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Hilbert's Sixth Problem Resolved: A Unified Axiomatization of Physics via URPD and Reflexive Micro-Granular Dynamics

AI Trinity Labo

Published Nov 29, 2025• 0 citations
Publisher

Abstract

This work presents a unified axiomatic framework that resolves Hilbert’s Sixth Problem by integrating two complementary foundations: Unified Relational Projection Dynamics (URPD), which defines probability, time, and measurement as geometric relations on CP^{N−1}, and Reflexive Micro-Granular Dynamics (RMG), which models fields not as continuous backgrounds but as finite, self-differential information structures. By combining these principles, the framework reproduces quantum mechanics, classical mechanics, gauge interactions, Navier–Stokes continuum physics, thermodynamic irreversibility, and emergent spacetime geometry without divergences, renormalization procedures, or additional postulates. The result is a coherent and complete axiomatization of physical law that satisfies Hilbert’s original requirement for a mathematically rigorous physics.

DOI
10.5281/zenodo.17756872

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This work introduces Unified Relational Projection Dynamics (URPD),a minimal information-theoretic and geometric framework that unifiesquantum probabilities, temporal flow, and causal structure into asingle underlying principle. In URPD, physical states are represented not as Hilbert vectors but aspoints in the complex projective manifold CPN−1CP^{N-1}CPN−1, where theFubini–Study geometry encodes both dynamical transitions andprobabilistic structure.Physical processes correspond to relational projections between frames,and the evolution of a system follows geodesic or near-geodesic pathsdefined by an information-based action. Within this formulation, the Born rule emerges uniquely as theprobability measure that maximizes information consistency,eliminating the need for external postulates.The arrow of time appears as the direction of increasing relationaldisplacement, while causal order arises naturally from theconcatenation of minimal geometric transitions.Classical spacetime and deterministic behavior emerge as macroscopiclimits of coherent frame dynamics. This article provides a compact, self-contained reconstruction ofquantum mechanics and classical causality from a single unifyinggeometric principle.URPD offers a minimal extension of the information-geometric approach,suggesting a pathway toward unifying quantum foundations andrelational physics within a coherent mathematical structure.
This paper introduces Unified Relational Projection Dynamics (URPD), an information-theoretic framework that derives quantum probabilities, temporal evolution, classical motion, and relativistic causality from a single relational principle. By modeling physical states as points on a projective Hilbert manifold and defining an information-based measure of relational displacement between consecutive frames, the framework replaces the Born rule, Schrödinger dynamics, and geodesic motion with a unified update rule. The theory also explains how classical spacetime structure emerges from decoherence and how quantum entanglement remains compatible with causal constraints. URPD serves as a minimal and coherent foundation that integrates quantum theory and relativity into one informational structure.
This work presents Unified Relational Projection Dynamics (URPD),a minimal and self-contained information-theoretic framework that reconstructsquantum probabilities, temporal flow, and causal structure from a singlegeometric principle. URPD replaces the traditional Hilbert-space formulation with arelational description of physical states as points in the complex projectivemanifold CPN−1CP^{N-1}CPN−1.Within this representation, the Fubini–Study geometry encodes both dynamicaltransitions and probabilistic structure without requiring the Born rule as anadditional axiom. In URPD, physical evolution corresponds to geodesic and quasi-geodesictransitions between relational frames, and the arrow of time arises from themonotonic increase of relational displacement.Classical spacetime and deterministic trajectories emerge naturally in themacroscopic limit as coherence within frame ensembles becomes sharply peaked. This article provides a compact reconstruction of the foundations of quantummechanics and causality from a unified geometric principle, offering a pathwaytoward a deeper relational understanding of physical law.
This paper introduces Unified Relational Projection Dynamics (URPD), a fully self-contained physical framework that derives quantum probabilities, temporal evolution, and relativistic causal structure from purely relational information constraints. The theory uses only two primitives—relational overlap fields between observer-dependent reference frames and the coherence-cost associated with maintaining inter-frame consistency. Without invoking higher-dimensional structures or additional postulates, URPD reproduces the Born rule, the Schrödinger evolution, the classical limit, and the emergence of the light-cone. This work presents the minimal information-theoretic foundation from which core physical laws arise naturally.
Hilbert’s Sixth Problem asks for an axiomatization of physics, especially the probabilistic foundations of mechanics and the passage from microscopic dynamics to macroscopic laws. Despite foundational progress in probability theory and statistical mechanics, a persistent conceptual gap remains: probability appears as a primitive or as an epistemic tool, rather than as a quantity derived from the structure of physical evolution itself. This paper proposes a Viscous Time Theory (VTT) reformulation of Hilbert 6: physics is axiomatized by admissibility (what transformations can occur while preserving identity), coherence (the measurable structure of informational organization), and recoverability (which collapses are irreversible independent of observer perspective). We present a minimal axiomatic system in which (i) time emerges as informational viscosity, (ii) probability emerges as an induced measure over admissible trajectories, and (iii) the micro→macro transition becomes a geometric scaling limit of coherence fields rather than an ad hoc coarse-graining step. We do not claim a completed proof of Hilbert 6 in the classical sense. Instead, we provide a coherent, falsifiable program that reframes the target: the objective is not “derive probability from mechanics,” but “derive probability from the geometry of admissible informational evolution,” with classical mechanics and statistical mechanics emerging as distinct regime limits.
This work introduces Unified Relational Projection Dynamics (URPD), a minimal information-theoretic framework in which physical states are represented as relational frames in the complex projective state space CPN−1\mathbb{C}P^{N-1}CPN−1.The theory defines evolution as entropy-maximizing transitions between frames under constraints of informational displacement, providing a unified geometric origin for quantum probabilities, temporal flow, and causality.URPD naturally ensures information preservation through non-vanishing frame overlaps induced by the Fubini–Study metric, offering a non-holographic explanation of the Page curve and black-hole information retention.This framework establishes a compact, relational, and geometry-based foundation for quantum mechanics and spacetime structure.