Entropy-Originated Topological Framework for a Unified Theory of Matter and Gravity – Bidirectional Projection Theory

1 Entropy-Originated Topological Framework for a Unified Theory of Matter and Gravity – Bidirectional Projection Theory Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, [email protected] Abstract We propose a unified conceptual framework in which space, time, and matter emerge as successive projections of a single underlying topological state space (the Diagram Hilbert Space, ℋ!). In this view: (i) spacetime arises through projection of coherent topological relations; (ii) matter and mass arise via further projection of stable topological energy modes; and (iii) gravitation appears as the dynamical response of emergent spacetime to topological energy gradients. We formalize this by introducing projectors Π"#,Π#,Π$ on ℋ!, describe their relations, and show how Einstein-like dynamics follow as a projective identity. Space, time, and matter thereby become distinct manifestations of one non-geometric substrate, suggesting a unified, entropic–topological origin of physical reality. 1 Introduction In conventional physics, spacetime provides the stage on which matter acts and interacts through fields and forces. We invert this hierarchy. We posit a pre-geometric Diagram Hilbert Space ℋ!, whose basis states are abstract, topological diagrams encoding relational connectivity without reference to geometry or temporal order. Space, time, and matter emerge only through projection—observables that select coherent, coarse-grained structures from ℋ!. This conception is inspired by converging developments: spacetime from entanglement and tensor networks [1–3]; emergent gravity and entropic approaches [4–8]; and quantumrelational models of geometry and matter [9–13]. We extend these into a unified projective ontology, in which spacetime, matter, and gravitation appear as different shadows of one topological reality. 2 The Diagram Hilbert Space ℋ! We define ℋ! as the complete Hilbert space whose orthonormal basis vectors correspond to diagrammatic/topological configurations—for instance, graphs, simplicial complexes, or knot diagrams. No metric or causal structure is presupposed. 2 The fundamental operator algebra acts on relations among diagram elements, not on spatial positions. A general state ∣Ψ!⟩=(𝑐% ∣Γ⟩ % represents a superposition of connectivity patterns. Operators on ℋ! measure topological invariants such as linking or Euler characteristics. An underlying generator 𝐻 ,! drives evolution in an internal parameter 𝜏, leading to an algebraic dynamic independent of geometric time. The usual notions of space, locality, and chronology arise only after projection. 3 Emergence of Spacetime: The Projection Π"# The first major projection, Π"#:ℋ!⟶ℋ"# selects those diagram states exhibiting spacetime coherence—a degree of relational regularity that allows manifold interpretation. Following entanglement-based constructions [1, 3, 14], a coherence functional 𝐶[Γ] quantifies how consistently a diagram encodes adjacency relations. Maximizing 𝐶[Γ] projects the subspace where a smooth pseudo-Riemannian geometry (𝑀,𝑔&') can be induced. Effective coordinates emerge as expectation values of diagram adjacency operators, 𝑋 8&=Π"#Γ 8&Π"#,𝑔&' = ⟨Ψ!∣𝑋 8&𝑋 8'∣ Ψ!⟩−⟨Ψ!∣ 𝑋 8&∣Ψ!⟩⟨Ψ!∣𝑋 8'∣Ψ!⟩ thereby defining emergent space and time collectively as coarse-grained relational order. 4 Emergence of Matter and Mass: The Projection Π$ Within ℋ!, certain diagrammatic modes are topologically stable under 𝐻 ,!. Projecting onto this subspace yields matter states: Π$:ℋ!⟶ℋ$ Each stable mode ∣Γ$⟩ satisfies 3 𝛿(ℰ[Γ]−𝜆𝒯[Γ])=0 where ℰ[Γ]=⟨Γ∣𝐻 ,!∣Γ⟩ is its diagram energy and 𝒯[Γ] a topological invariant. The mass of the emergent excitation follows as 𝑚𝑐(=⟨Ψ!∣ Π$ )𝐻 ,!Π$∣Ψ!⟩ Thus, mass measures the topological energy content of a projection, not a fundamental property. Matter is therefore the localized, energetically stable projection of ℋ!’s topological dynamics—akin to solitons or knotted defects within an emergent manifold [15–17]. 5 Emergence of Temporal Ordering: Diagram-Time and Relational Time 5.1 Diagram-time 𝜏 Though ℋ! lacks a geometric timeline, it supports an internal parameter 𝜏 ordering algebraic transformations: 𝑖ℏ 𝑑 𝑑𝜏 ∣Ψ!(𝜏)⟩=𝐻 ,!∣Ψ!(𝜏)⟩ This “diagram-time” is unobservable; it indexes reconfigurations of relational structure. 5.2 Relational time 𝑡 Once Π"# yields a coherent subspace, one emergent coordinate functions as time: 𝑡 =𝑓[𝜏,Ψ!(𝜏)] Hence physical time is a reparameterization of relational change within ℋ!, consistent with the Page–Wootters mechanism [18, 19] and the Connes–Rovelli thermal-time hypothesis [20]. 5.3 Dynamics and feedback Evolution in 𝜏 induces dynamical processes in 𝑡; topological energy gradients in ℋ! manifest as spacetime curvature responses. 4 5.4 Ontology Level Quantity Nature Observable Diagram Hilbert Space ℋ! 𝜏 (“diagram-time”) intrinsic algebraic order no Projected spacetime ℋ"# 𝑡 (“physical time”) emergent relational coordinate yes Relation 𝑡 =𝑓[𝜏,Ψ!(𝜏)] projection-dependent — The arrow of time encodes topological–entropic asymmetry in projection, not an intrinsic direction of ℋ! [4, 7, 21]. 6 Gravitation as Response to Topological Energy Gradients With matter and time established, gravitation emerges as the response of spacetime projection to topological energy gradients: ∇*+,𝐸 = 𝛿ℰ[Γ] 𝛿Γ ,ℛ-. ∝ ∇*+,𝐸-. Protectively, 𝑅&' −1 2𝑔&'𝑅 =𝜅 𝑇&' ⟺ ℛ-. = 𝜅 ∇*+,𝐸-. 7 Higgs Mechanism as Projective Symmetry Alignment The Higgs field functions as a mediator between ℋ! and ℋ"#. It aligns diagrammatic topological modes with spacetime vacuum expectation values, generating mass while preserving gauge symmetry at the fundamental level: 𝑚-𝑐(=⟨Ψ!∣Π$ )𝐻 ,!Π$∣Ψ!⟩=𝑔𝑣 Here 𝑣 is the vacuum expectation value of the projected Higgs operator Φ"# =Π"#Φ ,!Π"#. Mass arises as a topological energy misalignment, not an intrinsic property of particles. 5 8 Bidirectional Higgs Coupling and Projective Information Conservation 8.1 Forward projection: mass emergence Topological energy in ℋ! projects via Π"# to spacetime mass. Φ"# =Π"#Φ ,!Π"#,𝑚-𝑐(=𝑔-𝑣 8.2 Backward projection: experimental feedback Experimental manipulation of Higgs-related observables updates ℋ!: 𝑑 𝑑𝜏 ∣Ψ!(𝜏)⟩=𝐻 ,![ Φ"#(𝑡) ] ∣Ψ!(𝜏)⟩ 8.3 Conservation of projective information 𝑑𝑆! 𝑑𝜏 +𝑑𝑆"# 𝑑𝑡 =0 with the Higgs operator as a Lagrange multiplier maintaining global consistency. 8.4 Hermitian correspondence 𝒞/=Π"#Φ ,!+Φ ,! )Π"#,[𝒞/,𝐻 ,!] = [𝒞/,𝐻 ,"#] = 0 8.5 Physical implications The Higgs mediates reciprocal emergence of matter and geometry, ensuring both ℋ! and ℋ"# evolve consistently and entropically optimally. 6 References 1. B. Swingle, Annu. Rev. Condens. Matter Phys. 9, 345 (2018). 2. M. Van Raamsdonk, arXiv:1005.3035 (2010). 3. J. Maldacena & L. Susskind, Fortschr. Phys. 61, 781 (2013). 4. T. Padmanabhan, arXiv:1208.1375 (2012). 5. E. Verlinde, SciPost Phys. 2, 016 (2017). 6. S.-S. Lee, Phys. Rev. D 108, 024054 (2023). 7. L. Smolin, Time Reborn, Houghton Mifflin (2013). 8. C. Rovelli, Quantum Gravity, Cambridge Univ. Press (2004). 9. B. L. 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