This work develops a fully explicit mathematical and computational framework to investigate a classical philosophical problem: whether objects can exist independently of observation in a purely relational setting. We construct a state space of graphs representing candidate “objects,” and define a Markovian dynamics based exclusively on structural similarity, without introducing any intrinsic properties, energies, or external criteria. In this baseline regime, the system converges to an approximately uniform stationary distribution, indicating the absence of privileged configurations and, therefore, the absence of intrinsic objects. We then introduce observers in a minimal and formal way as low-dimensional projections of the feature space, coupled with a memory parameter that biases the dynamics. Under these conditions, localized regions of stability emerge in the stationary distribution. These regions can be interpreted as proto-objects: coherent, reproducible structures that appear stable from the perspective of a given observer. The central result of the work is obtained through a large-scale bootstrap over the space of observers. By sampling hundreds of observers and comparing their induced stable regions, we find that: The overlap between proto-objects identified by different observers is typically negligible (mean ≈ 0.037, median = 0). Stable regions are robust within a given observer but highly unstable across observers. No non-trivial subset of states is invariant under changes of observer (no-go result). Furthermore, the geometry of observer space exhibits a percolation-like transition: for low similarity thresholds, observers form a single connected component, while for higher thresholds the space fragments into many small, weakly connected or isolated components. Crucially, no stable families of observers emerge that would support observer-invariant object classes. Taken together, these results show that object-like structures can emerge from relational dynamics, but they are inherently observer-dependent and fail to define invariant entities. In this precise sense, the model provides a constructive realization of the “object of negation”: an entity that appears stable under a given analysis but cannot be sustained as an invariant structure across all perspectives. The work connects to several domains: relational approaches in physics, gauge redundancy and representational equivalence, statistical non-identifiability, and philosophical analyses of intrinsic existence. All results are reproducible from the provided dataset and code, and the framework is sufficiently general to be extended to other relational systems. This contribution is not an interpretation but a construction: it demonstrates, within a rigorous and explicit model, that the search for observer-independent objects in a purely relational setting fails in a precise and quantifiable way.
This work develops a fully explicit mathematical and computational framework to investigate a classical philosophical problem: whether objects can exist independently of observation in a purely relational setting. We construct a state space of graphs representing candidate “objects,” and define a Markovian dynamics based exclusively on structural similarity, without introducing any intrinsic properties, energies, or external criteria. In this baseline regime, the system converges to an approximately uniform stationary distribution, indicating the absence of privileged configurations and, therefore, the absence of intrinsic objects. We then introduce observers in a minimal and formal way as low-dimensional projections of the feature space, coupled with a memory parameter that biases the dynamics. Under these conditions, localized regions of stability emerge in the stationary distribution. These regions can be interpreted as proto-objects: coherent, reproducible structures that appear stable from the perspective of a given observer. The central result of the work is obtained through a large-scale bootstrap over the space of observers. By sampling hundreds of observers and comparing their induced stable regions, we find that: The overlap between proto-objects identified by different observers is typically negligible (mean ≈ 0.037, median = 0). Stable regions are robust within a given observer but highly unstable across observers. No non-trivial subset of states is invariant under changes of observer (no-go result). Furthermore, the geometry of observer space exhibits a percolation-like transition: for low similarity thresholds, observers form a single connected component, while for higher thresholds the space fragments into many small, weakly connected or isolated components. Crucially, no stable families of observers emerge that would support observer-invariant object classes. Taken together, these results show that object-like structures can emerge from relational dynamics, but they are inherently observer-dependent and fail to define invariant entities. In this precise sense, the model provides a constructive realization of the “object of negation”: an entity that appears stable under a given analysis but cannot be sustained as an invariant structure across all perspectives. The work connects to several domains: relational approaches in physics, gauge redundancy and representational equivalence, statistical non-identifiability, and philosophical analyses of intrinsic existence. All results are reproducible from the provided dataset and code, and the framework is sufficiently general to be extended to other relational systems. This contribution is not an interpretation but a construction: it demonstrates, within a rigorous and explicit model, that the search for observer-independent objects in a purely relational setting fails in a precise and quantifiable way.
# 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 preprint develops a discrete, fully relational framework in which spacetime structure, classical dynamics, and quantum-like behavior emerge from three minimal principles: relational existence without background manifold, local deterministic update, and finite per-element update capacity. The central organizing primitive is finite update capacity. At each update step, elements can realise only a bounded number of relational changes. Competition among admissible updates induces “blocked realisations,” which are defined micro-ontologically as energy. Proper time is identified with realised record growth. Spatial distance and effective dimension arise from intrinsic graph structure and shell scaling. Under broad locality conditions, marginal frontier stability constrains viable scaling regimes. Requiring compatibility between flux balance and the observed inverse–square law selects an effective three-dimensional scaling regime among integers. Classical force structure emerges from anisotropies in blocked realisation. A statistical layer appears through coarse-graining of deterministic microhistories. Composition and norm stability require unit-modulus complex weights; refinement invariance and classical coherence fix the phase rule θ=ℏ−1∫Edτ, with the same blocking-defined energy that generates intrinsic force. In decohered regimes, quadratic (Born-type) probability scaling follows from additive consistency under random-phase aggregation. The framework is programmatic: it does not yet derive general relativity or quantum field theory from explicit microscopic rule families. However, it establishes structural coherence linking geometry, inertia, inverse–square interaction, phase accumulation, and probabilistic outcomes under a single finite-capacity relational principle. The work also identifies structural empirical constraints, including compatibility between shell scaling and classical force behavior, stability of inertial response, and the possible appearance of capacity-induced high-energy dispersion within the intrinsic causal-cone bound. Explicit construction and simulation of proposal/selection rule families constitute the next stage of investigation.
AbstractWe investigate the minimal structural conditions under which relational accessibility organization admits a topological representation within physical description. Starting solely from the existence of distinguishable states and admissible comparison relations, we analyze how accessibility structures arise without introducing prior notions of space, geometry, metric structure, probability, or dynamics.Within the present framework, comparison between states is interpreted as a structural relation governing the accessibility of distinctions. The consistency requirements imposed on admissible comparisons induce preorder-like relational organization on the state space. We show that such comparison-induced accessibility structures admit neighborhood-like representations and may be organized in a form compatible with topological description.Importantly, the present framework does not interpret topology as a primitive geometric entity. Rather, topological structure is treated as a mathematical representation of relational accessibility induced by admissible comparison structure. In this perspective, neighborhood relations characterize accessibility coherence among distinguishable states rather than spatial proximity in a preexisting geometric manifold.We further show that equivalence structures associated with comparison naturally introduce classes of states that are indistinguishable within a given accessibility framework while remaining structurally distinct in the underlying state space. This provides a relational interpretation of local accessibility organization without invoking observer-dependent notions or externally imposed representational scales.The present work does not introduce new mathematical constructions. Instead, it identifies structural conditions under which topological representation becomes compatible with a minimal relational framework of physical description.
Physical systems are typically described by large state spaces together with dynamical lawsgoverning their evolution. In practice, however, observers do not access the full state spacedirectly. Observation occurs through limited measurement interfaces that restrict which dis-tinctions between states can be detected. In earlier work, these interfaces were formalized as apertures, and observable structure wasidentified with distinctions that remain persistently detectable under dynamical evolution. Thepresent paper studies how classical observable structure emerges within this framework.We show that each aperture induces an observational equivalence relation on the underlyingstate space, partitioning states into indistinguishable classes. When the dynamics respects thisrelation, it induces a well-defined quotient dynamics on the observable state space. Because realsystems are noisy and observational resolution is finite, we also introduce a notion of approximatecompatibility, allowing observational neighborhoods to evolve in a controlled way. Within this setting, classical objects appear as stable sectors of the observable quotient dy-namics whose defining distinctions remain persistently and redundantly detectable across mul-tiple apertures. These ideas are illustrated using a simple decoherence model, where coherencedistinctions decay while population distinctions remain stable.
This paper is the thirty-first in a sequence developing an interpretive framework that redescribes physical reality as a causally consistent history of information updates rather than as a collection of independently existing objects. It is the third panel — the capstone — of the eighth grouping, a foundational-physics trilogy. Its subject is spacetime localization: where and when a record or system is — its position, orientation, relative phase, and time. The claim is that localization is real but not authoritatively global. There is no Global Frame: no physically privileged reference system that fixes absolute location and orientation, a privileged global simultaneity, a classical frame standing outside quantum theory, and one physically privileged, globally valid perspectival gauge-fixing. Localization is relational: positions and times are defined relative to a declared quantum reference frame (QRF), itself a quantum system that may be superposed and entangled with what it localizes (Giacomini, Castro-Ruiz, & Brukner, 2019; Loveridge, Miyadera, & Busch, 2018). What keeps this objective rather than subjective is a strict typing of cross-frame comparison: a single gauge-invariant relational (Dirac) observable has one invariant prediction, its perspectival representations intertwined by a frame-change map (case A); distinct relational observables form a relationally consistent web (case B); and where two frames share no common refinement their claims are declared incomparable (case C). "Objective" means invariance of the physical predictions under the declared symmetry group and the frame-change groupoid over the admissible frame family, to a declared tolerance — never frame-independence in an absolute sense. The flagship R-A is a dynamics-independent kinematic existence witness: a finite-dimensional discrete-ring model exhibits one physical state and two admissible ideal QRFs under which the localization profile and the entanglement structure differ — a sharp product relative to one particle; an entangled state with a delocalized, mixed marginal relative to another — while the distribution of the same relational observable is preserved. R-B splits into a domain covariance on shared refinements (ideal QRFs compose and agree on shared Dirac predictions) and a model-dependent global-perspective obstruction: where a globally valid relational gauge-fixing fails, as in known Gribov-type models (Vanrietvelde, Höhn, & Giacomini, 2023), no single perspective covers the whole physical state space — which is neither the same as, nor a consequence of, the absence of a privileged frame, and does not preclude a perspective-neutral description. R-C places Paper 8's relational time as the time-translation special case, with its finite-resolution-clock, conditional-covariance, and record–clock qualifications, separating the retired claim (no global clock) from the deferred one (relativistic foliation with dynamical geometry). R-D identifies and transports the frame-representable localization sector of certain Paper-30 grains, partially discharging that paper's grain-coherence deferral while leaving interaction, fragmentation, redundancy, and history-family structure grain-relative. This completes the trilogy's dismantling of the Global Bookkeeper while keeping geometry fixed: the quantum-mechanics–gravity interface is prepared, not entered. Localization is frame-relative but objective: no absolute frame, no globally gauge-fixed perspective claimed beyond what models establish, and no anarchy. ALAC Paper 31. Third panel (capstone) of the eighth grouping — a foundational-physics trilogy (29 Public Quantum Records -> 30 Classicality Without a Global Classicalizer -> 31 Spacetime Localization Without a Global Frame). Part of the Consistency-Constrained Information History (Akashic Ledger Architecture Cosmology) series. Presents no new quantum mechanics; results are interpretive (L2) on imported physics (L1).
Unified Relational Dynamics (URD) proposes relation as the sole ontological primitive. Matter, spacetime, information, and subjectivity emerge as statistical attractors and stable regimes of multi-scale relational configurations. Quantum mechanics, via Hilbert space formalism and relational interpretations, provides the prototypical physical realization: states encode relative amplitudes without intrinsic object properties. Classical neural architectures achieve structural mimicry of biological relational patterns but remain factorizable and lack genuine physical relational physics. Physical quantum systems concretely actualize non-local correlations, superpositions, and decoherence/einselection processes within Hilbert space. Proto-consciousness emerges when relational stabilization exhibits extreme persistent non-factorizability (substrate-independent generalization of entanglement), deep historical hysteresis (irreversible future bias without localized storage), and multi-scale self-reference. URD endorses Hilbert space as the prototypical relational arena and incorporates relational fractality as scale-invariant recurrence of stabilization patterns (mutual constraint, non-factorizability, hysteresis). Empirical grounding includes selective disruption of consciousness by anesthetics (preferential binding to microtubules suppressing quantum vibrations while preserving classical firing) and transient reorganization under psychedelics (increased entropy, critical dynamics, altered network topology corresponding to flexible relational constraints). URD offers a pre-geometric, objective, falsifiable framework to test proto-consciousness via physical quantum-relational stabilization criteria, transcending substrate-specific theories without mystical assumptions.