Historical content. This page belongs to the former “solved problems” section. The Tratado de Fundamentos (v35, Jun 2026) does not present these problems as solved: it develops them in full, classifies several results as conditional and keeps seven open fronts. The values of S and the thresholds quoted here may correspond to the earlier formulation. The current classification of each result is in the status of the results.
CLAY PRIZE PROBLEM

Mass Gap (Yang-Mills)

One of the seven prize problems of the Clay Mathematics Institute: proving that Yang-Mills theory has a nonzero mass gap.

This page is a translation of the Spanish original. Where they differ, the Spanish text prevails.

The Problem

In Yang-Mills quantum field theory (the basis of QCD), to prove rigorously that a nonzero mass gap exists between the vacuum and the first excited state. This explains why gluons, although they have no mass in the Lagrangian, form massive bound states.

Formal Statement

For any simple compact gauge group G (such as SU(3) in QCD):

$$\exists \Delta > 0 : \quad E_1 - E_0 \geq \Delta$$

Where E₀ is the vacuum energy and E₁ is the energy of the first excited state. This Δ (the “mass gap”) must be mathematically proven to exist.

Classical QCD

  • Massless gluons in the Lagrangian
  • Nonlinear self-interaction
  • Confinement observed but not proven
  • Lattice QCD: numerical evidence

Observations

  • Lightest hadrons: pions (~140 MeV)
  • Nucleons: ~940 MeV
  • There are no massless glueballs
  • Estimated gap: ~1 GeV

The MCMC Solution

The MCMC provides a physical mechanism for the mass gap based on the entropic seal S = 1.001 and the field transition Φ_Ad → Φ_H:

The Third Seal: S = 1.001

The seal S = 1.001 is a critical threshold where three simultaneous phenomena occur:

  • Birth of time: The 3+1D metric emerges
  • Field transition: Φ_Ad → Φ_H (Adrián Field → Higgs)
  • Mass gap: Entropic discretization generates the gap
$$S_{gap} = 1.001 \quad \Rightarrow \quad \Phi_{Ad} \rightarrow \Phi_H$$

Entropic Discretization

At the seal S = 1.001, there is a minimum entropy increment that generates the quantum of energy:

$$\Delta S_{min} = 0.001 \quad \Rightarrow \quad E_{min} = \frac{\hbar c}{l_{lat}} \cdot f(\Delta S_{min})$$

This entropy quantum translates directly into the observed mass gap.

Modified Beta Function

The beta function of the renormalization group includes an entropic term:

$$\beta(g) = -\frac{11 N_c}{48\pi^2} g^3 + \beta_S(S) \cdot g$$

Where βS(S) is the entropic contribution that “freezes” the coupling at low energies, generating the gap.

Function β(S) and Mass Gap

β(g) with Entropic Correction

The entropic term stops the running at g* ≈ 1, generating Δ ≈ 1 GeV

Calculation of the Mass Gap

The gap emerges from the scale where βS dominates:

$$\Delta = \Lambda_{QCD} \cdot e^{1/\beta_S} \approx 200 \text{ MeV} \cdot e^{1.6} \approx 1.0 \text{ GeV}$$
Parameter Value Origin
Λ_QCD ~200 MeV Confinement scale
ΔS_min 0.001 S₄ - S₃
β_S ~0.6 Derived from k(S)
Δ (mass gap) ~1.02 GeV MCMC Prediction
Δ ≈ 1.02 GeV at S = 1.001
Mass gap emerging from the Φ_Ad → Φ_H transition

The same seal where time is born generates the mass gap in QCD

Comparison with Lattice QCD

Lattice QCD (Numerical)

  • Discretizes spacetime artificially
  • Continuum limit a → 0 problematic
  • Numerical evidence of a gap
  • It is not a rigorous proof

MCMC

  • Physical discretization (entropy)
  • l_lat is physical, not artificial
  • Gap emerges from ΔS_min
  • Explanatory mechanism

Ontological Connection

The Gap as a Consequence of the Lattice

In the MCMC, the mass gap is not a mathematical mystery but a physical consequence of the discrete structure of spacetime at the Planck scale:

  • The entropic lattice has a minimum spacing l_lat
  • This imposes a natural UV cutoff (not artificial)
  • The minimum excitable energy is E_min = ℏc/l_lat
  • Confinement emerges from the topology of the lattice

Implications

  • Confinement: Quarks are confined because the lattice does not allow fractional excitations of color charge
  • Asymptotic freedom: At high energies (small distances), β_S → 0 and we recover perturbative QCD
  • Hadronization: The transition to hadrons occurs when the energy falls below Δ
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