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D0: A Pregeometric Relational Framework for Emergent Spacetime Priority Note – Version 1.2 M. L¨utzel December 10, 2025 Abstract D0 is a fully pregeometric framework in which spacetime, gravity, and temporal ordering emerge from a timeless relational structure defined on a countably infinite set of quantum configurations. No geometric, causal, or topological notions exist fundamentally. A strictly positive similarity kernel R(α, β) provides the basis for reconstructing emergent spatial geometry via spectral methods. Energetic content induces structured distortions of this kernel, manifesting as curvature in the emergent macroscopic geometry. Temporal ordering arises from entropy (entropic time). This Priority Note establishes the axiomatic foundation of D0, introduces a declarative dynamical principle, and outlines structural implications such as fundamental singularity exclusion. Outstanding mathematical and physical questions are identified for future work. 1 Axiomatic Framework (A–G) Axiom A — Pregeometric Fundamentality No geometric, topological, causal, or temporal structure exists fundamentally. The only fundamental objects are the elements of a countably infinite set of configurations: C={α1, α2, α3,...}. No metric, adjacency relation, or ordering is defined on C. All emergent structure arises solely from the relational kernel R(α, β). 1
Axiom B — Quantum Configurations Each configuration is a pair α= (ρα, Hα), where ραis a density operator on a finite-dimensional Hilbert space H(positive semidefinite, unit trace) and Hαis a Hermitian operator with bounded spectrum. The set Cis assumed dense in D(H)×Herm(H), enabling a continuum limit for emergent geometry. Axiom C — Fundamental Relational Structure The similarity kernel is defined as R(α, β) = exp−DBures(ρα, ρβ)−λ∥Hα−Hβ∥2 HS, λ > 0. The kernel R(α, β) is strictly positive and a positive semidefinite Mercer kernel. No locality, adjacency, or causality exists at the fundamental level. Strict positivity guarantees finite proto-distances and excludes fundamental singularities. Axiom D — Emergent Distance Define the proto-distance: D(α, β) = −log R(α, β). While not a metric microscopically, after coarse-graining and spectral embedding it converges to a smooth Riemannian metric gµν (x) on an emergent spatial manifold. Combined with entropic time, this yields emergent spacetime structure. Axiom E — Energy–Geometry Coupling Energetic content is defined as Φ(α) = Tr(ραHα). Variations in Φ deform the kernel R. In the continuum limit, these distortions manifest as curvature of the emergent metric. Gravity thus appears as the macroscopic effect of energetic deformations of the relational kernel. 2
Axiom F — Entropic Time Time is not fundamental. Temporal ordering emerges from entropy along dynamically realized trajectories {αt}: t1< t2⇐⇒ S(αt1)< S(αt2), where S(α) = −Tr(ραlog ρα) is the von Neumann entropy. The arrow of time is emergent and thermodynamic. Axiom G — Dynamical Principle (Priority Form) Physical trajectories in Cand the induced geometric structures are determined by a variational principle applied to the relational kernel. There exists an action functional S[R, Φ] whose extrema define the allowed evolution of the similarity kernel and the physically realized configurations. The explicit form of S, its Euler–Lagrange equations, and the induced evolution laws for α(t) will be presented in the extended version of this work. 2 Structural Implications for Black Holes Because R(α, β) is strictly positive, the proto-distance D(α, β) is finite for all configurations. Thus the D0 framework excludes fundamental geometric singularities: distances never diverge. Curvature in the emergent geometry may grow extremely large but remains finite. The timeless and unitary foundation preserves information fundamentally. However, detailed black-hole phenomenology—including horizon formation and Hawking-like thermality—requires the full dynamical principle S[R, Φ]. D0 provides a structural basis for singularity-free, informationpreserving black-hole analogs, while the emergence of horizon thermodynamics remains an open research direction. 3 Limitations and Open Problems •The continuum limit from a discrete similarity kernel to a smooth manifold requires formal specification of the graph Laplacian and conver3
gence theorems. •Axiom F defines a time ordering but not a continuous temporal metric; the dynamical principle is expected to supply this structure. •Axiom G establishes a variational principle but not the explicit action or evolution equations; these will be given in D0 Version 2.0. •The Mercer positivity of Ris adopted as a foundational requirement; future work will characterize conditions on (ρ, H) ensuring it. Acknowledgments The conceptual framework, physical interpretation, and theoretical claims presented in this work were developed independently by the author. Automated computational tools were employed for algebraic verification, symbolic consistency checks, and linguistic refinement. All scientific content, mathematical structures, and theoretical assertions remain solely the responsibility of the author. 4 References 1. A. Uhlmann, “The metric of Bures and the geometric phase,” Springer (1992). 2. D. Bures, “An extension of Kakutani’s theorem,” Trans. Amer. Math. Soc. (1969). 3. M. Belkin, P. Niyogi, “Laplacian Eigenmaps,” Neural Computation (2003). 4. R. Coifman, S. Lafon, “Diffusion Maps,” Appl. Comput. Harm. Anal. (2006). 5. A. Connes, C. Rovelli, “Time–thermodynamics relation,” Class. Quant. Grav. (1994). 6. B. Swingle, “Holographic spacetimes via entanglement renormalization,” arXiv:1209.3304. 4
7. J. Maldacena, L. Susskind, “Cool horizons for entangled black holes,” arXiv:1306.0533. 8. S. Hawking, “Breakdown of Predictability,” Phys. Rev. D 14 (1976). 9. N. Aronszajn, “Theory of Reproducing Kernels,” Trans. Amer. Math. Soc. (1950). 5
