Block 4: Lattice-Gauge

Historical content. Part of the technical content of this block comes from the corpus prior to the Tratado de Fundamentos (v35) and may use threshold values from that formulation. The current classification of each result is in the status of the results.

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

The Seal S₄: Observable Big Bang (earlier formulation)

S₄ = 1.001 — QCD Scale

The seal S₄ marks the moment when the universe reaches the QCD energy scale (~200 MeV), where protons, neutrons, and observable baryonic matter emerge.

This is the “Observable Big Bang” of the MCMC: the moment at which the physics that we can measure directly begins.

MCMC Beta Function

The Yang-Mills coupling constant evolves with S:

$$\beta(S) = \beta_0 + \beta_1 \cdot e^{-b_S(S - S_3)}$$

Where S₃ = 1.000 is the seal of the electroweak transition.

Mass Gap

The minimum energy of the vacuum (mass gap) is calculated as:

$$E_{min} = \alpha_H \times \Lambda_{QCD}$$

With α_H = 0.1 and Λ_QCD = 0.2 GeV, this gives E_min ≈ 20 MeV.

Parameters

Parameter	Value	Description
β₀	6.0	Base coupling
β₁	2.0	MCMC correction
b_S	10.0	Decay rate
S₃	1.000	Electroweak seal
α_H	0.1	Mass gap factor
Λ_QCD	0.2 GeV	QCD scale

Simulation: β(S) and Mass Gap

Value of S: 1.000

β(S)

8.00

E_min

20 MeV

Phase

QCD

Connection with the Higgs

Electroweak Transition at S ≈ 1.001

The earlier formulation of the MCMC proposed a unified description of:

The electroweak symmetry breaking (S₃ = 1.000)
The QCD transition (S₄ = 1.001)
The Higgs mechanism as a consequence of the entropic collapse

In that formulation, the Higgs scale (~246 GeV) was associated with the seal S₃. In the Tratado de Fundamentos (v35), the electroweak scale is calibrated with the C¹ Higgs splice: it is not derived.

Physical Interpretation

What Is the Mass Gap?

What this block obtains is an effective spectral floor on the lattice in the emergent gauge theories of the model, tied to the mass scale by entropic quantization.

It is not a solution to the Yang–Mills problem in the strict mathematical sense, which demands a proof in the continuum: the errata FE-08-2026 (E10) corrects in that sense the scope that the earlier corpus gave to this calculation. A spectral floor that disappeared upon refining the lattice would refute it.

Philosophical Connection

Block 4 represents the complete manifestation: the point where the primordial tension has dissipated enough to allow the existence of stable matter and conscious observers.

It is the closing of the cycle that began in the Primordial State.

Implementation in Python

import numpy as np class LatticeGauge: """Block 4: Lattice-Gauge (Yang-Mills) of the MCMC.""" def __init__(self): self.beta0 = 6.0 self.beta1 = 2.0 self.bS = 10.0 self.S3 = 1.0 self.alpha_H = 0.1 self.Lambda_QCD = 0.2 # GeV def beta_MCMC(self, S): """MCMC beta function.""" return self.beta0 + self.beta1 * np.exp(-self.bS * (S - self.S3)) def mass_gap(self, S=None): """Compute the mass gap of the vacuum (GeV).""" return self.alpha_H * self.Lambda_QCD def fase(self, S): """Determine the phase according to S.""" if S < 1.0: return "Pre-electrodebil" elif S < 1.001: return "Electrodebil" else: return "QCD" # Usage lattice = LatticeGauge() print(f"β(S=1.0) = {lattice.beta_MCMC(1.0):.2f}") print(f"Mass gap = {lattice.mass_gap()*1000:.0f} MeV")

📁 File: lattice/bloque4_ym_lattice.py

Summary: The Computational Blocks

The Complete Journey

Block	Name	Function
0	Primordial State	Initial configuration (Mp = 1, Ep ≈ 0)
1	Pre-geometry	Collapse rate k(S), mass→space conversion
2	Cosmology	Modified Friedmann, expansion of the universe
3	N-body	Entropic friction, halo formation
4	Lattice-Gauge	Yang-Mills, mass gap, particle physics

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Computational Blocks