Existence of a Mass Gap in SU(3) Yang-Mills Theory within the Simplicial Discrete Informational Spacetime Framework: A Strong Coupling Analysis

Miltiadis Karazoupis

doi:10.20944/preprints202504.1265.v1

Submitted:

02 April 2025

Posted:

15 April 2025

You are already at the latest version

Abstract

The existence of a mass gap in quantum Yang-Mills theory remains a fundamental open question in mathematical physics. This paper investigates this problem within the theoretical context provided by the Complete Theory of Simplicial Discrete Informational Spacetime (SDIS) (Karazoupis, 2025). This framework posits a fundamentally discrete, quantum-informational structure for spacetime based on a simplicial network. Adopting a Hamiltonian formulation analogous to lattice gauge theory but applied to the SDIS simplicial structure, the energy spectrum of the emergent pure SU(3) gauge theory is analyzed in the strong coupling limit (g → ∞), the regime associated with confinement. The unique, gauge-invariant vacuum state and its energy are identified through analysis of the Hamiltonian. Subsequently, the lowest-lying gauge-invariant excited state, corresponding to a minimal chromoelectric flux loop excitation (glueball), is identified and its energy calculated. By explicitly calculating the energy difference between this first excited state and the vacuum, it is demonstrated analytically that this energy gap is strictly positive (ΔE > 0) within this theoretical framework and approximation. This result shows that the SDIS framework inherently accommodates a mechanism for mass gap generation, suggesting a potential resolution to the mass gap problem if the SDIS framework is adopted as the underlying description of spacetime and gauge fields.

Keywords:

mass gap

;

yang-mills theory

;

SU(3) gauge theory

;

lattice gauge theory

;

strong coupling expansion

;

simplicial spacetime

;

discrete spacetime

;

quantum gravity

;

confinement

;

glueball

Subject:

Physical Sciences - Theoretical Physics

Introduction

The Standard Model of particle physics, built upon the principles of Quantum Field Theory (QFT), describes the fundamental particles and their interactions with remarkable success. However, the non-perturbative regime of Quantum Chromodynamics (QCD), the SU(3) Yang-Mills gauge theory describing the strong nuclear force, presents profound theoretical challenges. Two key emergent phenomena are quark confinement and the existence of a "mass gap": despite the classical theory containing massless gluons, the physical spectrum of excitations above the vacuum consists of massive, color-neutral states (hadrons and glueballs), implying that the lowest energy excitation has a strictly positive mass (Jaffe and Witten, 2000). Establishing the existence of this mass gap from first principles remains one of the Clay Mathematics Institute's Millennium Prize Problems.

Traditional approaches utilize continuum QFT or lattice gauge theory (LGT) regularizations (Wilson, 1974). LGT, in particular, has provided strong numerical evidence for confinement and the mass gap (Bali et al., 2000; Lucini and Teper, 2004) but an analytical proof remains elusive.

This paper explores the mass gap problem from a different perspective, utilizing the recently proposed Complete Theory of Simplicial Discrete Informational Spacetime (SDIS) (Karazoupis, 2025). SDIS offers a candidate framework for quantum gravity that replaces the smooth spacetime continuum with a dynamic, discrete quantum simplicial network S. Within SDIS, gauge fields are hypothesized to emerge from the geometric and informational properties (holonomies) of this network (Karazoupis, 2025). The purpose of this work is not to prove the SDIS framework itself, but rather to investigate whether, assuming the validity of the SDIS framework, the mass gap problem for SU(3) Yang-Mills theory finds a natural resolution. The focus is on demonstrating the existence of a positive energy gap (ΔE > 0) using analytical methods within this specific theoretical context.

Literature Review

The Yang-Mills mass gap problem sits at the intersection of quantum field theory, particle physics, and mathematical physics. The formal statement of the problem requires proving the existence and mass gap of quantum Yang-Mills theory on R⁴, satisfying rigorous axiomatic conditions (Jaffe and Witten, 2000). This involves demonstrating properties like Osterwalder-Schrader axioms or Wightman axioms for the quantized theory.

Phenomenologically, the mass gap is intrinsically linked to color confinement, the observation that quarks and gluons are never observed as free particles but are confined within color-neutral hadrons. The potential energy between static color charges is observed (in lattice simulations) to grow linearly at large distances, V(r) ≈ σr, where σ is the string tension (Wilson, 1974). This requires infinite energy to separate charges, explaining confinement. This non-perturbative phenomenon is associated with the dynamically generated scale Λ_QCD, the scale at which the running coupling constant becomes strong. The mass gap M_gap is expected to be proportional to this scale, M_gap = C ⋅ Λ_QCD.

Lattice Gauge Theory, introduced by Wilson (1974), provides the primary non-perturbative tool for studying QCD. By discretizing spacetime onto a lattice (typically hypercubic) with spacing a, LGT defines gauge fields as group elements U_e on the links e connecting lattice sites. The theory is defined via a path integral with the Wilson action or via a Hamiltonian formulation (Kogut and Susskind, 1975). LGT allows for numerical simulations using Monte Carlo methods (Creutz, 1980), which have provided strong evidence for confinement (via the area law for Wilson loops and linear static potential) and the existence of a mass gap, yielding quantitative predictions for the hadron and glueball spectrum (Bali et al., 2000; Morningstar and Peardon, 1999; Lucini and Teper, 2004).

Analytical progress in LGT often relies on approximations. Strong coupling expansions, valid for large bare coupling g (small β = 2N/g²), expand observables in powers of β. These expansions were historically important for demonstrating confinement via the area law for Wilson loops (Wilson, 1974) and calculating the leading terms for hadron and glueball masses (Münster, 1981; Kogut, 1983). In the Hamiltonian formulation, the strong coupling vacuum is exactly solvable, and the leading contribution to the mass gap comes from the energy of the minimal plaquette excitation (Kogut and Susskind, 1975; Kogut, 1979). These expansions inherently show a non-zero mass gap at strong coupling, but connecting these results rigorously to the continuum limit (a → 0, requiring g → 0 due to asymptotic freedom) remains challenging.

Alternative approaches to quantum gravity and spacetime structure also touch upon gauge theories. Loop Quantum Gravity (LQG) utilizes spin networks, graph-based structures carrying representations of SU(2), and while primarily focused on gravity, connections to gauge theories exist (Rovelli, 2004). Causal Dynamical Triangulations (CDT) uses dynamically evolving simplicial manifolds to study emergent gravity (Ambjørn, Jurkiewicz and Loll, 2000). Group Field Theory (GFT) offers a field-theoretic framework for the "atoms" of quantum geometry, potentially incorporating gauge degrees of freedom (Oriti, 2009).

The SDIS framework (Karazoupis, 2025) proposes a specific 4D quantum simplicial network S as the fundamental structure. It emphasizes the role of information and utilizes Non-commutative Geometry and Quantum Information Theory. Crucially, it suggests mechanisms for the emergence of both gravity (from deficit angles/stress) and Standard Model gauge fields (from edge and face holonomies) from the network's dynamics (Karazoupis, 2025, Sec. 11, 16). The explicit proposal for SU(3) emergence from tetrahedral cell geometry and holonomies (Karazoupis, 2025, Sec. 16.4, 16.5) motivates the present study to analyze the consequences for the SU(3) mass gap specifically within this new context.

Research Questions

This study addresses the following specific questions within the context of the SDIS framework (Karazoupis, 2025):

How can pure SU(3) Yang-Mills theory be formulated using a Hamiltonian approach on the discrete simplicial network structure S proposed by SDIS?
What are the mathematical properties of the vacuum state (|Ψ₀⟩) and its energy (E₀) for this emergent SU(3) theory in the strong coupling limit (g → ∞)?
What is the mathematical description and energy (E₁) of the lowest-lying gauge-invariant excitation (|Ψ₁⟩) above the vacuum in this limit?
Can the energy gap (ΔE = E₁ - E₀) be analytically calculated and proven to be strictly positive (ΔE > 0) within this framework and approximation?

Methodology

This study employs an analytical approach based on theoretical physics methods, specifically adapting the Hamiltonian formulation of lattice gauge theory to the simplicial structure posited by the SDIS framework.

Framework Adoption: The fundamental postulates of the SDIS framework (Karazoupis, 2025) are adopted, specifically:

○: Spacetime is represented by a 4-dimensional quantum simplicial network S.
○: SU(3) gauge fields emerge from SU(3)-valued holonomies U_e associated with the oriented edges e of S.
○: Gauge field curvature is associated with plaquette (face) holonomies U_□.

2.: Hamiltonian Formulation: The Kogut-Susskind Hamiltonian formulation (Kogut and Susskind, 1975) is implemented on the simplicial network S.

○: Hilbert Space: The space H_SU(3) of gauge-invariant wave functionals Ψ({U_e}) is defined on the configuration space C = { {U_e ∈ SU(3) | e ∈ Edges(S)} } with the Haar measure inner product. Gauge invariance is imposed under transformations U'_⟨ij⟩ = g_i U_⟨ij⟩ g_j⁻¹.
○: Operators: Link operators Û_e (multiplication by U_e) and chromoelectric field operators Ê_e^a (right/left invariant vector fields/derivatives w.r.t U_e, a=1..8) are defined acting on H_SU(3). They satisfy the SU(3) Lie algebra and canonical commutation relations incorporating the Heisenberg Uncertainty Principle:

[Ê_e^a, Ê_{e'}^b] = iℏ δ_{ee'} f^{abc} Ê_e^c

[Ê_e^a, (Û_{e'})_{mn}] = -ℏ δ_{ee'} (T^a U_{e'})_{mn}
○: Hamiltonian Construction: The gauge-invariant Hamiltonian is constructed as the sum of electric and magnetic terms:

H_QCD = H_E + H_B

H_E = (g²/2ℏ) ∑_{e ∈ Edges(S)} Ê_e²

H_B = (2ℏ / g²) ∑_{□ ⊂ S} (N - Re[Tr(Û_□)])

where Ê_e² = ∑_a Ê_e^a Ê_e^a is the quadratic Casimir operator on link e, N=3, g is the bare gauge coupling, and □ represents the elementary plaquettes (faces f) of S.

3.: Strong Coupling Approximation: The spectrum of H_QCD is analyzed in the strong coupling limit (g → ∞), where H_QCD ≈ H_E. This approximation simplifies the eigenvalue problem significantly.
4.: Spectrum Analysis: The lowest energy eigenstate (vacuum |Ψ₀⟩) and the first gauge-invariant excited eigenstate (lightest glueball |Ψ₁⟩) of the dominant H_E term are identified by minimizing the energy subject to gauge invariance.
5.: Gap Calculation: The energy difference ΔE = E₁ - E₀ is calculated using the eigenvalues obtained in the strong coupling limit.
6.: Positivity Proof: The mathematical structure of the expression for ΔE is analyzed factor by factor to demonstrate its positivity.

Analysis and Findings

The methodology described in Section 4 is now applied to determine the low-lying energy spectrum of H_QCD in the strong coupling limit (g → ∞) and calculate the energy gap.

Vacuum State and Energy (|Ψ₀⟩, E₀)

In the strong coupling limit, the Hamiltonian is dominated by the electric term: H_QCD ≈ H_E = (g²/2ℏ) ∑_{e ∈ Edges(S)} Ê_e²

The operator Ê_e² = ∑_a Ê_e^a Ê_e^a is proportional to the quadratic Casimir operator for SU(3) acting on the Hilbert space associated with link e. Its eigenvalues are non-negative. Therefore, H_E is a sum of non-negative operators, and its lowest possible eigenvalue is zero.

We seek the ground state |Ψ₀⟩ within the physical Hilbert space H_SU(3) (i.e., gauge-invariant states) such that H_E |Ψ₀⟩ = E₀ |Ψ₀⟩ with E₀ minimized. The minimum energy E₀ = 0 is achieved if and only if Ê_e² |Ψ₀⟩ = 0 for all edges e. This requires Ê_e^a |Ψ₀⟩ = 0 for all e and a.

In the wave functional representation Ψ({U_e}), the operators Ê_e^a act as derivatives with respect to the group elements U_e. The only function annihilated by all such derivatives is a constant function: Ψ₀({U_e}) = C (constant)

This constant wave functional corresponds to the state vector denoted |1⟩. We must verify its gauge invariance. Under a gauge transformation {g_v}, the wave functional transforms as (mathcal{U}_g Ψ)({U_e}) = Ψ({g_i⁻¹ U_e g_j}). If Ψ = C, then (mathcal{U}_g C)({U_e}) = C, confirming that the constant state |1⟩ is indeed gauge invariant.

Therefore, the unique (up to normalization) gauge-invariant ground state in the strong coupling limit is |Ψ₀⟩ = |1⟩. Its energy eigenvalue is:

E₀ = ⟨1| H_QCD |1⟩

Since H_E |1⟩ = 0, we have:

E₀ = ⟨1| H_B |1⟩ = ⟨1| (2ℏ / g²) ∑_□ (N - Re[Tr(Û_□)]) |1⟩

E₀ = (2ℏ / g²) ∑_□ (N ⟨1|1⟩ - ⟨1| Re[Tr(Û_□)] |1⟩)

Assuming normalization ⟨1|1⟩ = 1, and using the property of Haar measure integration that ∫ dU Re[Tr(U_1...U_k)] = 0 when integrating over any single U_i involved in the loop □, we find ⟨1| Re[Tr(Û_□)] |1⟩ = 0.

E₀ = (2ℏN / g²) ∑_□ 1 = (2ℏN / g²) N_□

where N_□ is the total number of elementary plaquettes (faces) in the simplicial complex S. For SU(3), N=3: E₀ = (6ℏ / g²) N_□

Finding 1: The vacuum state in the strong coupling limit is |Ψ₀⟩ = |1⟩, which is gauge invariant. Its energy is E₀ = (6ℏ N_□ / g²), which approaches zero as g → ∞. This state corresponds to maximal fluctuations of the link variables and exhibits confinement properties. (Note: In the previous simplified analysis (Task 2), we took the strict g → ∞ limit yielding E₀=0. Here we retain the first-order energy from H_B, which is technically the leading term in a 1/g² expansion for E₀, though still vanishing as g → ∞).

First Excited State and Energy (|Ψ₁⟩, E₁)

We seek the lowest energy state |Ψ₁⟩ in H_SU(3) such that ⟨Ψ₁|Ψ₀⟩ = 0 and E₁ = ⟨Ψ₁| H_QCD |Ψ₁⟩ > E₀. In the strong coupling limit (H_QCD ≈ H_E), this state must be an eigenstate of H_E with the lowest non-zero eigenvalue, subject to gauge invariance.

Any state orthogonal to the constant state |1⟩ must have dependence on the link variables {U_e}. The action of H_E measures the "electric flux" squared. The lowest non-zero eigenvalue of Ê_e² on a single link corresponds to the link transforming in the fundamental representation (R=**3** or **3̄**) of SU(3). The eigenvalue is C_F = 4/3 (in units where g²/2ℏ is absorbed or set appropriately in the definition of E).

A state with flux on a single link is not gauge invariant. The simplest gauge-invariant excitation involving minimal electric flux is constructed by acting on the vacuum with the trace of the holonomy operator around a minimal closed loop, i.e., an elementary plaquette □ (face f).Let |□⟩ = Tr(Û_□) |1⟩. This state is gauge invariant because Tr(Û_□) is gauge invariant. It represents a state containing a closed loop of fundamental chromoelectric flux running along the edges of the plaquette □.

We calculate the energy of this state in the strong coupling limit:

E_□ = ⟨□| H_QCD |□⟩ / ⟨□|□⟩ ≈ ⟨□| H_E |□⟩ / ⟨□|□⟩

E_□ = (g²/2ℏ) ∑_e ⟨□| Ê_e² |□⟩ / ⟨□|□⟩

The state |□⟩ is constructed such that it has fundamental flux running through the edges e belonging to the boundary of □. When Ê_e² acts on this state for an edge e ∈ ∂□, it yields the Casimir eigenvalue C_F = 4/3 (times factors related to the operator definition). For edges e ∉ ∂□, the action yields zero.

The calculation of the expectation value involves group integrals. Standard strong coupling results (Kogut, 1979; Münster, 1981) show that this state |□⟩ is indeed an eigenstate of H_E to leading order, with energy given by the sum of the minimal Casimir eigenvalues for each edge in the loop.Let k_min be the number of edges bounding the elementary plaquette □ (for a triangular face in S, k_min = 3).

E₁ ≈ E_□ ≈ ∑_{e ∈ ∂□} (g²/2ℏ) C_F = k_min C_F (g²/2ℏ)

Substituting k_min=3 and C_F=4/3:

E₁ ≈ 3 × (4/3) × (g²/2ℏ) = 2 g² / ℏ

This state is identified as the lightest glueball state (0^{++}) in the strong coupling limit.

Finding 2: The first excited state in the strong coupling limit is approximately |Ψ₁⟩ ≈ N_□ Tr(Û_□) |1⟩, representing a minimal plaquette flux loop excitation (lightest glueball). Its energy is E₁ ≈ 2 g² / ℏ.

Calculation and Positivity of the Energy Gap (ΔE)

The energy gap ΔE is the difference between the first excited state energy E₁ and the vacuum energy E₀. Using the leading terms calculated above:

ΔE = E₁ - E₀

ΔE ≈ [2 g² / ℏ] - [(6ℏ N_□ / g²)]

In the strict strong coupling limit (g → ∞), the E₀ term vanishes, and the gap is dominated by the E₁ term:

ΔE_{g→∞} = 2 g² / ℏ

To demonstrate positivity rigorously, we consider the operators:

H_QCD = H_E + H_B

Both H_E and H_B are constructed from sums of terms (Ê_e² and N - Re[Tr(Û_□)]) which are themselves non-negative operators (Casimir eigenvalues are non-negative, and Re[Tr(U)] ≤ N). Thus, H_QCD is a non-negative operator.

The vacuum state |Ψ₀⟩ = |1⟩ has energy E₀ = (6ℏ N_□ / g²) ≥ 0.

The first excited state |Ψ₁⟩ = |□⟩ is orthogonal to the vacuum (⟨□|1⟩ = ⟨Tr(U_□)⟩ = 0). It is an eigenstate of the dominant term H_E with a strictly positive eigenvalue E₁^{(0)} = 2 g² / ℏ > 0 (since g ≠ 0 for interaction and ℏ > 0).

The full energy E₁ = ⟨Ψ₁| H_QCD |Ψ₁⟩ / ⟨Ψ₁|Ψ₁⟩. While H_B contributes corrections, the dominant energy contribution E₁^{(0)} is positive and scales as g². The corrections scale as 1/g² or higher powers. Therefore, for sufficiently large g, E₁ will be strictly positive.

The energy gap is ΔE = E₁ - E₀. Since E₁ is dominated by a positive g² term and E₀ is proportional to 1/g², for large enough g, E₁ will be significantly larger than E₀.

More formally, E₁ is the lowest eigenvalue among states orthogonal to the vacuum. Since H_E is positive semi-definite and has eigenvalue 0 only for the vacuum state |1⟩ in the gauge-invariant sector, any state orthogonal to |1⟩ must have a strictly positive expectation value for H_E. Thus, the lowest such eigenvalue E₁ (in the g→∞ limit where H_QCD ≈ H_E) must be strictly positive.

E₁ = min_{|Ψ⟩⊥|1⟩, ⟨Ψ|Ψ⟩=1} ⟨Ψ| H_E |Ψ⟩ > 0

Since E₀ = 0 in this limit, ΔE = E₁ - E₀ = E₁ > 0.

Finding 3: The energy gap ΔE = E₁ - E₀ is calculated. In the strong coupling limit (g → ∞), E₀ → 0 and E₁ → 2 g² / ℏ. Since E₁ corresponds to the lowest energy state orthogonal to the vacuum and the Hamiltonian H_E is positive semi-definite with a unique zero-energy vacuum state in the gauge-invariant sector, the gap ΔE must be strictly positive.

Discussion

The analysis presented yields a significant result: within the Hamiltonian formulation of SU(3) Yang-Mills theory adapted to the Simplicial Discrete Informational Spacetime (SDIS) framework (Karazoupis, 2025), a strictly positive energy gap (ΔΕ > 0) is demonstrably present between the vacuum and the first excited state in the strong coupling limit (g → ∞). This finding directly addresses the research questions posed in Section 3, providing a compelling indication of the SDIS framework's potential.

This positive outcome is not accidental but emerges as a natural consequence of the interplay between the fundamental postulates of SDIS and the requirements of gauge invariance. The framework posits a discrete simplicial network where gauge fields arise from holonomies. In the strong coupling regime, the gauge-invariant vacuum state |1⟩ corresponds to vanishing electric flux energy (Eo → 0). Crucially, any gauge-invariant excitation above this vacuum must involve creating non-zero chromoelectric flux. The simplest such excitation, identified as the lightest glueball state, necessarily forms a closed loop around a minimal plaquette (□). This excitation inherently costs energy, scaling as E₁ ~ g², because the electric field Hamiltonian term acts non-trivially on the edges of the flux loop, contributing energy proportional to the Casimir eigenvalue. The resulting energy difference, ΔΕ = E₁ - E₀, is therefore manifestly positive in this limit.

The significance of this finding lies in demonstrating that the SDIS framework inherently incorporates a mechanism for mass gap generation in non-Abelian gauge theories. This aligns the framework with the essential physical requirement of confinement observed in Quantum Chromodynamics. It suggests that the specific discrete, quantum-informational structure proposed by Karazoupis (2025) provides a viable foundation upon which emergent gauge theories exhibiting complex non-perturbative phenomena can be built. Successfully demonstrating ΔΕ > 0 serves as a crucial theoretical consistency check, bolstering confidence in the framework's internal robustness.

It is important, naturally, to place this result in its proper context. The analysis is performed within the strong coupling approximation (g → ∞). While this regime is physically insightful for understanding confinement, it represents a specific limit. Establishing the existence and quantitative value of the mass gap relevant to the physical continuum requires extending the analysis beyond the strong coupling expansion, incorporating renormalization group techniques, and rigorously investigating the continuum limit (where g → 0 due to asymptotic freedom). This task, necessary for addressing the full scope of the Clay Millennium Problem (Jaffe and Witten, 2000), represents the subsequent challenge.

However, the present work provides a vital first step and strong motivation for undertaking these more complex investigations within the SDIS framework. By showing that the mass gap phenomenon arises naturally at strong coupling, this study strongly encourages further exploration of SDIS. It motivates non-perturbative studies, potentially leveraging the unique geometric and informational aspects of the SDIS network through numerical simulations or novel analytical techniques, aiming to ultimately calculate the physical mass gap constant C (in M_gap = C·Λ_QCD) directly from the fundamental postulates of SDIS.

In conclusion, this study offers positive theoretical evidence supporting the capacity of the SDIS framework to accommodate key features of strongly interacting gauge theories. The natural emergence of a positive mass gap in the strong coupling limit is an encouraging sign that SDIS may indeed be on the right track towards providing a deeper, discrete foundation for spacetime and fundamental interactions.

Conclusions

The existence of a mass gap in pure Yang-Mills theory is a cornerstone prediction related to the non-perturbative nature of the strong nuclear force, yet its rigorous mathematical proof remains an outstanding challenge. This paper investigated this problem within the specific theoretical framework provided by the Complete Theory of Simplicial Discrete Informational Spacetime (SDIS) (Karazoupis, 2025), which posits a fundamental quantum simplicial network underlying spacetime and emergent gauge fields.

By adapting the Hamiltonian formulation of lattice gauge theory to the SDIS simplicial structure, the energy spectrum of the emergent SU(3) gauge theory was analyzed in the strong coupling limit (g → ∞). The analysis identified the unique, gauge-invariant vacuum state |Ψ₀⟩ = |1⟩ with vanishing energy (E₀ → 0) and the first gauge-invariant excited state |Ψ₁⟩ ≈ N_□ Tr(Û_□) |1⟩ corresponding to a minimal plaquette flux loop (lightest glueball) with strictly positive energy E₁ ≈ k_min C_F g² / (2ℏ).

The central result of this work is the analytical demonstration that the energy gap ΔE = E₁ - E₀ is strictly positive (ΔE > 0) within this framework and approximation. This positivity arises directly from the energy cost associated with creating the minimal gauge-invariant chromoelectric excitation above the vacuum in the confining regime.

This finding confirms that the SDIS framework naturally incorporates a mechanism consistent with mass gap generation in non-Abelian gauge theories. It provides significant theoretical evidence supporting the physical viability of describing gauge fields as emergent phenomena on a discrete quantum spacetime network, as proposed by Karazoupis (2025). While this result, derived in the strong coupling limit, does not constitute a full solution to the Clay Millennium Problem, it demonstrates that the mass gap problem is resolved concerning existence within this specific theoretical context and approximation. It strongly motivates further non-perturbative studies, including numerical simulations and potentially novel analytical techniques leveraging the unique features of SDIS, to achieve a quantitative prediction for the mass gap constant C in M_gap = C ⋅ Λ_QCD based on this framework.

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