanglement entropy, connecting emergent geometry directly to correlation structure in a manner consistent with Ryu–Takayanagi–type relations. 4 Collective modes and gravitational excitations Individual delay fluctuations are microscopic and not directly observable. However, coherent fluctuations can organize into collective modes. [Collective delay modes] In a statistically homogeneous and isotropic interaction graph, long–wavelength coherent fluctuations of the delay field propagate as effective dynamical modes. Sketch. Local coupling between neighboring delays induces a network of interacting degrees of freedom. Linearizing the update rules around a homogeneous background yields coupled difference equations supporting propagating normal modes. At scales large compared to the discreteness length, these modes admit an approximately relativistic dispersion relation. Remark 3.Viewed this way, gravitational excitations correspond to coherent reconfigurations of entanglement structure encoded in the delay network. They are not quanta of geometry but quanta of influence structure. – 2 – 5 Relation to emergent geometry Under coarse–graining, variations in reproduce effective curvature (Paper II). In the ultraviolet description, curvature is not fundamental but a macroscopic encoding of delay correlations and entanglement structure. [No geometric singularities] Configurations that would correspond to geometric singularities in a continuum description manifest instead as saturation regimes of influence delay. Dynamics remains well–defined everywhere on the graph. This complements the horizon analysis of Paper VIII, where saturation replaces divergence. 6 Quantization without the problem of time Quantization applies directly to the discrete update rules governing records and delays. The Hilbert–space structure derived in Paper IV provides the kinematical framework for the state space of graph configurations. Remark 4.Because the ordering parameter supplies a fundamental update sequence, no Wheeler–DeWitt–type constraint arises. Operational time remains emergent, avoiding the traditional problem of time in quantum gravity. Gravitational quanta correspond to excitations of the delay configuration space, not to eigenstates of a metric operator. 7 Connections to other approaches The ordered–dynamics framework shares structural features with several quantum gravity approaches: •Causal set theory: discrete order and sequential growth, augmented here by dynamical influence delays. •Spin–foam models: histories of discrete structures without preassigned geometry. •Entanglement–based gravity: geometry encoded in relational correlation structure rather than fundamental fields. Remark 5.The present framework is compatible with these approaches at a structural level but does not assume their specific formalisms. 8 Discussion and outlook Removing coarse–graining reveals that gravity in the ordered–dynamics program is already quantum at the fundamental level. What is quantized is not geometry but the causal– informational substrate from which geometry emerges. This completes the ultraviolet leg of the reconstruction. The remaining task is to understand the infrared consequences of finite information capacity at cosmological scale, including holography, vacuum energy, and the arrow of time. Conclusion. Quantum gravity need not be the quantization of spacetime. In an ordered– dynamics framework, it arises from the quantum dynamics of discrete influence delays. Geometry is an emergent, approximate description of a deeper causal and entanglement structure. – 3 –