D0 v1.41: A Pregeometric Relational Framework for the Emergence of Spacetime, Gravitation, Time, and Holography

D0 v1.41 A Pregeometric Relational Theory of the Emergence of Spacetime, Gravitation, Time, and Holography M. L¨utzel Independent Researcher December 15, 2025 Abstract We present D0 v1.41, a consolidated and defensively clarified formulation of a pregeometric, relational framework for the emergence of spacetime, gravitation, time, and holographic behavior. The theory is formulated without presupposing spacetime, dimensionality, metric structure, topology, causal order, or time at the fundamental level. The ontology consists of a finite set of abstract configurations and a strictly positive relational kernel encoding all physical information. From this minimal structure, effective geometric notions arise only in regimes where the relational structure admits a stable spectral embedding. Information is defined as a purely spectral, relational quantity, leading necessarily to a universal upper bound on information density and to holographic organization of emergent geometry. Black holes are reinterpreted as relational saturation objects (“black spheres”) characterized by maximal information density on a two-dimensional correlation surface, without interior spacetime or singularities. Lorentz boost invariance of relational information density excludes boost-induced horizon formation. Time emerges as an entropic ordering parameter rather than as a fundamental variable. Version v1.41 introduces no new dynamics. Its purpose is to consolidate the axiomatic core of D0, eliminate hidden geometric assumptions and circular reasoning, and clearly delimit the scope of current claims. The framework is intentionally incomplete but structurally consistent, providing a minimal foundation for future dynamical extensions. 1 Introduction Modern theoretical physics rests on a hierarchy of effective descriptions whose empirical success is beyond dispute, yet whose conceptual foundations remain in tension. General relativity describes gravitation as the dynamics of spacetime geometry, while quantum theory presupposes a fixed background on which states and operators evolve. Attempts to quantize gravity or geometrize quantum theory have revealed persistent structural problems, including spacetime singularities, non-renormalizability, and ambiguities surrounding the status of time and information. 1 These difficulties motivate the hypothesis that spacetime itself is not fundamental, but emergent. Various approaches—holography, tensor networks, causal sets, loop quantum gravity—suggest that geometry may arise from deeper relational or informational structures. However, many such approaches retain implicit geometric or temporal assumptions at the fundamental level. The D0 framework adopts a more radical stance. It assumes that no spacetime, geometry, dimensionality, metric, causal order, or time exists fundamentally. Instead, physics is formulated on a purely relational substrate consisting of abstract configurations and a similarity kernel between them. All familiar structures are emergent and contingent. The aim of this work is not to present a complete theory of quantum gravity, but to establish a minimal, non-circular, and internally consistent foundation upon which such a theory could be built. D0 v1.41 consolidates earlier versions by making all assumptions explicit, eliminating hidden imports of geometry or time, and clearly stating the limits of what is currently claimed. 2 Fundamental Ontology 2.1 Pregeometric Fundamentality Axiom A (Pregeometric Fundamentality). At the fundamental level, there exists no spacetime, no dimension, no topology, no metric, no causal structure, and no time. These concepts do not exist even in latent or approximate form. This axiom is ontological, not methodological. Geometry and time are not approximated away; they are absent. 2.2 Configurations Axiom B (Configurations). The fundamental entities are configurations α= (ρα, Hα), where ραis a density matrix on a fixed, finite-dimensional Hilbert space, and Hαis a bounded self-adjoint operator on that space. No spatial, temporal, or geometric degrees of freedom are associated with configurations. 2.3 Relational Kernel Axiom C (Relational Kernel). For every pair of configurations α, β there exists a strictly positive, symmetric relational kernel R(α, β)∈(0,1], R(α, α)=1. The kernel Ris the only primitive relational structure and encodes all physical information. Strict positivity implies that the induced proto-distance D(α, β) := −log R(α, β) 2 is finite for all configuration pairs. Fundamental singularities are therefore structurally excluded. 2.4 Proto-Distance and Emergent Geometry Axiom D (Proto-Distance). The proto-distance D(α, β) is not a metric at the fundamental level. Only after suitable coarse-graining and in regimes where the relational structure admits a stable spectral embedding can an effective metric description emerge. No geometric structure is assumed in defining D. 2.5 Energy–Information Content Axiom E (Energy–Information Content). The energy–informational content of a configuration is defined as Φ(α) := Tr(ραHα). Variations in Φ induce structured distortions of the relational kernel. In appropriate emergent regimes, these distortions manifest as gravitational phenomena. 2.6 Entropic Time Axiom F (Entropic Time). Time is not fundamental. A temporal ordering emerges only as an entropic ordering of configurations: αi≺αj⇐⇒ S(ραi)< S(ραj), where S(ρ)=−Tr(ρlog ρ). No external time parameter or fundamental evolution variable is assumed. 2.7 Finiteness of Relational Information Axiom G (Information Finiteness). For any finite subset Ω ⊂ C, the number of effectively independent relational degrees of freedom is finite. This implies the existence of a universal upper bound on relational information density. 2.8 Relational Action Principle Axiom H (Relational Action Principle). Physically realized relational structures correspond to extrema of a timeless action functional S[R, Φ], subject to positivity and finiteness of R. No explicit form of Sis specified in v1.41. 3 Relational Information For a finite cluster Ω ⊂ C, let RΩdenote the restricted kernel with eigenvalues {λi}. Define normalized weights pi:= λi Pjλj . 3 Definition (Spectral Information). I(Ω) := X i p2 i!−1 . I(Ω) measures the effective number of distinguishable relational degrees of freedom and is invariant under relabelings and Lorentz boosts. 4 Emergent Surface and Information Density 4.1 Level Separation An effective surface A(Ω) is not a fundamental object. It is defined only for clusters whose relational structure admits a stable, low-dimensional spectral embedding. For clusters that do not satisfy this condition, A(Ω) is undefined. This reflects a breakdown of geometric description rather than a physical singularity. 4.2 Effective Surface In an admissible embedding, the covariance matrix of embedded coordinates defines an effective ellipsoidal geometry. The associated scalar surface area A(Ω) is invariant when reconstructed in the cluster’s rest frame. 4.3 Information Density ρI(Ω) := I(Ω) A(Ω). ρIis a Lorentz-invariant scalar. 5 Regimes of Representability and Stability under Coarse-Graining The D0 framework admits a vast class of mathematically allowed relational kernels R(α, β). However, not all such kernels correspond to physically realizable situations. In this section, we introduce a minimal, non-dynamical selection principle that distinguishes physically admissible relational structures without presupposing spacetime, geometry, or temporal evolution. 5.1 Coarse-Graining Operations We consider a class Gof coarse-graining operations acting on finite clusters Ω ⊂ C. These operations may include, but are not limited to: •removal or aggregation of subclusters, •averaging over relationally equivalent configurations, •block-diagonalization and controlled truncation of RΩ, •stochastic subsampling preserving normalization and positivity. 4 Crucially, coarse-graining is understood as an informational operation on relational data, not as a geometric or spatial procedure. 5.2 Physical Admissibility Definition (Physical Admissibility). A relational kernel Ris said to be physically admissible if, for sufficiently large clusters Ω, the following relational observables are stable under the action of G: •the spectral information I(Ω), •the effective spectral rank and gap structure of RΩ, •any emergent quantities defined purely in terms of spectral invariants. Stability here means that these quantities vary only within controlled bounds under admissible coarse-graining operations. This definition introduces no new ontological primitives and does not assume any form of geometric or temporal structure. It merely restricts attention to those relational configurations whose informational content is robust under loss of microscopic detail. 5.3 Emergence as a Stability Regime In general, arbitrary relational kernels are unstable under coarse-graining: their spectral properties change dramatically under small perturbations or reductions of detail. Such kernels do not admit a consistent macroscopic description. However, a non-empty subclass of physically admissible kernels exhibits an additional property: a low-dimensional, stable spectral structure characterized by a clear separation of scales in the eigenvalue spectrum. Only in this regime do clusters admit a consistent effective embedding and hence a geometric interpretation. We emphasize that this emergence of geometry is not postulated. It arises as a representational phase in which the relational structure can be compressed into a small number of stable collective modes. 5.4 Breakdown at Saturation: Black Spheres Physical admissibility alone does not guarantee geometric representability. As relational information density increases locally, the spectral structure of RΩchanges. When the bound ρI(Ω) →imax is approached, volumetric geometric descriptions become unstable: additional relational degrees of freedom no longer increase distinguishability within the cluster. At this point, the effective geometric representation collapses to a two-dimensional correlation surface, defining a black-sphere configuration. This transition is not a dynamical collapse but a loss of representational stability at maximal informational density. 5 5.5 Return to the Fundamental Description Beyond the saturation regime, no geometric encoding—neither volumetric nor surface-based—remains stable under coarse-graining. The effective geometric description ceases to be meaningful, and the system must be described directly in terms of the fundamental relational kernel R. Thus, the sequence D0 (non-geometric) →geometric regime →black-sphere saturation →D0 (non-geometric) is understood as a sequence of representational regimes rather than a fundamental temporal evolution. 5.6 Interpretational Summary Geometric spacetime descriptions arise in D0 not as fundamental entities but as stable, compressible representations of relational information. Black-sphere formation marks the boundary of this representational validity. D0 itself remains the universal description at both low-density and saturation extremes. 6 Holography and Information Bound There exists a universal bound ρI(Ω) ≤imax. The theory requires only the existence of such a bound. Its numerical value may be matched to black-hole entropy densities, but no derivation is claimed here. Holographic scaling follows necessarily: once saturation is reached, additional information requires an increase of effective surface area rather than volume. 7 Black Spheres A black sphere is defined as a relational cluster Ω satisfying ρI(Ω) = imax. The boundary is a correlation-saturation surface. There is no interior spacetime, no singularity, and no volumetric information storage. The area law follows directly: δA ≥δI imax . 8 Boost Invariance Lemma (No Boost-Induced Horizons). Since I(Ω) is purely spectral and A(Ω) is reconstructed as a rest-frame scalar, ρI(Ω) is boost invariant. No Lorentz boost can induce horizon formation. 6 9 Gravitation and the Role of G Gravitation is an emergent manifestation of relational distortion near saturation. The constant Gappears only as a macroscopic stiffness or scale parameter. In v1.41, Gis: •not a microscopic coupling, •not part of a field equation, •not derived from an action, •meaningless outside emergent geometric regimes. 10 Explicit Non-Claims D0 v1.41 does not claim: •an explicit action S[R], •Einstein field equations, •quantum field theory, •Standard Model embedding, •Hawking radiation. 11 Conclusion and Outlook D0 v1.41 provides a minimal, non-circular, and singularity-free relational foundation for emergent spacetime physics. Its strength lies in conceptual clarity rather than completeness. Future work will introduce dynamics only where they can be formulated without violating the pregeometric core. References References [1] A. 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