Can Five Pregeometric Invariants Generate Gravity's Dynamical and Topological Sectors?

From Einstein-Cartan emergence to a complete topological classification

The pregeometry program holds that spacetime metric and gravitational dynamics are not fundamental but emerge from a more primitive gauge theory. Earlier work established that the Einstein-Cartan theory—general relativity extended with torsion—can arise from spontaneous symmetry breaking of an SO(1,4) or SO(3,2) gauge symmetry in a four-dimensional spacetime without a fundamental metric [3]. That precursor identified the effective metric and spin connection after symmetry breaking and showed that diffeomorphism invariance and the equivalence principle emerge in the broken phase [3]. Independently, Wetterich formulated Euclidean pregeometry as an SO(4) Yang-Mills theory with a vector field, demonstrating that a well-defined Euclidean functional integral exists with propagators free of tachyonic or ghost modes at short distances, and that general relativity emerges at long distances [2]. These works established the foundational claim that gravity can emerge from gauge-theoretic pregeometry, but neither systematically classified all possible pregeometric invariants nor addressed the topological sector.

The new paper advances this lineage by identifying the complete set of five irreducible pregeometric building blocks—L_MM, L_W, L_J, L_theta, and L_vartheta—that constitute the minimal generating set for any 4D pregeometric action consistent with general covariance in the unbroken phase [1]. The authors show that any functional of these invariants reduces upon symmetry breaking to combinations of effective couplings generated by the five building blocks [1]. This classification is the essential new contribution: it establishes that the pregeometric framework is not open-ended but has a finite atomic basis. The central result is that spontaneous symmetry breaking of the complete pregeometric Lagrangian yields an emergent gravitational theory containing the Einstein-Hilbert action, the cosmological constant term, and all four 4D topological invariants: Gauss-Bonnet, Pontryagin, Holst, and Nieh-Yan terms [1]. This unification of gravity's dynamical and topological sectors from a common pregeometric source is described by the authors as unprecedented in the literature [1].

Coupling relations, seesaw mechanism, and the Barbero-Immirzi parameter

The paper derives explicit relations between the seven pregeometric constants (k_MM, k_W, k_J, k_theta, k_vartheta, v, and m) and six emergent gravitational couplings (M_P, Lambda, theta_GB, theta_P, gamma_0, and theta_T) [1]. Several structural results emerge: the reduced Planck mass receives contributions only from L_MM and L_W, implying that at least one of these is indispensable for a realistic emergent theory [1]. The cosmological constant receives contributions from L_MM, L_W, and L_J, with L_J contributing exclusively to Lambda [1]. The Gauss-Bonnet coupling theta_GB receives contributions only from L_MM, while the Pontryagin coupling theta_P receives contributions only from L_theta [1]. The Barbero-Immirzi parameter gamma_0 depends on the difference between k_vartheta v^2 and 4k_theta, linking L_theta and L_vartheta, though it is also related to L_MM and L_W through the Planck mass definition [1].

In a special case where only two pregeometric coupling constants are independent (k_PG and k_Theta), the authors uncover a seesaw mechanism linking the Planck mass and the cosmological constant: M_P^2 / |Lambda| = 8 k_PG v [1]. In this same special case, the Gauss-Bonnet coupling is set by the ratio of Planck mass to cosmological constant, theta_GB = -M_P^2 / (2 Lambda) [1]. The Pontryagin and Nieh-Yan couplings are proportional, with theta_NY / theta_P = (4/3) Lambda [1]. The Barbero-Immirzi parameter can be expressed as gamma = 3 M_P^2 / (4 k_Theta |Lambda|) = 3 theta_GB / theta_P, effectively canceling its dependence on the symmetry-breaking scale v [1]. These relations are theoretical derivations within the pregeometric framework; they have not been tested against observational data [1].

How gauge-theoretic pregeometry differs from entropic and thermodynamic emergence

The pregeometry approach is not the only framework claiming that gravity is emergent. Verlinde's entropic gravity proposes that gravity arises as a thermodynamic force driven by entropy gradients rather than from gauge symmetry breaking [10]. This competing framework has been developed phenomenologically: a nonrelativistic master equation modeling entropic gravity as an open quantum system interaction has been shown to maintain quantum coherence in the strong coupling limit and to reproduce conventional free-fall dynamics [5]. This model was tested against the qBounce experiment with ultracold neutrons, showing agreement when the dimensionless coupling constant sigma exceeds approximately 250 [5]. The framework was later extended to Dirac fermions, demonstrating that spin is unaffected by entropic gravity and that apparent antigravity from free-falling antiparticles originates from zitterbewegung rather than a violation of the equivalence principle [6].

The key methodological difference is that entropic gravity derives gravitational dynamics from thermodynamic principles and has been connected to specific experimental tests [5][6], whereas the pregeometry paper derives gravitational dynamics from gauge symmetry breaking and remains at the level of theoretical construction [1]. The pregeometry approach also differs from Wetterich's Euclidean pregeometry, which formulates the theory as an SO(4) Yang-Mills theory with a vector field and emphasizes the well-defined functional integral and stability of propagators [2]. The new paper builds on the Minkowski-signature SO(1,4)/SO(2,3) framework developed in the precursor work [3] and extends it to include topological invariants. These are complementary rather than directly competing approaches: entropic gravity addresses the thermodynamic origin of gravitational force, while gauge-theoretic pregeometry addresses the gauge-theoretic origin of gravitational dynamics and topology.

Topological sector, Gauss-Bonnet physics, and the limits of current evidence

The topological terms generated by the pregeometric construction have been studied independently in other contexts. The Gauss-Bonnet invariant has been analyzed in teleparallel formulations, where its expression in terms of torsion, non-metricity, and Levi-Civita covariant derivative has been derived, and the number of invariant terms compared with effective field theory expectations [7]. Dynamical complexity in teleparallel Gauss-Bonnet gravity has been explored through phase-space analysis, identifying stable critical points and cosmological viability for models describing different phases of cosmic evolution [8]. Gauss-Bonnet effects on vacuum decay have been investigated, showing that in four dimensions the Gauss-Bonnet term impacts topology-changing transitions and can either suppress or enhance tunneling depending on the sign of the coupling [9]. These studies provide independent validation that the topological terms appearing in the pregeometric construction have well-defined physical effects in other gravitational frameworks.

However, the pregeometry paper's conclusions are strictly theoretical. The authors state that the work establishes pregeometric foundations from which all aspects of gravitation can dynamically emerge, providing a unified starting point for quantum gravity, dark energy phenomenology, and topological phases [1]. No experimental or observational evidence directly verifying the pregeometric mechanism is presented [1]. The paper notes that a complete understanding of the quantum dynamics, including the fate of the Barbero-Immirzi parameter under renormalization, requires a programmatic research approach beyond the current work [1]. Additionally, the special case yielding the seesaw mechanism assumes a specific relation among pregeometric couplings (k_PG = ±k_MM = k_W v^2 = ±k_J v^4 and k_Theta = ±k_theta = k_vartheta v^2), which may not hold in the general theory [1]. The observational boundary is further illustrated by constraints on related emergent gravity ideas: muonic hydrogen and deuterium spectroscopy have placed stringent limits on large extra dimension models, with calculated Planck scale values several orders of magnitude below the experimental lower limit of 2 TeV [4]. While this constraint applies to extra-dimension gravity rather than pregeometry directly, it illustrates the type of experimental scrutiny that emergent gravity frameworks may eventually face.

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This research page is built on 10 peer-reviewed studies — published from 2016 to 2026, 3 from 2024 or later, 4 in Q1 journals, collectively cited 1,136 times — selected as the most relevant from 13 studies that passed quality screening, drawn from 190 papers retrieved from a database of over 500 million.

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