MicroscopicDiagrammatic Entropy as the Origin of Emergent Gravity: Extending Verlinde's Framework via Combinatorial Microstate Counting

! 1! Microscopic Diagrammatic Entropy as the Origin of Emergent Gravity Extending Verlinde’s Framework via Combinatorial Microstate Counting Borros Arneth, Philipps University Marburg, Justus Liebig University Giessen, Germany, [email protected] Abstract Entropic and emergent approaches to gravity posit that gravitational dynamics arise from changes in microscopic information associated with matter and spacetime. While these theories successfully recover Newtonian and relativistic gravity, the underlying microstates whose entropy drives gravitational behavior remain unspecified. Here we introduce a microscopic framework based on a diagrammatic Hilbert space that assigns combinatorial and topological microstates to elementary matter. These microstates correspond to internal configurations of gluonic vertices, interaction motifs, and preonlevel structures inspired by QCD, Dyson–Schwinger dynamics, and compositeness models. The multiplicity of such configurations defines a microcanonical entropy whose spatial variation generates entropic gravitational forces, thereby providing a microscopic realization of Verlinde’s emergent gravity. We show how diagrammatic state counting yields particle masses, internal energies, and gravitational charges, and how these microphysical entropies couple naturally to holographic screens and thermodynamic derivations of Einstein’s equations. This unified approach embeds emergent gravity in the microstructure of matter, connecting nonperturbative QCD, holography, and entropic spacetime dynamics. 1. Introduction The thermodynamic origin of gravitational dynamics is a profound idea with roots in black-hole physics, quantum information, and non-equilibrium statistical mechanics. The discovery that black holes possess entropy proportional to horizon area [1,2] and radiate thermally [3] first established a link between geometry and information. Jacobson later demonstrated that Einstein’s equations can be derived as an equation of state by assuming the Clausius relation for local Rindler horizons [4]. These insights catalyzed the broader viewpoint that gravity may emerge from microscopic degrees of freedom via thermodynamic or entropic principles [5–7]. ! 2! Verlinde proposed an explicit entropic-force picture in which gravity arises from changes in information associated with matter displacements relative to holographic screens [8]. A later extension accounted for galactic rotation curves without dark matter by relating entropy displacement to de Sitter temperature [9]. Although conceptually compelling, Verlinde’s approach deliberately leaves open the nature of the microscopic degrees of freedom whose entropy gradients produce gravitational forces. Similar ambiguities appear in holography [10,11], tensor-network models of spacetime [12–15], and elastic/thermodynamic gravity frameworks [16–18]. A central open question therefore persists: What are the microstates whose entropy gives rise to gravitational dynamics? Here we propose a concrete answer by introducing a diagrammatic microstate framework that extends Verlinde’s emergent gravity by specifying the microscopic origin of entropy. The key idea is that elementary particles are not featureless points but possess rich internal structures describable as combinatorial configurations of diagrammatic interactions. These include gluon–gluon vertices, fermion–gluon couplings, color-flow patterns, and preonic substructure proposed in compositeness models [19–22]. Nonperturbative QCD studies, Dyson–Schwinger equations, and lattice simulations show that the majority of hadronic masses emerge from gluonic self-interaction and vacuum structure [23–28], hinting that internal interaction combinatorics encode a nontrivial microstate structure. We formalize these ideas by introducing a diagram Hilbert space whose basis vectors correspond to allowed internal micro-configurations of matter. A particle's internal entropy is defined from the multiplicity of such states. We show that spatial displacements modify the allowed diagrammatic microstates via holographic-screen constraints, reproducing Verlinde’s entropic-force law. Furthermore, diagrammatic entropy naturally scales with particle mass, providing a microscopic expression for gravitational charge and the equivalence principle. This establishes a direct link between nonperturbative QCD microstructure and macroscopic gravitational dynamics. Our framework thus supplies what Verlinde’s entropic gravity lacked: a physically grounded, microscopic entropy-counting mechanism. ! 3! 2. Diagrammatic Microstructure of Matter 2.1 Motivation from QCD and nonperturbative interactions QCD is dominated by complex internal interactions: triple-gluon and quartic-gluon vertices [23], color-rearrangement processes [24], dynamically generated masses [25], and confinement phenomena [26–28]. The internal vacuum structure is highly nontrivial, with strong evidence for large combinatorial degeneracy of gluonic configurations [27]. These results motivate a picture in which diagrammatic interaction patterns represent distinct microscopic states. For instance: • A quark–gluon vertex contributes a primitive diagrammatic unit. • A 3-gluon vertex adds combinatorial branching. • A 4-gluon vertex introduces topologically distinct motifs. • Color-flow diagrams encode permutation symmetries. • Preonic interactions add further combinatorial depth [19–22]. 2.2 Microcanonical entropy of internal diagrams We model particle internal structure using a microcanonical ensemble. A diagrammatic microconfiguration is specified by discrete counts 𝑛! of interaction types 𝑖, each assigned a multiplicity weight 𝜔!. The number of microstates is Ω=&𝜔! "! ! . The internal entropy is 𝑆 =𝑘#ln,Ω (1) This entropy is intrinsic to the particle and encodes nonperturbative interaction complexity. 2.3 Diagram Hilbert space Let ℋ$%&' be the span of all admissible microconfigurations. An orthonormal basis is denoted {∣Γ⟩}, where Γidentifies interaction graphs. Operators include: • Interaction-counting operators 𝑁 4! ! 4! • Topological-degree operators 𝜒6 • Projectors onto physical sectors 𝑃 8()*+ • Entropy operator 𝑆 9=𝑘,ln,Ω 4 This construction parallels spin networks, tensor-network geometries, and group-field theories [12–15,29–31], but grounded explicitly in QCD microstructure. 3. Emergent Gravity from Diagrammatic Entropy 3.1 Verlinde’s entropic-force law revisited Verlinde showed that a force can arise from maximizing entropy under energy constraints, yielding 𝐹 =𝑇∂𝑆 ∂𝑥 (2) for a displacement 𝑥 relative to a holographic screen. The temperature 𝑇 is associated with Unruh-like or de Sitter horizons. However, the entropy source 𝑆 is unspecified. We propose that the microscopic entropy (1) supplies this ingredient. 3.2 Spatial variation of diagrammatic microstates A matter particle near a holographic screen imposes boundary conditions on the allowed diagrammatic configurations due to entanglement and information-transfer constraints [10,12]. As the particle is displaced, the accessible microstate multiplicity varies: ∂𝑆 ∂𝑥 =𝑘,∂ ∂𝑥ln,Ω (3) Because Ω depends on holographic surface area and entanglement geometry, this yields a nonzero entropic force. 3.3 Newtonian gravity from diagrammatic entropy By identifying the temperature of a holographic screen with the Unruh relation 𝑘,𝑇 = ℏ𝑎 2𝜋𝑐 (4) ! 5! and substituting (3), Newton’s law emerges: 𝐹 =𝑚 𝑎(5) where the inertial mass 𝑚 ∝ ∂𝑆 ∂𝑥 (6) is directly proportional to the diagrammatic entropy gradient. This establishes a microscopic origin of gravitational charge. 3.4 Relativistic and MOND-like regimes Using extensions of Verlinde’s de Sitter analysis [9], the diagrammatic entropy also reproduces: • relativistic gravitational equations via Jacobson’s thermal derivation [4]; • MOND-like acceleration at galactic scales due to entropy deficits [9]. Thus, the framework incorporates Newtonian, relativistic, and modified-gravity regimes. 4. Diagrammatic Entropy and Mass Generation 4.1 Mass as internal entropy Strong-interaction mass generation is known to arise from gluon interactions and dynamical chiral symmetry breaking [23–27]. We propose: 𝑚 =𝛼 Ω 𝐸-(7) where 𝐸is a universal internal energy scale and 𝛼 depends on vertex topology. This mirrors the fact that quark bare masses contribute little to hadron masses [24]. 4.2 Equivalence of inertial and gravitational mass Because inertial mass (7) depends on Ω and gravitational mass (6) depends on ∂.ln,Ω, they coincide whenever changes in Ω scale proportionally with Ω itself, consistent with multiplicative diagrammatic structure. This offers a microscopic explanation of the equivalence principle. ! 6! 5. Relation to Holography and Quantum Information Diagrammatic microstates naturally integrate with: • entanglement-based spacetime emergence [12–15], • the Ryu–Takayanagi formula for holographic entropy [10], • ER=EPR duality linking geometry and entanglement [32], • tensor-network models of emergent geometry [13–15]. The diagrammatic state degeneracy determines entanglement entropy between matter and holographic screens, suggesting that spacetime geometry arises from underlying interaction-information structure. 6. Discussion We have supplied Verlinde’s emergent gravity with a concrete microscopic entropy source: diagrammatic microstates of matter. This framework unifies several previously unrelated structures: • nonperturbative QCD diagrams, • preonic combinatorics, • holographic screens, • entropic forces, • and gravitational dynamics. It embeds gravitational behaviour within particle-internal microstructure, creating a bridge between Standard Model physics and emergent spacetime. 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