We discuss the emergence of topology from a consideration of set extensions in General Systems Theory. Boundaries arise in a natural way, separating independent elements or regions of the system. Our aim is a unification of Etter theory, Kron's method of Tearing and Jessel's formulation of Huygens' Principle. This should make explicit the equivalence between the objective, structural, holographic and the subjective, relative definitions of information, sought in Bowden (1994b), reprinted in this Special Issue. It connects the abstract generalisations of Schrodinger's equation and Bom's rule derived in probabilistic Etter theory with the real world of electrical and other physical phenomena in General Physical Systems Theory. This paper can be considered as a continuation of Bowden (1990; 1994a) and as a response to Bowden (1994b), reprinted in this issue. We review the ideas behind Kron's Method of Tearing and Jessel's Principle of Secondary sources (both special cases of the above theory) and their equivalence. We follow Hiley's argument in Hiley (1996) to show how Schrodinger's equation can be thought of as specifying the evolution of (a series of) tearings in continuous space. These can be shown on a commutative diagram as a series of similarity transforms. We compare this with Etter's derivation (Etter, 1998). We describe briefly a recently published derivation of Maxwell's equations from a non-commutative algebra and show how they fit onto a related commutative diagram. Finally we make some comments on applications of the general theory to computer systems. This paper is a series of vignettes of work in progress. It is designed to point the direction of work to come in Constructive Physics.
We discuss the emergence of topology from a consideration of set extensions in General Systems Theory. Boundaries arise in a natural way, separating independent elements or regions of the system. Our aim is a unification of Etter theory, Kron's method of Tearing and Jessel's formulation of Huygens' Principle. This should make explicit the equivalence between the objective, structural, holographic and the subjective, relative definitions of information, sought in Bowden (1994b), reprinted in this Special Issue. It connects the abstract generalisations of Schrodinger's equation and Bom's rule derived in probabilistic Etter theory with the real world of electrical and other physical phenomena in General Physical Systems Theory. This paper can be considered as a continuation of Bowden (1990; 1994a) and as a response to Bowden (1994b), reprinted in this issue. We review the ideas behind Kron's Method of Tearing and Jessel's Principle of Secondary sources (both special cases of the above theory) and their equivalence. We follow Hiley's argument in Hiley (1996) to show how Schrodinger's equation can be thought of as specifying the evolution of (a series of) tearings in continuous space. These can be shown on a commutative diagram as a series of similarity transforms. We compare this with Etter's derivation (Etter, 1998). We describe briefly a recently published derivation of Maxwell's equations from a non-commutative algebra and show how they fit onto a related commutative diagram. Finally we make some comments on applications of the general theory to computer systems. This paper is a series of vignettes of work in progress. It is designed to point the direction of work to come in Constructive Physics.
# 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.
There is increasing evidence that information may be the basic stuff of the universe. We consider this proposition in the light of Bohm and Hiley's (1993) quantum potential, the work of the ANPA group on the combinatorial hierarchy, and the natural philosophies of G. Kron (1963) and M. Jessel (1962). We compare and contrast the philosophical backgrounds of both these and the more conventional Copenhagen interpretation. In conclusion, we suggest that our approach should be termed 'constructive postmodern physics'.>
Holographic Informational Ontology Framework - Information Network (HIOF•IN)A Relational Reading of Quantum Mechanics This is a short, non-technical companion to the six-paper HIOF- IN (Information Network) series. It is written for readers with a background in physics, philosophy of science, or information theory — not for readers already familiar with the author's wider theoretical framework. The core idea, stated plainly Quantum mechanics is usually taught by first drawing a stage — a box, a particle, a coordinate, an observer — and then adding uncertainty on top of it. This series proposes reading the quantum the other way round: as a relation between two sides that has not yet been settled, prior to any box, coordinate, or observer. Nothing mystical is claimed; the series draws on the same discipline used in relational and information-theoretic approaches to physics. What differs is the insistence that naming — settling a relation into something that can be pointed to — always comes after the relation itself, never before it. Why "the field" is not where the physics actually lives Fields, force lines, and vacuum fluctuations are usually treated as the bedrock of physical reality. This series argues instead that force appears once two sides have paired, and the field is only the geometric bookkeeping drawn afterwards, in order to compute that force. The vacuum is not "a point trembling"; it is the print-out, in field language, of a simpler fact — that a state with no relation at all cannot be sustained. How to read quantum mechanics without mistranslating it Rather than declaring quantum-mechanical terms secretly identical to relational ontology, the series proposes a four-step reading rule: take the statement, strip away the stage furniture, ask whether what remains is an open question, an unsettled naming, or a settled naming, and only then write one sentence back into relational vocabulary. Entanglement survives this test almost intact; the picture of a particle "running down every path at once" mostly does not. Testing the method against a live physics paper Rather than relying only on textbook examples, one paper applies the same four-step rule to a genuine, contemporary gauge-theory report on centre-vortex confinement, published in 2026. The purpose is not to referee that paper's mathematics, but to show the reading method survives contact with current, technical physics literature — without joining that literature's own unresolved debates. Where this series agrees, and disagrees, with existing ideas Two companion papers place the series carefully alongside neighbouring research programmes. One compares it with Relational Quantum Mechanics and QBism, after both traditions independently deny that observed events are absolute — the series states precisely which shared assumption is dropped, and no more. The other compares it with causal sets, spin networks, and quantum graphity, approaches in quantum gravity that also treat relation as prior to spacetime, and states exactly which basic building blocks — events, vertices, pre-existing points — the series still refuses to assume. What the full series covers The six papers proceed from naming the relational locus, to demoting the field to a computational layer, to a four-step method for reading quantum mechanics, to a case study against a 2026 gauge-theory report, to a boundary against relational interpretations of quantum theory, and finally to a boundary against combinatorial models of emergent spacetime. A note on method The series separates, explicitly, what it names and prohibits from what it computes. It does not derive new dynamical equations, does not challenge the calculational apparatus of quantum mechanics, and does not claim to resolve open problems — such as where time comes from, or why spacetime has three spatial dimensions — that remain open across comparable research programmes. Readers are encouraged to treat its ontological claims as a disciplined reading exercise, not as a replacement for physical theory. Keywords: quantum mechanics, relational ontology, information theory, quantum foundations, field theory, entanglement, decoherence, quantum gravity, causal sets, spin networks, philosophy of physics AuthorWai-Hung Tam (Pan), Independent ResearcherORCID: 0009-0002-7789-8464Email: panxtam@protonmail.com
## Title The Relational Closure Framework: From Relations to Matter via Causal Consistency ## Authors Mingdong He (Independent Researcher, ORCID: 0009-0009-5176-7571) ### Overview This deposit contains the complete source code and compiled PDF of the Relational Closure Framework (RCF) version 2.0C, a relational approach to the foundations of physics developed as a candidate contribution to Hilbert's sixth problem (the axiomatization of physics). The framework proposes that the universe is constituted by **relations** rather than objects. Stable relations form **closures** (self-consistent cycles); closures generate topological invariants (holonomies); and these invariants underpin all physical structures—conservation laws, the least-action principle, the Born rule, dimension, geometry, and matter—as emergent phenomena of the closure network at different scales. ### What is new in v2.0C Version 2.0C introduces a **causal consistency principle**: physical existence requires causal consistency as a precondition, not as a consequence of continuous spacetime. What appears as continuity is reinterpreted as **pseudo-continuity**—the statistical artifact of mutually constraining discrete closures. This reframing dissolves (rather than solves) the continuum-limit problem that has obstructed discrete approaches to physics for decades. Key new results in v2.0C: 1. **Causal consistency as physical precondition** — The speed of light is derived as the enforcement mechanism for causal consistency (not an arbitrary postulate). 2. **Randomness from constraint gaps** — Quantum probability arises naturally from the incompleteness of discrete constraints, not from an axiom. 3. **Wave function as constraint-gap envelope** — The wave function exists because the network is discrete; it "waves" because holonomy phases interfere; it is non-classical because it carries probability, not energy. 4. **Schrödinger equation from pseudo-continuity** — Via the Madelung transformation, the Schrödinger equation is shown to emerge from: (i) holonomy action stationarity (Hamilton–Jacobi), (ii) constraint-gap-induced probability density, and (iii) Noether conservation (continuity equation). The quantum potential is interpreted as the signature of pseudo-continuity failure. 5. **Fields as long-range constraint mechanisms** — All fundamental fields are intrinsically long-range because causal consistency is a global requirement. The de Sitter entropy (~10^122) is conjectured as the information-theoretic capacity bound on causal constraints. ### Core results (retained from v1.2) - A **generalized Noether theorem**: conservation is identified with the kernel of the exchange map, derived from axioms without assuming symmetry. - A **least-action principle**: the action is proved to be necessarily a function of closure holonomies; δS = 0 is equivalent to a non-abelian Kirchhoff law; the continuum limit recovers Maxwell and Yang–Mills equations. - The **Born rule**: stochasticity arises from the observer as an open dynamical system (Nakajima–Zwanzig); the square law |α|² is forced by the Gleason–Busch theorem under finite additivity. - The **Z₃–γ₅ algebraic correspondence**: a strict algebraic identity linking the minimal relational closure (3-cycle) to the Dirac chirality operator. - A **quantum error correction correspondence**: the Gauss-law operator is structurally identical to the toric-code stabilizer; vacuum stability is topologically protected. ### AI use disclosure This manuscript was prepared with the assistance of AI language models (Claude/ChatGPT/Gemini families) for typesetting, reference formatting, and consistency checking. All theoretical content—axioms, proofs, conjectures, and the causal consistency principle—is the sole intellectual contribution of the author. AI tools were used as a writing assistant, not as a source of scientific ideas.
Holographic Informational Ontology Framework - Information Network (HIOF•IN)A Relational Reading of Quantum Mechanics This is a short, non-technical companion to the six-paper HIOF- IN (Information Network) series. It is written for readers with a background in physics, philosophy of science, or information theory — not for readers already familiar with the author's wider theoretical framework. The core idea, stated plainly Quantum mechanics is usually taught by first drawing a stage — a box, a particle, a coordinate, an observer — and then adding uncertainty on top of it. This series proposes reading the quantum the other way round: as a relation between two sides that has not yet been settled, prior to any box, coordinate, or observer. Nothing mystical is claimed; the series draws on the same discipline used in relational and information-theoretic approaches to physics. What differs is the insistence that naming — settling a relation into something that can be pointed to — always comes after the relation itself, never before it. Why "the field" is not where the physics actually lives Fields, force lines, and vacuum fluctuations are usually treated as the bedrock of physical reality. This series argues instead that force appears once two sides have paired, and the field is only the geometric bookkeeping drawn afterwards, in order to compute that force. The vacuum is not "a point trembling"; it is the print-out, in field language, of a simpler fact — that a state with no relation at all cannot be sustained. How to read quantum mechanics without mistranslating it Rather than declaring quantum-mechanical terms secretly identical to relational ontology, the series proposes a four-step reading rule: take the statement, strip away the stage furniture, ask whether what remains is an open question, an unsettled naming, or a settled naming, and only then write one sentence back into relational vocabulary. Entanglement survives this test almost intact; the picture of a particle "running down every path at once" mostly does not. Testing the method against a live physics paper Rather than relying only on textbook examples, one paper applies the same four-step rule to a genuine, contemporary gauge-theory report on centre-vortex confinement, published in 2026. The purpose is not to referee that paper's mathematics, but to show the reading method survives contact with current, technical physics literature — without joining that literature's own unresolved debates. Where this series agrees, and disagrees, with existing ideas Two companion papers place the series carefully alongside neighbouring research programmes. One compares it with Relational Quantum Mechanics and QBism, after both traditions independently deny that observed events are absolute — the series states precisely which shared assumption is dropped, and no more. The other compares it with causal sets, spin networks, and quantum graphity, approaches in quantum gravity that also treat relation as prior to spacetime, and states exactly which basic building blocks — events, vertices, pre-existing points — the series still refuses to assume. What the full series covers The six papers proceed from naming the relational locus, to demoting the field to a computational layer, to a four-step method for reading quantum mechanics, to a case study against a 2026 gauge-theory report, to a boundary against relational interpretations of quantum theory, and finally to a boundary against combinatorial models of emergent spacetime. A note on method The series separates, explicitly, what it names and prohibits from what it computes. It does not derive new dynamical equations, does not challenge the calculational apparatus of quantum mechanics, and does not claim to resolve open problems — such as where time comes from, or why spacetime has three spatial dimensions — that remain open across comparable research programmes. Readers are encouraged to treat its ontological claims as a disciplined reading exercise, not as a replacement for physical theory. Keywords: quantum mechanics, relational ontology, information theory, quantum foundations, field theory, entanglement, decoherence, quantum gravity, causal sets, spin networks, philosophy of physics AuthorWai-Hung Tam (Pan), Independent ResearcherORCID: 0009-0002-7789-8464Email: panxtam@protonmail.com
The mathematical description of quantum systems univocally identies their nature. In other words we treat a system as quantum if we describe its behaviour adopting Hilbert spaces and structures thereof, as prescribed by the postulates of quantum theory. The choice of using quantum systems as the elementary systems of physics can be justied in terms of informational principles, thanks to results of the last decade. Such results come as the conclusion of a research program that lasted almost one century, with the aim of reformulating quantum theory in terms of operational principles. This achievement now poses a new challenge, that we face here. If the systems of quantum theory are thought of as elementary information carriers in the rst place, rather than elementary constituents of matter, and their connections are logical connections within a given algorithm, rather than space-time relations, then we need to nd the origin of mechanical concepts—that characterise quantum mechanics as a theory of physical systems. To this end,we will illustrate howphysical laws can be viewed as algorithms for the update of memory registers that make a physical system. Imposing the characteristic properties of physical laws to such an algorithm, i.e. homogeneity, reversibility and isotropy, we will show that the physical laws thus selected are particular algorithms known as cellular automata. Further assumptions regarding maximal simplicity of the algorithm lead to two cellular automata only, that in a suitable regime can be described by Weyl’s dierential equations, lying at the basis of the dynamics of relativistic quantum elds. We will nally discuss how the same cellular automaton can give rise to both Fermionic eld dynamics and to Maxwell’s equations, that rule the dynamics of the electromagnetic eld. We will conclude reviewing the discussion of the relativity principle, that must be suitably adapted to the scenario where space-time is not an elementary notion, through the denition of a change of inertial reference frame, and whose formulation leads to the recovery of the symmetry of Minkowski space-time, identied with Poincar´e’s group. Space-time thus emerges as one of the manifestations of physical laws, rather than the background where they occur, and its features are determined by the dynamics of systems, necessarily equipped with dierential equations that express it. In brief, there is no space-time unless an evolution rule requires it.