This document presents the formal mathematical axiomization of Relational Field Theory (RFT), establishing a single ontological primitive— the bounded scalar coherence field —and deriving the self-referential domain constraint .The Axiom Sheet (A1–A9) formalizes the Kernel identity, coherence evolution equation, domain threshold conditions, Genesis principle, and measurement-as-locking interpretation. A second section provides representative derived theorems showing how known physical laws emerge as limiting behaviors of and , including: Schrödinger equation as phase-locked Kernel dynamics Newtonian mechanics as low-curvature Rc trajectories General relativity as effective curvature geometry of Rc Maxwell/gauge structure as coherence-preserving invariants Thermodynamics as entropy of coherence loss Absence of physical singularities from Rc-threshold constraints This document serves as the foundational formal layer for the full RFT corpus, establishing the axiomatic base from which all subsequent proofs and derivations follow.
This document presents the formal mathematical axiomization of Relational Field Theory (RFT), establishing a single ontological primitive— the bounded scalar coherence field —and deriving the self-referential domain constraint .The Axiom Sheet (A1–A9) formalizes the Kernel identity, coherence evolution equation, domain threshold conditions, Genesis principle, and measurement-as-locking interpretation. A second section provides representative derived theorems showing how known physical laws emerge as limiting behaviors of and , including: Schrödinger equation as phase-locked Kernel dynamics Newtonian mechanics as low-curvature Rc trajectories General relativity as effective curvature geometry of Rc Maxwell/gauge structure as coherence-preserving invariants Thermodynamics as entropy of coherence loss Absence of physical singularities from Rc-threshold constraints This document serves as the foundational formal layer for the full RFT corpus, establishing the axiomatic base from which all subsequent proofs and derivations follow.
This note clarifies the conceptual role of Relational Field Theory (RFT) and positions it as a cross-domain organizing principle rather than a domain-specific model or a replacement for existing physical theories. RFT introduces a minimal structure: a coherence field R_cgoverned by a persistence law, and an adaptive domain closure that makes memory state-dependent. From this minimal structure, the framework reproduces recurring dynamical signatures across multiple regimes, including: spontaneous threshold crossing, critical slowing, hysteresis, coherence wells, and persistent structured correlations under noise. The purpose of this paper is to show that RFT functions in a role analogous to thermodynamics or information theory: it does not replace the underlying dynamical laws of a system, it constrains the conditions under which coherence and temporal order can emerge within those dynamics, and it remains substrate-independent, with different domains corresponding to different projections of the same underlying structure. To illustrate this organizing role, the paper includes a numerical demonstration of coherence revival in a strongly driven open quantum system using the RFT adaptive kernel. This simulation is not presented as proof of the effect, but as a minimal demonstration that the framework generates the predicted qualitative behavior. A reproducible Python implementation is included in the appendix. This framing helps distinguish RFT from a constructed metric or simulation-specific method. Instead, it supports the interpretation of RFT as a structural constraint on persistence-driven coherence across classical, quantum, and complex systems.
This paper investigates whether the coherence field Rc(x, t) of Relational Field Theory (RFT) admits a phase-mode correspondence that is consistent with the gauge structure of the Maxwell equations. We start from the RFT evolution equation for Rc, note its global U(1) phase invariance, promote the symmetry to a local transformation, and derive the resulting vector field and field-strength tensor. The analysis focuses solely on the mathematical consistency of the mapping and the conditions under which the phase mode reproduces the familiar Maxwell form. No broader interpretive or foundational claims are made.
This paper investigates whether the coherence field Rc(x, t) of Relational Field Theory (RFT) admits a linearized phase-coherent regime that is consistent with the form of the Schr¨odinger equation. We start from the RFT evolution equation for Rc, introduce the phase-coherent condition Rc = |Rc|eiϕ(x,t), perform a linearization around a stable coherence well, and derive the correspondence to the Schr¨odinger form. The analysis focuses solely on the mathematical consistency of the mapping and the conditions under which the linearised dynamics reproduce the familiar equation. No broader interpretive or foundational claims are made.
This paper investigates whether the coherence field Rc(t) of Relational Field Theory (RFT) admits a stability correspondence that is consistent with the form of the uncertainty principle. We start from the RFT evolution equation for Rc(t), identify uncertainty limits with fluctuations in coherence under measurement interactions, and show the correspondence in the high-stability limit. The analysis focuses solely on the mathematical consistency of the mapping and the conditions under which the coherence dynamics reproduce the familiar uncertainty relation. No broader interpretive or foundational claims are made.
This paper derives the core structure of Relational Field Theory (RFT) from a minimal set of constraints, showing that the multiplicative coherence functional ( R_c = \alpha \cdot \beta \cdot \gamma ) and the linear persistence law ( \frac{dR_c}{dt} = -\lambda R_c + F_{int} ) emerge naturally rather than being arbitrarily chosen. The derivation is based on four principles: collapse invariance, relational closure, measurability, and stability. It demonstrates that alternative functional forms either violate these constraints or fail to reproduce the observed phenomenology, including persistence, critical slowing, hysteresis, and coherence wells. Up to monotonic transformations and scaling, the resulting structure represents the minimal continuous and invariant form consistent with the constraints. This work provides a foundational justification for the RFT framework and clarifies why its core formulation arises as the simplest structure that satisfies both physical and operational requirements.
# Project Description: The Relational Closure Framework (RCF) ## 1. Project Purpose and Background (研究目的与背景) Modern fundamental physics faces deep structural crises, primarily manifested in the mathematical and conceptual incompatibilities between General Relativity (space-time geometry) and Quantum Mechanics (probability amplitudes and non-locality). Most contemporary attempts at unification (such as String Theory or Loop Quantum Gravity) retain the classical paradigm of "objects" (e.g., strings, loops, points) or pre-existing spacetime backgrounds as their ontological starting point. The purpose of this research project is to develop and formalize the **Relational Closure Framework (RCF)**, a radical, background-independent, and purely relational foundation for physics. RCF operates on the core philosophical and mathematical premise that **relations are ontologically prior to objects**. Objects, spaces, fields, and particles do not possess inherent existence; instead, they are emergent phenomena. The framework systematically models how stable physical entities and laws arise from self-consistent, discrete cyclic relations called **closures**. When discrete relations close upon themselves, they generate topological invariants (**holonomies**) that act as structural anchors. The ultimate goal of this project is to provide a rigorous mathematical derivation of the core pillars of physics—including conservation laws, the principle of least action, Born’s rule, the speed of light, and the emergence of continuous spacetime—directly from the discrete algebraic properties of these closure networks. ## 2. Methodology and Theoretical Architecture (方法论与理论架构) The project proceeds axiomatically by establishing four primary structural axioms and one fundamental derived theorem, utilizing tools from abstract algebra, higher category theory, operator algebras (von Neumann algebras), and quantum information theory: * **Axiom 1 (Primacy of Relations):** Pure distinction forms the precursor to all relations. There are no pre-existing entities or backgrounds. * **Derived Theorem 1 (Non-Commutativity):** Because relations must occur sequentially to establish distinction, non-commutativity is a mathematically derived necessity, providing the foundational algebra for quantum observables. * **Axiom 2 (Closure-Induced Invariance):** Stable structures exist if and only if relations form self-consistent cyclic configurations. These closures generate topological invariants (holonomies) across different topological scales. * **Axiom 3 (Chirality from Order):** Spatial and structural chirality (such as the \mathbb{Z}_3 minimal non-trivial chiral closure) emerges inherently from the algebraic ordering of relations. * **Axiom 4 (Bidirectional Group Exchange):** Observation and physical measurement are modeled not as passive recordings, but as bidirectional group-theoretic exchanges between overlapping relational systems forming a joint closure. ## 3. Expected Outcomes and Key Derivations (预期数学成果与推导) The primary expected outcome of this project is a self-consistent mathematical physics framework that derives standard physics as a macroscopic statistical limit of the discrete network. Specifically, the framework delivers the following five principal theoretical results: * **A Generalized Noether Theorem:** The project proves that physical conservation laws are equivalent to the non-trivial kernel of the relational exchange map. In the continuous limit, this reduces to standard Noether currents, but in the discrete regime, it predicts strictly discrete, quantized conserved numbers. * **Derivation of the Least Action Principle:** The action functional (S) is derived as a direct function of the network’s closure invariants (holonomies). The classical principle of least action (\delta S = 0) is shown to be equivalent to a non-Abelian Kirchhoff's law enforcing relational consistency across the network, recovering Maxwell and Yang-Mills fields in the continuous limit. * **Emergence of the Speed of Light (c) and Causality:** Utilizing the **Lieb-Robinson bound** on discrete networks, the project demonstrates that the propagation of structural constraints has a finite, maximum velocity (v_{LR}). This maximum velocity is identified as the physical speed of light (c = v_{LR}). Consequently, causal consistency (the absence of closed timelike curves) emerges as a mathematical corollary of network locality. * **Derivation of the Born Rule:** Quantum randomness is explained via the **Nakajima-Zwanzig projection method** acting on the information limits of localized observers. By applying the **Gleason-Busch theorem**, the quadratic probability rule (|\alpha|^2) is derived rigorously from finite additivity and non-contextuality within the closure space. * **Origin of Spacetime and Wave Functions:** Physical spacetime is reinterpreted as a "pseudo-continuity"—a statistical artifact arising from mutually constraining discrete closures. Similarly, wave functions and the Schrödinger equation are derived as the statistical envelopes of underlying discrete constraint-gap dynamics. ## 4. Empirical Predictions and Observational Significance (经验预测与实验检验) While primarily a foundational and theoretical physics project, RCF yields distinct, falsifiable empirical predictions that differentiate it from standard quantum mechanics and general relativity, providing a clear pathway for future experimental validation (e.g., via multi-ion traps or superconducting qubit benchmarks): * **Observer-Capacity Dependent Decoherence:** RCF predicts that a quantum system's phase dephasing rate (\gamma_{deph}) is explicitly bounded by, and scales non-linearly with, the total information storage/processing capacity (C) of the local physical observer. * **The Quantum "Locking Phase Transition":** The framework predicts that when the information required to specify a quantum state (I_{req}) approaches the maximum capacity of the environment/observer (C), the system will undergo a sharp, non-linear phase transition, dramatically freezing its lifetime within its pointer basis. * **Non-Linear S-Shaped Trigger Curves:** In quantum erasure or interferometry tracking experiments, the interference visibility (V) as a function of information saturation (\Delta = I_{req} - C) is predicted to follow a distinct Sigmoid (S-shaped) curve, rather than the linear or pure exponential decay predicted by standard environmental decoherence models. * **Cosmological Entropy Bounds:** The framework provides a localized, boundary-driven explanation for de Sitter space entropy, predicting the maximum cosmological observer complexity to be bounded near S_{dS} \approx 10^{122} based on the global finiteness of the constraint network.