Relational Theory of Expression (RTE) proposes a new conceptual framework in which physics is understood not primarily as a theory of independently existing objects, but as a theory of the conditions under which physical content becomes expressible within a quasilocal domain. The central idea is that the physically accessible object is not an isolated state x, but an operational equivalence class [x]ᴰ, induced by an expression map Eᴰ : Sphys → Sobs(D). This naturally leads to x₁ ∼ᴰ x₂ ⇔ Eᴰ(x₁) = Eᴰ(x₂) and to the factorization of physical descriptions through operationally meaningful quotient spaces. The framework separates definitions, theorems, physical postulates, hypotheses, and interpretations, allowing philosophical motivation and mathematical structure to evolve independently while remaining logically consistent. Beyond operational equivalence, RTE introduces a rigorous distinction between manifestation, reconstruction, and ontology. Past and future are treated as compatible reconstructions, represented by the space of histories Γᴰ(d) rather than directly observable entities. Local physical descriptions are organized through presheaf and sheaf structures, where restriction maps connect overlapping domains and failures of gluing provide a natural mathematical language for contextuality, boundary effects, and reconstruction limits. Within this framework, auxiliary constructions such as B⁴ are treated with explicit type discipline: they may exist as mathematical fillings or variational tools without being promoted to physically existing regions of spacetime. The programme therefore distinguishes mathematical existence from physical existence in a precise and operationally useful way. The present work is intended as the foundational paper of a broader research programme connecting quasilocal cosmology, information theory, observer-dependent reconstruction, and categorical mathematics. Rather than proposing a new dynamical law, it establishes a minimal structural language in which later developments—such as the DRD information framework, quasilocal B⁴ cosmology, boundary variational principles, and observational cosmology—can be formulated within a common conceptual architecture. The guiding principle is that physics studies invariant structures of expressibility, while the objects of physics emerge as stable relational manifestations rather than self-subsistent entities. This first version is therefore conceived as the opening movement of a larger mathematical programme, providing a disciplined foundation for subsequent theoretical and empirical developments.
Relational Theory of Expression (RTE) proposes a new conceptual framework in which physics is understood not primarily as a theory of independently existing objects, but as a theory of the conditions under which physical content becomes expressible within a quasilocal domain. The central idea is that the physically accessible object is not an isolated state x, but an operational equivalence class [x]ᴰ, induced by an expression map Eᴰ : Sphys → Sobs(D). This naturally leads to x₁ ∼ᴰ x₂ ⇔ Eᴰ(x₁) = Eᴰ(x₂) and to the factorization of physical descriptions through operationally meaningful quotient spaces. The framework separates definitions, theorems, physical postulates, hypotheses, and interpretations, allowing philosophical motivation and mathematical structure to evolve independently while remaining logically consistent. Beyond operational equivalence, RTE introduces a rigorous distinction between manifestation, reconstruction, and ontology. Past and future are treated as compatible reconstructions, represented by the space of histories Γᴰ(d) rather than directly observable entities. Local physical descriptions are organized through presheaf and sheaf structures, where restriction maps connect overlapping domains and failures of gluing provide a natural mathematical language for contextuality, boundary effects, and reconstruction limits. Within this framework, auxiliary constructions such as B⁴ are treated with explicit type discipline: they may exist as mathematical fillings or variational tools without being promoted to physically existing regions of spacetime. The programme therefore distinguishes mathematical existence from physical existence in a precise and operationally useful way. The present work is intended as the foundational paper of a broader research programme connecting quasilocal cosmology, information theory, observer-dependent reconstruction, and categorical mathematics. Rather than proposing a new dynamical law, it establishes a minimal structural language in which later developments—such as the DRD information framework, quasilocal B⁴ cosmology, boundary variational principles, and observational cosmology—can be formulated within a common conceptual architecture. The guiding principle is that physics studies invariant structures of expressibility, while the objects of physics emerge as stable relational manifestations rather than self-subsistent entities. This first version is therefore conceived as the opening movement of a larger mathematical programme, providing a disciplined foundation for subsequent theoretical and empirical developments.
# 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.
Physics has often advanced by revealing that concepts once regarded as fundamental are instead emergent manifestations of deeper organizing principles. Forces gave way to fields, gravitation became the geometry of spacetime, and quantum mechanics replaced deterministic particles with probabilistic wavefunctions. The Relational Primitive Hypothesis (RPH) asks whether this progression may continue one step further. Rather than treating particles, fields, spacetime, or geometry as the primitive constituents of physical reality, RPH proposes that evolving relational structures constitute the primitive ontology from which familiar physical quantities emerge. The objective of RPH is not to replace established physics, but to investigate whether a simpler primitive ontology can provide a common foundation from which established physical theories may naturally emerge. Within this framework, matter is interpreted as stable relational organization, mass as persistent relational stability, spacetime as emergent relational geometry, and physical interactions as consequences of evolving relational dynamics. This work does not present a completed physical theory. Instead, it establishes a conceptual framework and a structured research program. It defines a proposed ontology, formulates a minimal set of postulates, identifies the mathematical objectives required to develop a predictive theory, examines the physical consequences expected from the proposed framework, and establishes scientific criteria by which the hypothesis must ultimately be evaluated. The Relational Primitive Hypothesis neither replaces nor contradicts established physical theories. Rather, it investigates whether General Relativity, quantum mechanics, and other successful physical theories may themselves emerge as effective descriptions of a more fundamental relational dynamic. The ultimate success or failure of the hypothesis depends not upon its philosophical appeal, but upon the development of a mathematically consistent framework capable of reproducing established physics while generating new, experimentally testable predictions. Until such a framework is achieved, the Relational Primitive Hypothesis should be regarded as a structured research program whose validity remains an open scientific question.
Abstract This paper develops an observer-centric framework for analyzing how common physical structure may be reconstructed from finite observational perspectives. The starting point is operational rather than metaphysical: every physically situated observer or measuring apparatus has bounded discrimination, finite retention, limited temporal resolution, and finite capacity to register distinctions. Each observational regime is represented by a state space 𝑋𝑖, with deterministic reductions represented by bonding maps. More general stochastic and quantum reductions are treated as compatible map extensions rather than as ordinary set-valued inverse systems. The central claim is methodological. Inter-perspectival objectivity is formulated relative to an observational network through three distinct questions: does a compatible reconstruction exist, is it unique (up to a stated equivalence), and which properties remain invariant across all admissible reconstructions? The paper distinguishes local feasibility, overlap compatibility, and global reconstructibility, and introduces obstruction certificates for localizing compatibility failure. A classical phase space example establishes the kinematic architecture, while a worked Peres Mermin contextuality application provides a nontrivial foundational benchmark. There the appropriate mathematical object is a limit of an overlap diagram: each context is locally consistent, but the complete family has no compatible global section. The manuscript compares this reconstruction perspective with ordinary coarse graining, relational descriptions, and sheaf-theoretic contextuality, and states explicit limitations and criteria for future empirical or explanatory success.
This work presents a conceptual and representational framework aimed at clarifying the structural relationships among established physical theories across relativistic, quantum, and cosmological regimes. Rather than proposing new dynamical laws, particles, or experimentally testable predictions, the manuscript focuses on how existing physical descriptions may be consistently interpreted within a unified representational structure. The proposed Dual-Spacetime Projection Framework distinguishes between an underlying descriptive domain and an observed effective spacetime, connected through a scale-dependent projection. This distinction is intended as a formal and interpretative tool, not as an ontological claim about the existence of multiple physical spacetimes. Within this perspective, several foundational features of modern physics—such as Lorentz invariance, finite signal propagation, quantum uncertainty, and large-scale cosmological behavior—are discussed as consequences of representational structure rather than as independent postulates. The manuscript is positioned as a contribution to the foundations of theoretical physics, offering a coherent conceptual lens for organizing and interpreting existing theories. It is intended for readers interested in foundational questions, conceptual unification, and the representational aspects of physical law, and does not rely on experimental data or numerical simulation.
This work presents the Machwerk Relational Physics Framework, a non-Euclidean, operational approach to physical description grounded in observability, relational structure, and information-theoretic limits. The framework introduces a sharp boundary (M2) between physically observable geometry and mathematically definable but operationally inaccessible extensions. Classical physics, including Newtonian mechanics, is recovered exactly within the observable domain, while higher-dimensional and non-local effects are treated as relational projections rather than fundamental entities. Singularities, quantum phenomena, and apparent paradoxes such as superposition, entanglement, and tunneling are shown to arise naturally when three-dimensional constraints are relaxed, without invoking additional postulates or speculative dynamics. The work emphasizes structural consistency, reversibility, and empirical compatibility over untestable assumptions. It is intended as a conceptual and formal reference for researchers interested in foundational questions of spacetime, measurement, and the limits of physical description. This Zenodo record provides a citable archival version of the work. The associated formal core implementation and research framework is publicly available at: https://github.com/machwerk-formal-framework/machwerk-relational-physics
This work introduces the foundations of an observer-dependent informational theory of physics.Rather than proposing a closed or final model, the paper establishes the conceptual and methodological architecture of a unified research program in which physical phenomena are described through observer-accessible informational states. The theory treats the observer as an integral component of physical description, rather than as an external disturbance or purely abstract entity.Information is defined operationally, as a relational construct linking observation, structure, and temporal coherence. The paper does not present new mathematical derivations.Instead, it provides a stable foundational framework that unifies a set of modular technical works, each addressing specific physical, cognitive, or interpretative problems. Mathematical formulations, empirical models, and testable predictions are delegated to the corresponding modular papers, allowing the theory to evolve through falsification and refinement without requiring repeated foundational revision. This contribution is intended as a foundational reference for an evolving observer-centered informational approach to physics.