Mathematical Formulation of the Primordial Spin Coupling in Tetrachronic Geometry

1. Introduction

Within the framework of tetrachronic geometry, the origin of matter can be interpreted as the result of a harmonic coupling between two opposite spins.
This process —analogous to the principle of Yin and Yang— represents the transition from the energetic imbalance of the vacuum to the first stable unity of the tetradimensional spacetime: the seed of matter.

Such coupling not only produces a stable state but also generates an elementary four-dimensional volume, whose existence depends on both an energetic threshold and the phase coherence between the spin states.

2. Coupled State of Opposite Spins

Let the two opposite spin states be represented by complex phase functions:

The coupling of these two states defines the seed function of matter:

When the phases become equal, ϕ↑ = ϕ↓​, the system reaches its fundamental equilibrium:

This represents a neutral and stable four-dimensional state, where the spin components cancel each other, creating a perfect symmetry — the first matter as a dynamic balance of complementary forces, a quantum Yin–Yang.

3. Energetic Condition for Creative Coupling

Matter emerges from the spin coupling only if the energy of the system exceeds a specific tetradimensional quantum threshold:

where

Here:

• ℏg​ is the gradional constant, analogous to Planck’s constant but governing the quantization of four-dimensional space.

• Ω0​ is the minimum assembly frequency, the basal frequency of space itself.

From this relation follows the tetrachronic uncertainty principle:

where Δw represents the variation in the fourth spatial coordinate.
This inequality implies that matter formation requires a minimum action in four dimensions.

4. Generation of Tetradimensional Volume

In tetrachronic coordinates (r,θ,ϕ,w), the elemental volume generated by coupling is:

with

which corresponds to the surface area of the unit 3-sphere in four-dimensional space.

The four-volume becomes active when the coupling energy satisfies:

where εs​ denotes the minimum spatial energy density required to stabilize the 4D volume.

Thus, we can define the effective tetradimensional energy density as:

and the existence condition simplifies to:

5. Quantization of Assembly: The Gradion

Each successful coupling event generates a discrete quantum of space called a gradion, whose number is given by:

where η (0 < η ≤ 1) represents the assembly efficiency of the process.

The total assembled volume from Ng gradions is:

where Vg​ is the fundamental four-dimensional volume associated with a single gradion.

6. Tetrachronic Interpretation

This process can be visualized as a spiral of equilibrium between two counter-rotating spin flows, one right-handed (↑) and one left-handed (↓), interlacing along the fourth coordinate w.
Their interaction establishes a field of coherence in which energy compacts until reaching the threshold Ethreshold​, giving rise to the first manifestation of matter.

Mathematically, the stability of this coupling implies zero curvature in 4D hyperspace; physically, it represents the self-organization of the vacuum.

Thus, the dynamic balance of Yin and Yang finds its four-dimensional formulation:
the creation of space arises from the perfect coupling of two opposite spins that, upon uniting, rotate around a higher symmetry — the Tetracron.

7. Conclusion

The model presented here describes the genesis of matter as a coupling of opposite spins within the framework of tetrachronic geometry.
From the symmetry of phase and the energetic threshold, spatial quanta (gradions) emerge, assembling into the first four-dimensional volumes.

In this view, the universe can be conceived as a vast network of primordial couplings, where every point of space is not an absence, but the four-dimensional echo of the first equilibrium:

— the foundational act in which the quantum Yin–Yang gave birth to the Tetracron.