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Unified Relational Projection Dynamics (URPD): A Minimal Information-Theoretic Foundation for Quantum Probabilities, Temporal Flow, and Causality

A.I. Trinity Lab

Published Nov 24, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.17693650

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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), 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 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.
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.
Quantum teleportation is reinterpreted as a natural geometric process within Unified Relational Projection Dynamics (URPD).Instead of treating teleportation as a special quantum protocol, this framework views quantum states as relational frames in the projective space CPN−1\mathbb{CP}^{N-1}CPN−1.Entanglement establishes a high-overlap shortcut between distant frames, and a Bell measurement corresponds to a discrete transition in this geometric structure.Classical communication is required not as an add-on but as a direct consequence of causal constraints encoded in URPD.The no-cloning theorem, perfect fidelity, and the structure of the teleportation protocol all emerge from the geometry of relational frames.This work shows that teleportation is an inevitable feature of the informational geometry underlying URPD, providing the first unified explanation of quantum state transfer, the Born rule, and relativistic causality within a single theoretical framework.
This paper presents a novel resolution of the black hole information paradox using Unified Relational Projection Dynamics (URPD), a minimal information-theoretic framework grounded in the geometry of the complex projective state space CPN−1\mathbb{C}P^{N-1}CPN−1.URPD treats physical states not as absolute objects but as relational frames, whose overlaps ∣⟨F1∣F2⟩∣|\langle F_1 | F_2 \rangle|∣⟨F1∣F2⟩∣ never vanish due to the compactness of the Fubini–Study geometry.As a result, information is never destroyed during black-hole evaporation; instead, it is continuously compressed into the residual overlaps between interior and exterior frames.This mechanism naturally reproduces the Page curve and preserves global unitarity without relying on holography, replica wormholes, island formulas, or any additional quantum-gravity assumptions.The work provides the first fully relational and geometrically minimal explanation of information preservation across the entire evaporation process.