Historical content.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.

Block 1: Pre-geometry

Collapse rate k(S) and entropic integral ε(S) that determine the conversion of mass into space.

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

1

Pre-geometry — Pre-Big Bang Phase

This block covers the complete pre-geometric phase: from the Seal C₀ (S = 0.009) to the Big Bang (S = 1.001). It describes the four Collapse-Seal cycles and the progressive dimensional emergence:

V₀D → V₁D → V₂D → V₃D → V₃₊₁D

Complete Collapse-Seal Sequence

S value Type Dimension Emergent Attribute
0.001 Collapse C₀ V₀D start Embryonic gravity
0.009 Seal C₀ V₀D sealed Planck (~10¹⁹ GeV)
0.010 Collapse C₁ V₁D start PP/AP, motion, U(1)
0.099 Seal C₁ V₁D sealed GUT (~10¹⁶ GeV), c fixed
0.100 Collapse C₂ V₂D start Turn, rotation
0.999 Seal C₂ V₂D sealed EW (246 GeV), c² calibrated
1.000 Collapse C₃ V₃D start 3D volume
1.001 Seal C₃ V₃₊₁D BIG BANG (~200 MeV)

Collapse

Manifestation of a new dimension. The tension opens toward a new geometric freedom.

Seal

Stabilization. The potential V(Φ_Ad; S) reaches a stable minimum.

Collapse Rate k(S)

The collapse rate is a function of the structuring variable S that includes harmonics:

$$k(S) = k_0 \times [1 + a_1 \sin(2\pi S) + a_2 \sin(4\pi S) + a_3 \sin(6\pi S)]$$

Parameters

Parameter Value Description
k₀ 6.307 Base collapse rate
a₁ 0.15 First harmonic amplitude
a₂ 0.25 Second harmonic amplitude
a₃ 0.35 Third harmonic amplitude

Interactive Simulation: k(S)

k(S)
6.307
ε(S)
0.500
PME
0.000
Mp(S)
0.500

Entropic Integral ε(S)

The entropic integral accumulates the collapse rate and determines the fraction of converted mass:

$$\varepsilon(S) = \frac{\int_0^S k(s) \, ds}{\int_0^{S_4} k(s) \, ds}$$

With this, Primordial Mass and Primordial Space evolve as:

$$M_p(S) = M_{p0} \times (1 - \varepsilon)$$ $$E_p(S) = M_{p0} \times \varepsilon$$

Physical Interpretation

Why Harmonics?

The sinusoidal terms in k(S) are not arbitrary. They represent the natural resonances of the collapse process, similar to the normal modes of vibration in physics.

Each harmonic corresponds to a different energy scale in the evolution of the universe, from the Planck scale to the QCD scale.

Philosophical Connection

Pre-geometry represents the process of manifestation: how pure Being (Mass) is gradually transformed into form and extension (Space).

By structural analogy (a correspondence of function, not an identity), it is read alongside the passage from “Non-Thinking” (pure potential) to “Thinking” (manifestation).

Law of Cronos (Ontological Block I)

The Law of Cronos defines the functions that govern the pre-geometric evolution:

Expansion Function C(S)

$$\frac{d(\ln a)}{dS} = C(S)$$

Typical values: C_early = 2.2, C_mid = 1.7, C_late = 1.05

Chronification Function T(S)

$$T(S) = T_0 + T_{peak} \times \exp\left(-\frac{(S - S_{peak})^2}{2 \times T_w^2}\right)$$
T₀
1.0
T_peak
0.15
T_w
0.010

Tensional Field Φ_ten(S)

$$\Phi_{ten}(S) = \phi_{env}(S) + \phi_{bump}(S)$$

Envelope component:

$$\phi_{env}(S) = \phi_{env\_amp} \times S \times \exp\left(-\frac{S}{\phi_{env\_\tau}}\right)$$

Peak component:

$$\phi_{bump}(S) = \phi_{bump\_amp} \times \exp\left(-\frac{(S - S_{bump})^2}{2 \times \phi_{bump\_w}^2}\right)$$
φ_env_amp
0.7
φ_env_τ
0.6
φ_bump_amp
0.8
φ_bump_w
0.020

Lapse Factor N(S)

$$N(S) = \exp\left(\Phi_{ten}(S)\right)$$

The lapse determines the rate of relative time as a function of the tensional density.

Relative Time

$$t_{rel}(S) = \int_{S_{Planck}}^{S} T(s) \times N(s) \, ds$$

Ontological Mass Gap (earlier formulation)

In the formulation of the earlier corpus, the mass gap was presented as a consequence of the discrete structure of the structuring variable S. The Tratado de Fundamentos (v35) treats it as an effective spectral floor: a dictionary proposed under normalization, not the Yang–Mills problem (FE-08-2026, E10).

$$E_{min} = k \times \Delta S, \quad \text{donde } k \equiv M_{Pl} \times c^2, \quad \Delta S = 10^{-3}$$

Physical Meaning

In that formulation, the mass gap reflected that there are no excitations below the tension quantum. It manifested as:

  • Higgs mass: ~125 GeV (after the transition Φ_Ad → Φ_H at S = 1.001)
  • QCD scale: ~200 MeV (quark confinement)

It is not an added parameter, but an inevitable consequence of the discrete structure of the MCMC. The entropic discretization imposes a minimum energy gap.

The Big Bang (S = 1.001)

The Seal C₃ at S = 1.001 marks the birth of the observable spacetime V₃₊₁D:

What emerges at the Big Bang

  • Complete V₃₊₁D metric: spacetime with Lorentzian signature (-,+,+,+)
  • Relative time: the Law of Cronos is activated, emergent time enters the scene
  • Effective Higgs field: Φ_Ad → Φ_H, with VEV v = 246 GeV
  • Four fundamental forces: gravity, electromagnetism, weak force, strong force
  • QCD confinement: scale ~200 MeV (effective spectral floor, not the Yang–Mills problem)
$$\Phi_{Ad} \xrightarrow{S = 1.001} \begin{cases} \text{Faz escalar} \rightarrow \Phi_H & \text{(Campo de Higgs)} \\ \text{Faz tensorial} \rightarrow g_{\mu\nu} & \text{(Metrica gravitatoria)} \end{cases}$$

This is the observable Big Bang of the MCMC.

Implementation in Python

import numpy as np from scipy.integrate import quad class Pregeometria: """Block 1: Pre-geometry of the MCMC.""" def __init__(self): self.k0 = 6.307 self.a1 = 0.15 self.a2 = 0.25 self.a3 = 0.35 self.S4 = 1.001 self.Mp0 = 1.0 def k(self, S): """Collapse rate k(S).""" return self.k0 * (1 + self.a1 * np.sin(2 * np.pi * S) + self.a2 * np.sin(4 * np.pi * S) + self.a3 * np.sin(6 * np.pi * S)) def epsilon(self, S): """Normalized entropic integral.""" integral_S, _ = quad(self.k, 0, S) integral_S4, _ = quad(self.k, 0, self.S4) return integral_S / integral_S4 def Mp(self, S): """Primordial Mass as a function of S.""" return self.Mp0 * (1 - self.epsilon(S)) def Ep(self, S): """Primordial Space as a function of S.""" return self.Mp0 * self.epsilon(S) def P_ME(self, S): """Mass-space polarization.""" Mp = self.Mp(S) Ep = self.Ep(S) return (Mp - Ep) / (Mp + Ep)

📁 File: mcmc_core/bloque1_pregeometria.py

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