Historical content.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.

Fermion Mass Hierarchy

Why do fermion masses vary over 6 orders of magnitude, from the electron to the top quark?

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

The Problem

The masses of the Standard Model fermions range from me ≈ 0.5 MeV to mt ≈ 173 GeV, a range of 6 orders of magnitude. There is no fundamental explanation for this hierarchy.

The Mass Spectrum

Family Up Quark Down Quark Charged Lepton
1st u: ~2 MeV d: ~5 MeV e: 0.511 MeV
2nd c: ~1.3 GeV s: ~100 MeV μ: 106 MeV
3rd t: ~173 GeV b: ~4.2 GeV τ: 1.78 GeV

The Standard Model does not explain why there are exactly 3 families, nor why the masses follow this hierarchical pattern.

The MCMC Solution

The MCMC explains the hierarchy through the phases of the Adrián Field:

Phases of Φ_Ad

The Adrián Field has a phase structure that couples differentially to each generation:

$$m_f = y_f \cdot v \cdot \langle\Phi_{Ad}\rangle_n$$

Where n = 1, 2, 3 indicates the family, and the VEV of Φ_Ad depends on the entropic seal:

$$\langle\Phi_{Ad}\rangle_n \propto e^{-\lambda(S_n - S_1)}$$

Three Emergent Families

The three families emerge from the three “material” entropic seals S₁, S₂, S₃:

  • S₁ = 0.010: First family (e, u, d)
  • S₂ = 0.100: Second family (μ, c, s)
  • S₃ = 1.000: Third family (τ, t, b)

There is no fourth family because S₄ = 1.001 corresponds to the transition to QCD, not to new fermions.

Mass Ratios

The ratios between families scale exponentially:

$$\frac{m_3}{m_2} \approx \frac{m_2}{m_1} \approx e^{\lambda \Delta S} \sim 10-100$$
Ratio Observed MCMC
m_t / m_c ~130 ~120 (S₃-S₂)
m_c / m_u ~600 ~500 (S₂-S₁)
m_τ / m_μ ~17 ~15
m_μ / m_e ~207 ~200
3 Emergent Families
from 3 material entropic seals

The mass hierarchy emerges geometrically from the structure of Φ_Ad

Why Exactly 3 Families?

Standard Model

  • 3 families are an input
  • More could exist (Z' decay)
  • Yukawas are free parameters
  • No explanation of the hierarchy

MCMC

  • 3 “material” seals (S₁-S₃)
  • S₄ is the QCD transition, not a family
  • Yukawas emerge from Φ_Ad
  • Natural exponential hierarchy

Predictions

Neutrinos

The MCMC predicts that neutrino masses follow a similar pattern but suppressed by the entropic seesaw mechanism:

$$m_\nu \propto \frac{m_f^2}{M_{Pl} \cdot f(S)}$$

This gives masses in the sub-eV range, consistent with oscillations.

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