Towards a Relational Account of Spacetime Symmetries and Structural Scales: An Axiomatic Framework and Algebraic Minimal Realization Based on Observation–Feedback Closure
This work presents a two-stage first-principles construction for foundational physics. In the first stage, an axiomatic framework is introduced in which physical objecthood is defined exclusively through participation in observation–feedback (OF) processes. At this axiomatic level, no background spacetime, metric, Hilbert space, or symmetry group is assumed a priori. In the second stage, a minimal faithful realization of this framework is adopted, consisting of a set of explicitly stated, controlled algebraic choices that support finite operational distinguishability. Within this minimal realization, a hierarchy of physically significant structures is consistently represented: an internal O(3) symmetry with oriented SO(3) sectors; a Lorentz-type kinematical structure with invariant group SO⁺(3,1) (the proper orthochronous Lorentz group) and an associated invariant propagation scale that occupies the structural role of the conventional speed c; and a minimal closure scale that, under standard action-dimensional normalization, plays a structural role analogous to that of the reduced Planck constant ℏ. This construction thus provides a unified relational setting in which the core kinematical architectures of special relativity and quantum mechanics admit a consistent, operationally grounded representation, realized under explicitly stated minimal assumptions. No derivation of dynamical equations or direct identification with empirical spacetime geometry is claimed.
This work presents a two-stage first-principles construction for foundational physics. In the first stage, an axiomatic framework is introduced in which physical objecthood is defined exclusively through participation in observation–feedback (OF) processes. At this axiomatic level, no background spacetime, metric, Hilbert space, or symmetry group is assumed a priori. In the second stage, a minimal faithful realization of this framework is adopted, consisting of a set of explicitly stated, controlled algebraic choices that support finite operational distinguishability. Within this minimal realization, a hierarchy of physically significant structures is consistently represented: an internal O(3) symmetry with oriented SO(3) sectors; a Lorentz-type kinematical structure with invariant group SO⁺(3,1) (the proper orthochronous Lorentz group) and an associated invariant propagation scale that occupies the structural role of the conventional speed c; and a minimal closure scale that, under standard action-dimensional normalization, plays a structural role analogous to that of the reduced Planck constant ℏ. This construction thus provides a unified relational setting in which the core kinematical architectures of special relativity and quantum mechanics admit a consistent, operationally grounded representation, realized under explicitly stated minimal assumptions. No derivation of dynamical equations or direct identification with empirical spacetime geometry is claimed.
# 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.
This work presents a unified theoretical framework that aims to bridge relativity and quantum mechanics through a single structural principle: delayed realization. Instead of assuming spacetime, time evolution, or quantum states as primitive entities, the framework introduces a pre-geometric definability network in which relational differences are initially undefined and become stabilized through nonlinear dynamics. The emergence of spacetime is then reinterpreted as the stabilization of definable differences, while the present is not given a priori but selected via delayed consistency over histories. Within this formulation, quantum phenomena arise naturally when multiple realization candidates coexist, producing interference structures governed by a variational principle. In the low-defect limit, the framework recovers effective relativistic geometry and field-theoretic behavior, including emergent gravitational curvature and gauge structures. A central contribution of this work is the formulation of a unified definability action, combining stabilization, curvature minimization, delayed consistency, and informational structure into a single variational framework. Under appropriate coarse-graining, this action yields structures consistent with Einstein field equations, quantum amplitudes, and gauge invariance as different regimes of the same underlying mechanism. This paper does not claim a fully closed mathematical unification of all physical sectors. Instead, it provides a coherent structural framework in which spacetime emergence, present realization, quantum probability, and effective field dynamics are organized as manifestations of a single principle. The framework is intended as a foundational step toward a deeper unification of physics, where time, geometry, and probability are not assumed, but emerge from the stabilization and realization of definable relational structures.
Horizon Monism proposes a unified theoretical framework in which quantum structure, operational time, and spacetime geometry emerge from finite reference horizons embedded within a single, scale-free relational–interaction totality. Rather than treating spacetime, fundamental time, and quantum probabilities as primitive, the framework investigates their possible reconstruction through finite distinguishability, relational compression, and effective projection structures. The paper establishes a strict mathematical hierarchy of finite-horizon resolutions, deriving monotonic distinction deficits and a directed fine-to-coarse order as a candidate foundation for operational time. This construction is extended to multiple reference horizons through order-preserving transport and conditional shared-blindness theorems. Within a conditionally constructed complex groupoid algebra, the minimal noncommutative effective structure is identified as M₂(ℂ). Projection compression is shown to generate noncommuting effective operators, while smoothly varying projections induce connection curvature. Under specified geometric conditions, this curvature reconstructs the Levi-Civita curvature of an emergent Lorentzian spacetime. Explicit de Sitter and nonlinear Green-spectral constructions provide conditional examples of Einstein geometry and stable nonvanishing projected curvature. A further reconstruction of quantum probability establishes that normalized, context-independent, coarse-graining-consistent effect assignments admit a unique density-matrix representation, recovering the Born rule for pure states. The Principle of Minimal Sufficient Realization is additionally applied to a restricted class of Standard-Model-type representations, yielding conditional minimality results for color, fermion generations, and Higgs structure. The framework distinguishes exact mathematical theorems from conditional physical realizations. Its principal unresolved questions concern the microscopic origin of quantum probability consistency, physical state selection, continuum dynamics, and experimentally distinguishable predictions. Horizon Monism is presented as a mathematical and theoretical research program toward the possible co-emergence of quantum physics and gravitation.
Quantum Mechanics as Reversible Closure Representation and the Emergence of Relativistic Structure (Paper 2) Abstract This work develops a representation-theoretic extension of the closure-first discrete gravity framework established in Reality is What Misspoken Language Cannot Deliver (Rev17). Starting from finite primitive state spaces and reversible closure algebra, we construct a unitary representation of the closure group on a complex Hilbert space. Quantum evolution arises naturally as the continuous representation of discrete reversible transitions, with phase structure determined by restoration periods. Defect transport within the closure algebra yields dispersion relations that recover relativistic propagation in the macroscopic limit. Local flatness emerges as the tangent limit of the discrete curvature substrate, giving Special Relativity as a local geometric approximation of the underlying simplicial structure. General Relativity appears as the coarse-grained curvature density of defect-labelled simplicial geometry, preserving the Einstein field equations within their established effective domain of validity. Quantum mechanics is therefore not postulated independently, but arises as the representation layer of reversible finite closure, while relativistic structure emerges from geometric coarse-graining of the same primitive algebra. No new macroscopic gravitational equations are introduced; the present work concerns the structural origin of existing relativistic frameworks rather than their modification. Introduction In the preceding work (Rev17), finite entropy bounds were shown to exclude primitive continuum ontology and to imply a finite state space for bounded physical regions. Under assumptions of deterministic closure, no primitive erasure, and inverse realisability, the reversible sector of primitive transitions forms a group. Defect labels associated with group elements define discrete curvature quanta, which embed naturally into a Regge-type geometric framework. A minimal continuum-limit argument yields the Einstein–Hilbert action under refinement. Importantly, this construction does not alter the Einstein field equations at macroscopic scales. General Relativity remains valid within its established regime as an effective continuum description. The prior work addresses the structural substrate from which that continuum description emerges. The present work extends this foundation by addressing two remaining structural questions: How does quantum mechanics arise from finite reversible closure? How does relativistic kinematics emerge from the same primitive structure? Rather than introducing Hilbert space or Lorentz symmetry axiomatically, we derive them as representation and coarse-graining layers of the previously established closure algebra. The guiding principle is minimal structural extension: no new primitive entities are introduced. All additional structure follows from representation consistency and macroscopic limit coherence. Quantum theory appears as the unitary representation of reversible closure; relativistic structure emerges as the geometric compression of defect-labelled algebra under scale separation. References[1] J. Bloggs, Aligning GR with Reality – Fully and Finally, Zenodo (2026). DOI: 10.5281/zenodo.18673556.[2] T. Regge, General relativity without coordinates, Nuovo Cimento 19 (1961) 558–571. DOI:10.1007/BF02733251.[3] J. W. Barrett, D. Oriti, and R. M. Williams, Tullio Regge’s legacy: Regge calculus and discrete gravity, arXiv:1812.06193 (2018). DOI: 10.48550/arXiv.1812.06193.[4] R. Loll, Discrete approaches to quantum gravity in four dimensions, Living Rev. Relativ. 1 (1998) 13. DOI: 10.12942/lrr-1998-13.[5] B. Dittrich and S. Steinhaus, Path integral measure and triangulation independence in discrete gravity, Phys. Rev. D 85 (2012) 044032. DOI: 10.1103/PhysRevD.85.044032.
Several fundamental physical relations share a mathematical structure in which the product of two conjugate quantities equals an invariant constant. Representative examples include the energy-period relation ET = h (where T = 1/f is the oscillation period and h is Planck’s constant), the momentum-wavelength relation pλ = h (the de Broglie relation), and the frequency-wavelength dispersion relation fλ = c. Although these expressions originate from quantum mechanics, wave mechanics, and electromagnetism respectively, each belongs to the generalized class XY = K, where K is an invariant constant. Their shared algebraic form raises the question of whether a common organizational principle underlies physical representation. This paper proposes the Foundational Principle of Closure Representation as an interpretational framework. Within this framework, action is adopted as the foundational quantity, and closure is introduced as a representation-independent measure of cyclic completion, defined by the condition Δθ = 2π. Planck’s constant h is interpreted as the action associated with one completed closure—consistent with the Bohr–Sommerfeld quantization condition ∮p dq = nh. The Planck relation E = hf is accordingly understood as the product of action per closure and closure rate per unit time. A representation-transfer relation (RX)(Y/R) = XY is introduced, where R is a dimensionless representation coordinate. A measurable realization is given by R = λC/λdB = p/(mc), coinciding with the standard relativistic quantity γβ. This relation is a mathematical identity; its physical significance lies in the proposed interpretation of R as a representation coordinate characterizing the distribution of closure action between conjugate observables. The proposed framework does not introduce new physical laws, modify established equations, or claim to supersede existing theories. It provides a unified interpretational architecture within which several established reciprocal relations—connecting action, closure, wavelength, frequency, momentum, and propagation—may be understood as instances of a common closure-action structure.
This publication presents an extended axiomatic revision of the Probability-Space Axiomatic Foundation. The original minimal logical core (K1–K4), governing realization, temporal asymmetry, and constrained selection among unrealized possibilities, is preserved verbatim and unchanged. This revision introduces an Extended Core (v2.0) consisting of three additional axioms (K5–K7) that are additive rather than revisionary. These axioms are required to support observer-dependent projection of observables, non-signaling correlation via shared collapse ancestry (entanglement), and the emergence of persistent relational deformation interpreted as effective curvature. The extended axioms do not modify the original logical foundation; instead, they constrain how realized structure is accessed, correlated, and stabilized when instantiated as a physical system. The work is structured as a strict in-place extension of the original manuscript. All original text, numbering, and logical development are preserved word-for-word. New material is inserted only where logically required, and is explicitly labeled as part of the Extended Core. No experimental claims are made. No force-based or field-first primitives are introduced. Physical quantities such as mass, energy, or spacetime curvature are not assumed at the axiomatic level. This axiomatic framework is intended to function as a constraint layer for physical theories built on Probability-Space concepts, ensuring compatibility with observer locality, non-signaling correlation, and emergent large-scale structure without altering the minimal logical basis. The framework is falsifiable in principle: any physical realization must recover standard quantum statistics and empirically tested gravitational behavior in appropriate limits. This record supersedes and extends the author’s prior Zenodo publication by providing a formally defined Extended Core (v2.0), while maintaining full continuity with the original axiomatic foundation.