Emergent Gravity from Topological Quantum Field Theory: Stochastic Gradient Flow Perspective away from the Quantum Gravity Problem

Andrea Addazi; Antonino Marciano; Giuseppe Meluccio; Jinglong Liu; Salvatore Capozziello; Xuan-Lin Su

REVIEW 3 major objections 6 minor 39 references

Emergent Gravity from Topological Quantum Field Theory: Stochastic Gradient Flow Perspective away from the Quantum Gravity Problem

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A stochastic gradient flow on a pre-geometric gauge theory realizes General Relativity as an infrared fixed point and topological BF theory as an ultraviolet fixed point, making quantization of the Einstein-Hilbert action redundant.

desk verdict A clearly written scenario that ties Wilczek pre-geometry to stochastic Ricci flow and BF theory, but the UV fixed point rests on a flawed constraint action and the conclusions outrun the derivation. read the letter →

arxiv 2505.17014 v2 pith:KGAX2IZ2 submitted 2025-05-22 gr-qc hep-th

Andrea Addazi , Salvatore Capozziello , Jinglong Liu , Antonino Marciano , Giuseppe Meluccio , Xuan-Lin Su This is my paper · ORCID

classification gr-qchep-th MSC 83C4581T4553C44 PACS 04.60.-m04.20.-q11.15.-q

keywords emergentgravitystochasticgradientflowRiccitopologicalBFtheorypre-geometricspontaneoussymmetrybreakingquantum

topics Quantum Gravity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the problem of quantizing General Relativity disappears if gravity is emergent. The authors take a pre-geometric gauge theory, the one proposed in Ref. [13], and subject it to a stochastic gradient flow. The flow has an infrared fixed point at which stochastic fluctuations vanish and diffeomorphism symmetry is restored: the theory becomes classical Einstein-Hilbert gravity. The flow also has an ultraviolet fixed point at which the theory becomes a topological BF theory. Between the two fixed points, in the pre-geometric phase, the theory can be quantized perturbatively and generates corrections to the Einstein-Hilbert action.

What carries the argument

The machine is the stochastic pre-geometric flow equation (Eq. 3), written in terms of the composite pre-geometric fields $P_{\mu\nu} = \eta_{AB}\nabla_\mu \phi^A \nabla_\nu \phi^B$ and $w^A_\mu = \pm \epsilon^{ABCDE}\epsilon_{\mu\nu\rho\sigma}\nabla^\nu \phi_B \nabla^\rho \phi_C \nabla^\sigma \phi_D \phi_E$. By a correspondence principle borrowed from the spontaneous-symmetry-breaking analysis, this equation reduces under SSB to the stochastic Ricci flow (Eq. 4), $-\mathrm{i}\partial_{\bar n^0} g_{\mu\nu} = -2R_{\mu\nu} + 2\Lambda g_{\mu\nu} + \xi_{\mu\nu}$. In the BF formulation, the pre-geometric Plebanski action (Eq. 10), $S_{P\text{-}W} = \int \left(B^{AB}\wedge F_{AB} + \epsilon_{ABCDE} B^{AB}\wedge B^{CD}\phi^E\right)$, carries the simplicity constraints, and the key dynamical assumption is the decoupling (Eq. 15) of the scalar multiplet flow $\partial_s \phi^A - \xi_s \phi^A \approx 0$ near fixed points, which yields the infrared vacuum $\phi^A \sim (v/2)\delta^A_5$ and the ultraviolet vanishing $\phi^A \to 0$ that turns the theory topological.

What would settle it

A direct check would be to solve the full stochastic flow of the pre-geometric action with the scalar multiplet kept coupled to the frame field, and to test whether a nonzero vacuum $\phi^A \sim (v/2)\delta^A_5$ is a genuine attracting fixed point with vanishing noise while $\phi^A \to 0$ is reached in the ultraviolet. Failing to find both fixed points, or finding that the simplicity constraints cannot be imposed along the flow, would rule out the scenario.

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Extended reading notes

Core claim

The central claim is that the stochastic gradient flow of the pre-geometric theory possesses two fixed points, and that these fixed points realize both phases of gravity. In the infrared limit, spontaneous symmetry breaking of the parental $SO(1,4)$ or $SO(3,2)$ symmetry sends the Higgs-like multiplet to a vacuum value, the stochastic noise vanishes, and the flow equation reduces to the stochastic Ricci flow with a cosmological constant; the resulting classical theory is General Relativity in the time gauge. In the ultraviolet limit, the scalar multiplet vanishes and the simplicity constraints can no longer be imposed, so the pre-geometric Plebanski action degenerates into a topological BF action with no local degrees of freedom. Quantization therefore applies away from the infrared fixed point, perturbatively in the pre-geometric phase and non-perturbatively at the topological ultraviolet fixed point, while the Einstein-Hilbert action itself need not be quantized.

Load-bearing premise

The existence and exact form of the stochastic pre-geometric flow equation, and the assumed decoupling of the scalar multiplet from the other fields near the fixed points, are not derived from the underlying action; if either fails, the two-fixed-point picture collapses along with the claim that quantizing the Einstein-Hilbert action is redundant.

Editorial extensions

If this is right

If the fixed-point picture holds, quantizing the Einstein-Hilbert action is unnecessary: classical General Relativity is an equilibrium state, and quantum corrections come from stochastic fluctuations in the pre-geometric phase.
The ultraviolet fixed point supplies a topological BF theory that admits standard non-perturbative quantization methods, giving a well-defined ultraviolet completion of the emergent gravity scenario.
The scalar mode of the pre-geometric theory, the extra degree of freedom beyond the graviton, is geometrized as the gradient-flow direction and controls both phase transitions.
The stochastic flow automatically selects the time gauge $\bar n^0 = \bar n^{-1}$ at equilibrium, matching the gauge fixing used in loop-quantization phase-space constructions.
The two phase transitions, from topological BF to the pre-geometric phase and from the pre-geometric phase to General Relativity, are tied to the same scalar multiplet, so the mass and dynamics of that scalar are potentially observable near the Planck scale.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the central claim is right, the observed smallness of the cosmological constant fixes the thermal-time scale of the pre-geometric epoch, since the flow time is set by $1/\sqrt{\Lambda}$; the paper states this only implicitly.
A natural extension is that the scalar degree of freedom liberated by symmetry breaking, expected near the Planck scale, would appear as a massive scalar in the effective gravitational theory and could influence early-universe cosmology.
The scenario implies no graviton quanta in the deep infrared: gravitational effects there are coherent classical geometry, so low-energy searches for quantum-gravity signatures should find none.
If the stochastic noise originates in deterministic chaos within the pre-geometric gauge theory, the flow equation could in principle be derived rather than assumed, which would close the paper's weakest assumption.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague—here is my read of 2505.17014. The new thing is the interpolation: a stochastic gradient flow that runs between two known fixed points, Wilczek's pre-geometric action in the IR and a BF theory in the UV, with a transient pre-geometric phase. That synthesis is not in the cited literature, and the paper is clearly written and unusually transparent about what is being assumed. It is also honest that the central flow equation (3) is an ansatz, not a derivation. The canonical analysis summary is useful, giving three degrees of freedom (graviton plus a scalar) and a clean match to ADM in the time gauge.

The soft spots are real. The IR reduction from the pre-geometric flow to the stochastic Ricci flow (Eq. 3 to Eq. 4) is asserted via a 'correspondence principle' rather than computed. The UV fixed point rests on Eq. (15), which the paper says follows from imposing the simplicity constraints. That would be fine if the constraints were the full B^{(AB}∧B^{CD)}=0, since a decomposable B gives a vanishing ε-contracted product; but the action (10) couples φ^E to the ε-tensor, so extremizing in φ gives only the five conditions ε_ABCDE B^AB∧B^CD=0, which are much weaker than the full simplicity constraint. So the statement that the extremization imposes B^{(AB}∧B^{CD)}=0 is wrong as written. Note: the stress-test's claim that the Holst solution makes the ε-contraction nonzero is itself incorrect—ε contracted with four powers of the same vector vanishes by antisymmetry—but that does not rescue the action. This is a load-bearing flaw in the UV argument, though likely repairable by introducing a proper Lagrange multiplier with the symmetries of the Riemann tensor, as in 4D Plebanski.

Beyond that, the paper overreaches in the abstract: it concludes that quantizing GR is 'redundant' based on an ansatz plus the assumption that the UV fixed point is reached. That is a scenario, not a proof. As a proposal it is worth engaging; as a derivation it fails.

Who is this for? People working on pre-geometric gravity, stochastic Ricci flow, and topological BF reformulations. I would not cite it in my own work right now, but I would send it to a serious referee, expecting major revisions.

Referee Report

3 major / 6 minor

Summary. The manuscript proposes that a stochastic gradient flow on the configuration space of Wilczek's pre-geometric SO(1,4)/SO(3,2) gauge theory interpolates between two fixed points: an infrared fixed point at which spontaneous symmetry breaking yields classical General Relativity, and an ultraviolet fixed point described by a topological BF theory. The authors write a stochastic pre-geometric flow equation (3), assert that under SSB it reduces to the stochastic Ricci flow (4), recast the Wilczek action as a Plebanski-type constrained BF theory, write stochastic flow equations (12)–(14), and use a decoupling condition (15) to derive an exponential decay of the Higgs multiplet (16). They conclude that the Einstein-Hilbert action need not be quantized because GR is an emergent infrared fixed point and the UV phase is topological.

Significance. If the proposed flow and its fixed-point structure could be derived, the picture would be significant for quantum gravity: it would connect stochastic Ricci-flow quantization, pre-geometric gauge theories, and constrained BF theories, and would recast GR as an emergent low-energy phase. The manuscript is transparent about the speculative nature of several steps, which is a strength: Eq. (3) is explicitly labeled a 'natural ansatz,' and Eq. (15) is introduced as an imposition. However, the paper does not provide a derivation or a concrete test of these assumptions, and at least one auxiliary computation (Appendix B) is inconsistent. Because the central claims—emergent GR and the UV BF phase—rest on these unproven elements, the significance of the scenario is not yet established.

major comments (3)

[Sec. III, Eq. (3)] The central pre-geometric flow equation (3) is introduced as 'A natural ansatz,' and its reduction to the stochastic Ricci flow (4) under SSB is asserted through a 'correspondence principle' borrowed from [14]. The authors do not substitute the SSB configuration φ^A → v δ^A_5 into Eq. (3) and evaluate both sides; they also do not derive Eq. (3) from the Wilczek action or from stochastic quantization of S_P-W. Since this reduction is the only link between the pre-geometric dynamics and the claimed infrared Einstein-Hilbert fixed point, the emergent-GR conclusion is not established by the manuscript.
[Sec. IV, Eq. (15)] The ultraviolet fixed point and the transition to topological BF theory rely on Eq. (15), ∂φ^A/∂s − ξ_s φ^A ≈ 0, which the text states is obtained by imposing the simplicity constraints near the fixed points. However, the simplicity constraints (11) are B^{(AB} ∧ B^{CD)} = 0, an algebraic condition on the two-form B; they do not imply the vanishing of the ε-contracted combination C^A = ε^A_{BCDE} B^{BC} ∧ B^{DE} that appears in the φ^A flow (14). For the Holst-type solution B^AB_{Holst} ∝ ±∇φ^A ∧ ∇φ^B quoted in Sec. IV, one generically finds C^A ∝ ε^A_{BCDE} X^B X^C X^D X^E with X^I = ∇φ^I, which is nonzero for a generic five-component X^I. Thus Eq. (15) is an independent dynamical assumption rather than a consequence of the constrained BF dynamics, and the subsequent UV fixed point and BF phase do not follow from the stated premises.
[Appendix B, Eqs. (B3)–(B6)] The derivation of the exponential decay (16) contains a stochastic-calculus error. From Eq. (B3), dφ^A = (1/2) φ^A ds + φ^A dW_s (using dW_s ≡ ξ_s ds). Applying Itô's lemma to f = ln φ^A gives d ln φ^A = (1/2 − 1/2) ds + dW_s = dW_s, so the solution is φ^A = const × e^{W_s}, with no exponential decay factor. Equation (B4) writes d ln φ^A = −(1/2)(φ^A)^2(φ^A)^{-2} ds + dW_s, which retains a drift term that cancels identically in Itô calculus. Since the vanishing of φ^A for s → ∞ is used to reach the UV topological fixed point, this inconsistency removes the quantitative support for that fixed point.

minor comments (6)

[Author affiliations] The affiliation line contains a typo, 'It aly'; please proofread the manuscript.
[Sec. III] The notation ar{n}^0 ≡ n^0/ar{n} and its relation to the SSB of ar{w}^0_0 is introduced without a definition of ar n in the main text; all symbols should be defined before use.
[Eq. (17)] The infrared fixed point limit is written as φ^A_0(s) ∼ v/2 δ^A_5 [1 + θ(s − s_0)], which involves a step function and is not a well-defined smooth fixed-point limit; the meaning of this expression should be clarified.
[References] Reference [21] is cited as an arXiv preprint; if a published version exists, it should be cited instead of or in addition to the preprint.
[Abstract and Conclusions] The statements that quantizing GR 'becomes redundant' and 'becomes meaningless' are stronger than what the analysis supports; they should be explicitly qualified as conditional on the assumed stochastic flow and decoupling conditions.
[Eqs. (12)–(14)] The flow equations are said to be 'easily recovered' from the Plebanski action (10), but the explicit functional derivatives are not shown; including them would improve reproducibility.

Circularity Check

3 steps flagged · score 8.0 of 10

The two fixed-point claims (IR GR and UV BF) are each imposed by construction: Eq. (3) is an ansatz engineered to reduce to the stochastic Ricci flow, and Eq. (15) imposes the desired UV decoupling rather than following from the simplicity constraints.

self definitional [Sec. III, Eqs. (2)-(4)]
"A natural ansatz is that the stochastic pre-geometry flow equation ... must contain the stochastic noise and analogous combinations of fields than in Eq. (2). ... according to the correspondence principle [14], the stochastic pre-geometric flow can be written, using the variables specified by the action proposed by Wilczek, as [Eq. (3)]. The SSB of this equation ... reproduces the stochastic geometry (Ricci) flow [Eq. (4)]."

Equation (3) is not derived from the Wilczek action; it is an ansatz designed so that its SSB limit coincides with Eq. (4), the stochastic Ricci flow of [21]. The reduction to GR is therefore written into the flow equation by construction. The paper's infrared 'prediction' of General Relativity is just the already-known SSB limit of the Wilczek action [13,14], restated as a fixed point of a flow chosen to reproduce that limit.
self definitional [Sec. IV, Eqs. (14)-(16)]
"imposing (in proximity of the fixed points of the stochastic gradient flow) the simplicity constraints Eq. (11) to the pre-geometric theory, corresponds to demand the flow for φA to be un-coupled also from B, i.e. ∂φA/∂s − ξsφA ≈ 0 . Eq. (15) entails the existence of ... a second fixed point for φA, reached in the ultraviolet regime."

Via the flow equation (14), Eq. (15) is exactly the condition ε^A_{BCDE} B^{BC} ∧ B^{DE} = 0. The simplicity constraints (11) are B^{(AB} ∧ B^{CD)} = 0, which do not imply that epsilon-contracted product. For the paper's own Holst-type solution B ∝ ∇φ ∧ ∇φ, the epsilon product is generically nonzero. Thus the ultraviolet fixed point and the claimed flow to topological BF are imposed by Eq. (15), not derived, and then presented as consequences of the imposed decoupling.

1 more flagged steps

self citation load bearing [Sec. III, refs. [14] and [21]]
"we consider the stochastic (Ricci) geometry flow [21] ... according to the correspondence principle [14], the stochastic pre-geometric flow can be written, using the variables specified by the action proposed by Wilczek, as [Eq. (3)]."

The central IR reduction (3)→(4) rests on two inputs both traceable to the present authors: [21] (Lulli–Marcianò–Shan) supplies the stochastic Ricci flow equation and [14] (Addazi–Capozziello–Marcianò–Meluccio) supplies the 'correspondence principle'. Neither is an externally verified theorem or a machine-checked result; the load-bearing step therefore reduces to a self-citation chain rather than to an independent first-principles derivation.

full rationale

The paper is transparent that its central equation is a 'natural ansatz' and that its UV condition is obtained by 'imposing' decoupling. But that transparency does not remove the circularity: the IR fixed point of GR is built into Eq. (3), which was constructed to reproduce the stochastic Ricci flow and hence the Einstein-Hilbert limit already known from Wilczek's SSB mechanism; the UV fixed point of BF is built into Eq. (15), which is an independent dynamical condition rather than a consequence of the stated simplicity constraints. The derivation of both fixed points is therefore equivalent to the inputs by construction. If Eqs. (3) and (15) are regarded as defining axioms of a model, the consequences follow consistently, but they cannot be presented as first-principles predictions of emergent GR and a topological UV completion. The score of 8 reflects that the central claims of the paper are forced by these imposed equations and by the self-citation of the correspondence principle, rather than by an independent argument.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on an undereived stochastic flow equation and on assumptions about noise, decoupling, and symmetry breaking inherited from the authors' prior work. The free parameters are the noise variance and the constants of the Wilczek action; none is determined independently in this paper. No new entities are introduced beyond the pre-geometric fields and noise terms already present in the cited framework.

free parameters (6)

noise variance σ²_ξs (common for ξf, ξg, ξs) = not specified; set to unity in the main text, generalized in Appendix B
Controls the exponential decay of the scalar multiplet φA toward the UV fixed point and the flow time scale; no independent determination.
v (vev of Higgs multiplet φA) = related to Planck mass via M_P² = -8 kW v³ m²; not numerically fixed
Sets the symmetry-breaking scale at which GR emerges; inherited from Wilczek theory.
m (mass parameter setting cosmological constant) = Λ = ±6m²; not fitted in this paper
Determines the cosmological constant term in the emergent GR action.
kW (coupling in Wilczek action) = M_P² = -8 kW v³ m²
Overall coupling of the pre-geometric Lagrangian; free parameter of the parent theory.
kSSB (coupling in SSB potential) = not specified
Strength of the symmetry-breaking potential; free parameter of the parent theory.
s0 (thermal time of IR fixed point) = not specified
Reference time in the solution φA(s) = exp(-σ²(s-s0)/2) φ0(s); location of the GR fixed point.

assumptions (5)

ad hoc to paper The stochastic pre-geometric flow equation (3) has the form assumed, with Pμν and wμA coupling as in the Ricci flow projector.
Section III: 'A natural ansatz is that the stochastic pre-geometry flow equation ... must contain the stochastic noise and analogous combinations of fields than in Eq. (2).' This is not derived from the Wilczek action.
domain assumption Under SSB, the pre-geometric flow reduces to the stochastic Ricci flow (4) automatically selecting the time gauge.
Section III: 'according to the correspondence principle [14]' and 'The SSB of this equation ... reproduces the stochastic geometry (Ricci) flow, automatically selecting the time gauge.' The correspondence between Pμν, w0_0 and gμν, n0 is inherited from [14].
ad hoc to paper Near the fixed points, the simplicity constraints (11) decouple the φA flow from the B field, giving Eq. (15).
Section IV: 'imposing (in proximity of the fixed points of the stochastic gradient flow) the simplicity constraints Eq. (11) to the pre-geometric theory, corresponds to demand the flow for φA to be un-coupled also from B.' This is an assumption about the dynamics, not a consequence.
domain assumption Multiplicative scalar noise and Ito calculus apply to the pre-geometric fields.
Appendix A-B; standard stochastic quantization used in [21,34,35].
domain assumption The parent gauge group is SO(1,4) or SO(3,2) and SSB reduces it to SO(1,3).
Section II; from Wilczek [13,14].

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Cite this review

Pith. "Pith review of Emergent Gravity from Topological Quantum Field Theory: Stochastic Gradient Flow Perspective away from the Quantum Gravity Problem." pith.science (2026). https://pith.science/paper/KGAX2IZ2

@misc{pith2026250517014,
  author       = {Pith},
  title        = {Pith review of: Emergent Gravity from Topological Quantum Field Theory: Stochastic Gradient Flow Perspective away from the Quantum Gravity Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KGAX2IZ2}},
  note         = {Machine review of arXiv:2505.17014}
}

read the original abstract

We propose a scenario according to which the ultraviolet completion of General Relativity is realized through a stochastic gradient flow towards a topological BF theory. Specifically, we consider the stochastic gradient flow of a pre-geometric theory proposed by Wilczek. Its infrared limit exists, and corresponds to a fixed point where stochastic fluctuations vanish. Diffeomorphism symmetries are restored in this limit, where the theory is classical and expressed by the Einstein-Hilbert action. The infrared phase then corresponds to the classical theory of General Relativity, the quantization of which becomes meaningless. Away from the infrared limit, in the pre-geometric phase of the stochastic gradient flow, the relevant fields of the Wilczek theory undergo stochastic fluctuations. The theory can be quantized perturbatively, generating corrections to the classical Einstein-Hilbert action. The stochastic gradient flow also possesses an ultraviolet fixed point. The theory flows to a topological BF action, to which non-perturbative quantization methods can be applied. Two phase transitions occur along the thermal time dynamics, being marked by: i) the breakdown of the topological BF symmetries in the ultraviolet regime, which originates the pre-geometric phase described by the Wilczek theory; ii) the breakdown of the parental symmetries characterizing the Wilczek theory, from which General Relativity emerges. The problem of quantizing the Einstein-Hilbert action of gravity finally becomes redundant.

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