Entropy, Topology, and Information as Structural Foundations of Physical Law A Conceptual and Operator-Theoretic Framework for the Emergence of Spacetime, Fields, and Dynamical Structure

! 1! Entropy, Topology, and Information as Structural Foundations of Physical Law A Conceptual and Operator-Theoretic Framework for the Emergence of Spacetime, Fields, and Dynamical Structure Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, Germany [email protected] Abstract Contemporary physics increasingly suggests that spacetime, geometry, and even fundamental interactions may not be primitive constituents of nature but emergent structures arising from deeper informational or topological organization. Motivated by developments in entanglement-based emergent gravity, topological quantum field theory, and operator-algebraic approaches to quantum theory, this article proposes an information-theoretic and topological framework in which physical law arises from entropy-weighted projections on a diagrammatic Hilbert space. The aim is not to present a fundamental physical theory, but to articulate a coherent conceptual architecture clarifying how informational constraints, topological equivalence classes, and operator structures can jointly underpin geometric and dynamical regularities. A diagrammatic Hilbert space encodes possible topological configurations; an entropy functional over these structures defines a rule of informational preference; and entropy-weighted projection operators implement a form of epistemic coarse-graining that—when iterated across scales—gives rise to geometric, gauge-like, and dynamical structures through the algebra of commutators. We situate the proposal within the existing philosophical and scientific literature on emergence, information, and the ontology of spacetime, emphasize its conceptual motivations, and delineate the implications for the status of laws and physical modality. While speculative in scope, the framework is intended as a structured conceptual contribution to foundational discussions about the possible informational origin of physical law. 1. Introduction: The Question of Foundations Physics in the 21st century increasingly confronts the possibility that many of the structures once regarded as fundamental—spacetime geometry, gauge interactions, conservation laws—may instead arise from deeper informational or algebraic principles. Developments in quantum gravity, condensed matter, and quantum information theory have radically expanded the conceptual landscape. Entanglement entropy appears to ! 2! encode spatial connectivity in holographic dualities [1–3]; topological quantum field theories describe phases of matter whose behavior depends purely on global invariants [4–6]; tensor-network representations reconstruct emergent geometries from entanglement patterns [7–9]; and thermodynamic interpretations of gravitational dynamics suggest that the Einstein equation might be an equation of state, not a fundamental law [10–12]. Across these domains, a recurring idea has gained traction: Physical law may emerge from the structure, constraints, and organization of information. This idea is not new. Wheeler’s “it from bit” [13], Jaynes’s reconstruction of statistical mechanics [14], and structuralist approaches to quantum theory [15,16] all presaged an informational perspective. But contemporary work has revived and deepened the program with concrete mathematical tools. In this article, we propose a coherent conceptual framework designed to connect three elements: 1. Information — encoded through an entropy functional measuring the informational richness of diagrammatic structures. 2. Topology — embodied in equivalence classes of diagrams representing possible relational or connective configurations. 3. Operator structure — implemented via entropy-weighted projections acting on a Hilbert space of topological possibilities. The resulting scheme offers a structured answer to several foundational questions: • What is the ontological status of spacetime—fundamental entity, emergent structure, or organizational construct? • How can informational constraints shape dynamical behavior? • What is the meaning of physical law if geometry and interaction emerge from coarse-graining? • What counts as a “possible physical configuration,” and how is modality constrained? The proposal does not claim to deliver a new predictive physical theory. Rather, it articulates a conceptual architecture by which geometry and interaction may arise from informational and topological principles. As such, it resonates with the philosophical debates on emergence, fundamentality, and structural realism [17–21], while drawing mathematical inspiration from operator algebra and topological field theory. Foundations of Physics, as a journal interested in conceptual and philosophical aspects of physics, provides the appropriate venue for such a synthesis. ! 3! 2. Conceptual Motivation: Why Information and Topology? 2.1 The epistemic and structural roles of information Information has a dual role in contemporary physics: • Epistemic: encoding what can be known about a system • Structural: constraining which physical states are possible or meaningful The latter point is central. Many derivations of physical laws—Einstein’s equation from local Clausius relations [10], the Unruh effect from modular flow [22], holographic entanglement from boundary entropy [1,2]—show that information-theoretic consistency imposes dynamical constraints. Thus, a foundational framework that places an entropy functional at its center is not attributing physical reality to information per se. Rather, it recognizes that: Informational constraints are often the most primitive statements we can make about the relation between physical states. Topological entropy, in our context, quantifies the multiplicity of microconfigurations compatible with a given structural pattern. It thereby functions as a measure of informational richness, not thermodynamic disorder. 2.2 The conceptual necessity of topology Topology provides a natural combinatorial language for: • relational structure • connectivity • equivalence under deformation • invariants insensitive to metric details This makes it a powerful carrier of pre-geometric information, as seen in: • topological quantum field theories [4–6]; • spin networks and spin foams [23–25]; • topological quantum computing models [6,26]. A central insight motivating our framework is: Topological equivalence classes represent structural possibilities that remain stable when geometric details vary. ! 4! Thus, encoding the space of “possible structural configurations” naturally leads to a diagrammatic or topological basis. 2.3 Why projection operators? Projection operators embody selection, coarse-graining, and constraint. When weighted by entropy, they define: • which structures are informationally privileged, • how coarse-graining reduces the space of possibilities, • how emergent effective descriptions arise from restricted subspaces. A projection is inherently epistemic—choosing which distinctions matter and which do not. But when projections are organized across scales, their algebra gives rise to structures that mimic dynamical evolution or geometric change. Thus: Information + topology + projection = structural foundation of emergent law 3. Diagrammatic Possibility Spaces To formalize structural possibilities, we introduce a Hilbert space of topological diagrams. The purpose is not physical modelling per se, but philosophical clarification: such a space represents the domain of possible structural configurations, much like a configuration space represents possible classical states. 3.1 The diagrammatic Hilbert space Let 𝒟 be the set of finites, labelled diagrams representing topological or combinatorial relationships. These may encode: • connectivity graphs, • knot/link structures, • simplicial complexes, • tensor-network cells, • relational structures of fundamental degrees of freedom. We consider the quotient 𝒟/∼ under equivalence moves such as Reidemeister/Pachner transformations [27,28]. ! 5! Definition 3.1. The diagrammatic Hilbert space ℋ! is the separable Hilbert space spanned by orthonormal basis vectors ∣𝐷⟩, one for each equivalence class 𝐷 ∈ 𝒟/∼. This structure captures: • modality — what structures can exist • identity under deformation — structural realism • non-metric relationality — combinatorial ontology of connectivity 3.2 Interpretive significance (1) Structural Possibility Space The space ℋ! represents a landscape of relational configurations. It is akin to configuration space, but for topological rather than positional degrees of freedom. (2) No Geometry Assumed Geometry does not enter at the fundamental level. Distance, dimension, and continuity emerge—if at all—from the algebra of operators. (3) Possibility, not actuality The framework does not claim that the universe “contains” such diagrams. Rather: The diagrammatic Hilbert space encodes the logical space of possible relational structures compatible with informational constraints. This distinguishes ontology from epistemology in a manner aligned with the work of Ladyman and Ross [17], Wallace [18], and Knox [19]. 4. Entropy as a Principle of Structural Selection With a space of structural possibilities defined, we next ask: How are these possibilities constrained into effective physical law? Our answer uses an entropy functional as a structural principle. ! 6! 4.1 Topological entropy We define an entropy functional 𝑆:𝒟/∼ ⟶ ℝ which assigns an informational weight to each topological configuration. This quantity captures: • multiplicity of microconfigurations, • combinatorial richness, • informational cost of describing a structure. Interpretive claim: Entropy here is not thermodynamic entropy, but a measure of complexity or informational richness, analogous to: • topological entanglement entropy [29,30], • complexity measures in quantum circuits [31], • statistical complexity in computational mechanics [32]. 4.2 Entropic projection and emergent structure An entropy-weighted projection operator is defined as: 𝑃" = 1 𝑍1𝑒#"(!) ! ∣𝐷⟩⟨𝐷 ∣, 𝑍 =1𝑒#"(!) ! This operator selects and weights structural configurations according to informational criteria. Interpretation: • Low-entropy configurations are favoured when 𝑆(𝐷) counts redundancy or complexity. • High-entropy configurations are favoured when 𝑆(𝐷) measures diversity. The operator thus represents a form of epistemic coarse-graining, expressing which structures are salient for constructing effective physical laws. ! 7! 4.3 The role of projection in foundational physics Projection plays many roles in physics: • Decoherence as projection onto pointer states [33,34] • Constraints in classical/quantum theories as projectors onto subspaces [15] • Selection of relevant degrees of freedom in renormalization [35] In this conceptual framework: Physical law arises from the structure of informationally meaningful projections on the space of possibilities Thus, laws are neither arbitrary nor imposed; they reflect the structure of epistemic and structural constraints. 5. Operator Structure and the Emergence of Geometry Thus far, we have argued conceptually that a diagrammatic Hilbert space encodes structural possibilities, while an entropy-weighted projection selects informationally meaningful configurations. We now consider how geometric structure can emerge from the algebra of operators defined on this space. The key idea is this: Geometry is not postulated but reconstructed from the failure of informational coarse-graining to commute with diagrammatic deformation This connects the operator-theoretic structure to the philosophical view that geometry is an emergent organizational scheme rather than an ontological primitive [19,36]. 5.1 Coarse-derivative operators as structural change We introduce diagram-deformation operators ∇& representing small changes in relational structure. These do not correspond to motion in spacetime (there is no spacetime yet), but rather to transformations in the space of possible relations. Formally:88∇&∣𝐷⟩ =∣ 𝛿&(𝐷)⟩−∣ 𝐷⟩ where 𝛿& modifies diagram 𝐷 via: ! 8! • adding/removing an edge, • changing connectivity, • local reconnection, • contraction/expansion moves. Interpretation: ∇& represents structural variation within modality—a transformation to a neighboring possibility. These echoes work on configuration spaces for topological states [23,24]. 5.2 Commutator structure as proto-geometry Given the entropy-weighted projector 𝑃", define a curvature-like operator: 𝑅&' = [∇&,𝑃"] ∇'−[∇',𝑃"] ∇& This measures how much informationally meaningful coarse-graining interferes with structural change. Interpretation: • If coarse-graining and deformation commute, there is no curvature: the structure is “flat.” • If they fail to commute, curvature emerges as a measure of informational obstruction to rearranging structure. Thus: Curvature quantifies the obstruction to reordering structural change and informational selection This resembles: • entanglement-induced curvature in holography [1–3], • relative-entropy curvature in algebraic QFT [22], • modular flow geometry [37]. ! 9! 5.3 Topological covariance of curvature Curvature is invariant under topological equivalence transformations, reflecting structural realism: 𝑈(𝑅&'𝑈( )=𝑅&' Thus, geometric emergence does not depend on arbitrary diagrammatic choices but on equivalence classes—mirroring how geometry in GR is invariant under diffeomorphisms. 5.4 Philosophical significance of emergent curvature This construction supports a structural, relational, non-fundamental ontology of geometry. Rather than being a background stage on which physics happens, geometry arises from: • how informational constraints restrict relational configurations, • how these constraints behave under local structural changes. This resonates with: • relational views of spacetime (e.g. Barbour, Rovelli), • structuralist metaphysics (Ladyman & Ross), • Knox’s “inertial structure” view of spacetime [19], • emergent gravity programs [10–12]. Geometry becomes: a codification of the algebraic structure of informational relevance 6. Gauge-Like Structure and Constraint Emergence Emergent geometry is only one ingredient of physical structure. Another is interaction, often encoded by gauge symmetries in standard physics. In this framework, gauge-like structures arise from topological redundancy within the diagrammatic Hilbert space and from the structure of the projection operator. ! 16! Conflict of Interest The author declares no conflict of interest. References [1] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006). [2] V. E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 2007, 062 (2007). [3] M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42, 2323 (2010). [4] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121, 351–399 (1989). [5] M. Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68, 175–186 (1989). [6] A. Yu. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2–30 (2003). [7] G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99, 220405 (2007). [8] G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101, 110501 (2008). [9] R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics 349, 117–158 (2014). [10] T. Jacobson, Thermodynamics of spacetime: The Einstein equation of state, Phys. Rev. Lett. 75, 1260–1263 (1995). [11] E. P. Verlinde, On the origin of gravity and the laws of Newton, JHEP 2011, 029 (2011). [12] T. Padmanabhan, Gravity and the thermodynamics of horizons, Phys. Rep. 406, 49– 125 (2005). ! 17! [13] J. A. Wheeler, Information, physics, quantum: The search for links, in Complexity, Entropy, and the Physics of Information, ed. W. H. Zurek, Addison-Wesley (1990). [14] E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106, 620 (1957). [15] C. A. Fuchs and R. Schack, Quantum-Bayesian coherence, Rev. Mod. Phys. 85, 1693 (2013). [16] L. Hardy, Quantum theory from five reasonable axioms, arXiv:quant-ph/0101012 (2001). [17] J. Ladyman and D. Ross, Every Thing Must Go: Metaphysics Naturalized, Oxford University Press (2007). [18] D. Wallace, The Emergent Multiverse: Quantum Theory according to the Everett Interpretation, Oxford University Press (2012). [19] E. Knox, Physical relativity from a functionalist perspective, Stud. Hist. Phil. Mod. Phys. 67, 118–138 (2019). [20] R. W. Batterman, The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction and Emergence, Oxford University Press (2002). [21] T. Maudlin, The Metaphysics within Physics, Oxford University Press (2007). [22] H. Casini, M. Huerta and R. C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 2011, 036 (2011). [23] C. Rovelli and L. Smolin, Spin networks and quantum gravity, Phys. Rev. D 52, 5743 (1995). [24] A. Perez, Spin foam models for quantum gravity, Class. Quantum Grav. 20, R43 (2003). [25] D. Oriti, The microscopic dynamics of quantum space as a group field theory, in Foundations of Space and Time, Cambridge University Press (2011). [26] M. Freedman, M. Larsen and Z. Wang, A modular functor which is universal for quantum computation, Commun. Math. Phys. 227, 605–622 (2002). [27] L. H. Kauffman, On Knots, Princeton University Press (1987). [28] J. H. Przytycki and K. Taniyama, Pachner moves and triangulations, Contemp. Math. 233, AMS (1999). ! 18! [29] A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006). [30] M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96, 110405 (2006). [31] M. A. Nielsen, A geometric approach to quantum circuit lower bounds, arXiv:quantph/0502070 (2005). [32] J. P. Crutchfield and K. Young, Inferring statistical complexity, Phys. Rev. Lett. 63, 105 (1989). [33] W. H. Zurek, Decoherence and the transition from quantum to classical, Phys. Today 44, 36–44 (1991). [34] W. H. Zurek, Pointer basis of quantum apparatus: Into what mixture does the wave packet collapse?, Phys. Rev. D 24, 1516 (1981). [35] K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 773 (1975). [36] H. Halvorson and R. Clifton, Reconsidering the foundations of quantum field theory, in Philosophy of Physics, North-Holland (2007). [37] D. Harlow, The Ryu–Takayanagi formula from quantum error correction, Commun. Math. Phys. 354, 865–912 (2017). [38] J. Earman, Rough Guide to Spacetime, University of Chicago Press (2019). [39] J. Weatherall, Regarding the hole argument, Brit. J. Phil. Sci. 69, 329–350 (2018). [40] G. Riggs, Gauge symmetry and redundancy, Philos. Sci. 82, 1248–1259 (2015). [41] R. Ellis, The statistical theory of coarsening, J. Stat. Phys. 8, 1–12 (1973). [42] L. P. Kadanoff, Scaling laws for Ising models near 𝑇+, Physics 2, 263 (1966). [43] J. Woodward, Making Things Happen, Oxford University Press (2003). [44] M. Lange, Laws and Lawmakers, Oxford University Press (2009).