Warning to the Reader

This document presents the Theory of Sealed Universes (TUS) through a narrative construction. The reader is guided from a basic formalism toward the elaboration of an internal geometric model.

This progression aims to establish the coherence of the theory and reveal the unifying role of geometry. Each step is designed to logically follow from the previous one, presenting a complete and self-contained reasoning.

The Concept of a Sealed Universe

The Theory of Sealed Universes postulates that certain objects exhibiting intrinsic coherence can be modeled as a fundamental entity called a "Sealed Universe" (SU). An object’s coherence is defined by its ability to maintain its structure and energy autonomously and stably over time.

A Sealed Universe is not perceived as an inert entity but as a dynamic structure whose internal organization ensures its own permanence. This dynamics is confined within a volume that the theory designates as the "statistical sphere." It is within this sphere that the object’s energy is contained and its fundamental properties emerge.

Thus, TUS proposes to consider these objects not merely through their external interactions but through the internal logic that governs them. The integrity of the Sealed Universe is a direct consequence of this dynamics.

The Choice of the Electron as a Reference Sealed Universe

To initiate the exploration and formalization of the Theory of Sealed Universes, the electron emerges as a primary candidate due to several fundamental reasons rooted in its intrinsic properties and observed behavior:

1. Elementary Nature and Omnipresence: The electron is recognized as a stable, non-composite particle at our current observational scales. Its apparent simplicity and constitutive role in all matter make it foundational.

2. Quantifiable and Precisely Measured Properties: The electron possesses physical attributes defined with high precision: its rest mass (\(m_e = 9.109 \times 10^{-31}\) kg), elementary charge (\(e\)), spin (\(\hbar/2\)), and the reduced Compton radius (\(R_u = \hbar / (m_e \cdot c)\)) to which the electron is intrinsically linked. These provide a solid basis for characterizing the geometry of a Sealed Universe.

3. Evocative and Coherent Interaction Phenomena: The electron is involved in key interaction processes that suggest internal dynamics:

   - Electron-Positron Annihilation and Photon Emission:
   During annihilation with a positron, the electron participates in producing gamma photons of very specific energy.

   - Energy Restitution: The electron’s ability to re-emit energy it previously absorbed (e.g., as another photon) highlights an internal mechanism.

The Theory of Sealed Universes develops its formalism based on the electron and its well-documented properties.

Phenomenology of the Electron

An examination of critical experimental phenomena that provide fundamental insights into its dynamics.

Electron-Positron Annihilation

The first phenomenon to consider is electron-positron annihilation. When an electron (\(e^-\)) encounters its antiparticle, the positron (\(e^+\)), these particles mutually annihilate. This process involves the complete conversion of their mass into energy.

This energy typically manifests as the emission of two gamma photons. Each photon carries 511 keV of energy, corresponding precisely to the rest mass energy of an electron (or positron) via Einstein’s relation, \(E=mc^2\).

This phenomenon is remarkable for several reasons:

- Conservation of Energy and Momentum: The total energy of the initial particles is fully converted into photon energy. The emission of two oppositely directed photons ensures conservation of the system’s total momentum.

- Properties of Emitted Photons: The photons produced are not mere energy quanta. They travel at the speed of light (\(c\)), possess intrinsic spin (\(\pm\hbar\)), and exhibit helicity (alignment or anti-alignment of spin with motion). Their wavelength is precisely determined by their energy (511 keV).

Photon Emission Following Electron Excitation

The electron is involved in other energetic interactions, such as when a free electron at rest absorbs energy from an external source (e.g., a photon). The absorbed energy excites the electron, and the excess energy relative to its rest state is released via photon emission.
This process results in a photon with specific energetic properties (wavelength and frequency), demonstrating the link between absorbed and emitted energy.
Crucially, this process does not annihilate the electron, which returns to its rest-state characteristics.

Basic Mathematical Formalism of TUS

The Theory of Sealed Universes (TUS) is built on a set of fundamental quantities that characterize a Sealed Universe (SU) and their mathematical relationships. These definitions establish the initial quantitative framework for describing the coherence and dynamics of these entities. For a given SU, the basic properties are defined as follows:

- Characteristic Radius (\(R_u\)): Represents the statistical spatial scale of the SU. For the electron, it corresponds to the reduced Compton radius, in meters. \[R_u = \frac{\hbar}{m_e c} \label{eq:Ru}\] where \(\hbar\) is the reduced Planck constant, \(m_e\) is the rest mass of the electron, and \(c\) is the speed of light in a vacuum.

- Energy of the SU (\(E_u\)): The intrinsic energy of the Sealed Universe, directly related to its mass through the mass-energy equivalence relation, in Joules. \[E_u = m_e c^2 \label{eq:Eu}\]

- Characteristic Frequency (\(F_u\)): A fundamental frequency associated with the SU, dimensionless. \[F_u = \frac{c}{R_u} \label{eq:Fu}\]

- Proper Time (\(T_u\)): The period associated with the characteristic frequency of the SU, dimensionless. \[T_u = \frac{1}{F_u} = \frac{R_u}{c} \label{eq:Tu}\]

From these quantities, fundamental relationships for the speed of light and the mass of the SU follow:

- Speed of Light (\(c\)), in m/s, as a function of SU properties: \[c = R_u F_u \label{eq:c_RuFu}\]

- Mass of the SU (\(m_u\)), in kg, as a function of its properties: \[m_u = \frac{E_u T_u^2}{R_u^2} \label{eq:mu}\]

From these relationships, the "steps" characteristic of the SU are defined, representing space and energy quanta within its structure:

- Space Step (\(S_u\)), in meters: Represents the fundamental space quantum of the SU. \[S_u = \frac{R_u}{F_u} = \frac{R_u^2}{c} \label{eq:Su}\]

- Energy Step (\(N_u\)), in Joules: Represents the fundamental energy quantum of the SU. \[N_u = \frac{E_u}{F_u} = E_u T_u \label{eq:Nu}\]

These relationships form the mathematical foundation from which the Theory of Sealed Universes develops its description of coherent entities. They enable linking the electron’s mass and energy properties to intrinsic length scales, time scales, and action quanta.

Fundamental Calculations and Energy Phenomenology of the Electron in TUS

This section applies the mathematical formalism of TUS to the electron, detailing the calculations of characteristic quantities for the Sealed Universe (SU) of the electron, then modeling the interaction of a free electron with a photon.

Calculation of TUS Quantities for the Electron

Using fundamental constants and properties of the electron, we calculate the characteristic quantities of the electron’s SU:

- Fundamental Constants:

- (\(c\)): \(2.99792458 \times 10^8\) m/s

- (\(\hbar\)): \(1.05457170 \times 10^{-34}\) J\(\cdot\)s

- (\(m_e\)): \(9.1093837 \times 10^{-31}\) kg

- Calculation of the Characteristic Radius (\(R_u\)): \[R_u = \frac{\hbar}{m_e c} = \frac{1.05457170 \times 10^{-34}}{9.1093837 \times 10^{-31} \times 2.99792458 \times 10^8} \approx 3.86159268 \times 10^{-13} \text{ m}\]

- Calculation of the SU Energy (\(E_u\)): \[E_u = m_e c^2 = 9.1093837 \times 10^{-31} \times (2.99792458 \times 10^8)^2 \approx 8.1871057 \times 10^{-14} \text{ J}\] ()

- Calculation of the Characteristic Frequency (\(F_u\)): \[F_u = \frac{c}{R_u} = \frac{2.99792458 \times 10^8}{3.86159268 \times 10^{-13}} \approx 7.7634426 \times 10^{20} \text{ Hz}\]

- Calculation of the Proper Time (\(T_u\)): \[T_u = \frac{1}{F_u} = \frac{R_u}{c} = \frac{3.86159268 \times 10^{-13}}{2.99792458 \times 10^8} \approx 1.2880886 \times 10^{-21} \text{ s}\]

- Calculation of the Space Step (\(S_u\)): \[S_u = \frac{R_u}{F_u} = \frac{R_u^2}{c} = \frac{(3.86159268 \times 10^{-13})^2}{2.99792458 \times 10^8} \approx 4.9744910 \times 10^{-34} \text{ m}\]

- Calculation of the Energy Step (\(N_u\)): \[N_u = \frac{E_u}{F_u} = E_u T_u = 8.1871057 \times 10^{-14} \times 1.2880886 \times 10^{-21} \approx 1.0545717 \times 10^{-34} \text{ J}\] The energy step \(N_u\) is numerically equivalent to the reduced Planck constant \(\hbar\).

From these quantities, fundamental relationships for the speed of light and the SU mass follow:

- Speed of Light (\(c\)) as a function of SU properties: \[c = R_u F_u\]

- Mass of the SU (\(m_u\)) as a function of its properties: \[m_u = \frac{E_u T_u^2}{R_u^2}\]

Interaction of a Free Electron with a Green Photon

Consider the absorption and restitution of energy by a free electron, using a green photon of 540 nm as an example.

Energy Absorption

The energy of a photon is given by \(E_{\text{photon}} = hf = hc/\lambda\). For a green photon with \(\lambda = 540 \text{ nm}\): \[E_{\text{green photon}} = \frac{6.62607015 \times 10^{-34} \times 2.99792458 \times 10^8}{540 \times 10^{-9}} \approx 3.679 \times 10^{-19} \text{ J}\]

When a free electron absorbs this energy, TUS suggests this modifies its characteristic radius \(R_u\). The number of "space-time steps" (or energy quanta) corresponding to this absorption is the ratio of the photon energy to the electron’s energy step: \[\text{Number of steps} = \frac{E_{\text{green photon}}}{N_u} = \frac{3.679 \times 10^{-19}}{1.0545717 \times 10^{-34}} \approx 3.4886 \times 10^{15}\] This absorption modifies the SU radius of the electron. The adapted radius (\(R_{u, \text{adapted}}\)) is calculated by subtracting the product of the number of steps and the space step \(S_u\) from \(R_u\): \[\begin{aligned} \text{Radius change} &= \text{Number of steps} \times S_u \\ &= 3.4886 \times 10^{15} \times 4.9744910 \times 10^{-34} \\ &\approx 1.735 \times 10^{-18} \text{ m} \end{aligned}\] The adapted SU radius becomes: \[\begin{aligned} R_{u, \text{adapted}} &= R_u - \text{Radius change} \\ &= 3.86159268 \times 10^{-13} - 1.735 \times 10^{-18} \\ &\approx 3.86157533 \times 10^{-13} \text{ m} \end{aligned}\] This modification leads to a new characteristic frequency (\(F_{u, \text{adapted}}\)): \[F_{u, \text{adapted}} = \frac{c}{R_{u, \text{adapted}}} = \frac{2.99792458 \times 10^8}{3.86157533 \times 10^{-13}} \approx 7.7634771 \times 10^{20} \text{ Hz}\] This frequency exceeds the rest frequency, indicating the excited state.

Energy Restitution

During energy restitution, the electron returns to its initial state by releasing the absorbed energy. The radius change (\(\Delta R_u\)) is the difference between the initial and adapted radii: \[\Delta R_u = R_u - R_{u, \text{adapted}} \approx 1.735 \times 10^{-18} \text{ m}\] The restituted energy (\(E_{\text{restituted}}\)) equals the number of steps multiplied by \(N_u\): \[E_{\text{restituted}} = \text{Number of steps} \times N_u = 3.4886 \times 10^{15} \times 1.0545717 \times 10^{-34} \approx 3.679 \times 10^{-19} \text{ J}\]

Comparison of Results and Role of Steps

Comparing absorbed and restituted energy:

- : \(\approx 3.679 \times 10^{-19}\) J

- : \(\approx 3.679 \times 10^{-19}\) J

The results demonstrate energy conservation, with absorbed energy equal to restituted energy.

This model highlights the central role of the "space step" (\(S_u\)) and "energy step" (\(N_u\)) in electron interactions. These quanta act as fundamental units through which the SU absorbs and releases energy. Energy exchanges occur in discrete multiples of these steps, resulting in quantized geometric changes (radius) of the SU. The transformation between photon energy and SU radius illustrates a mechanism for energy management within the Sealed Universe.

The Limit Case of Zero Radius and the Photon’s Radius

The formalism of the Theory of Sealed Universes (TUS), as developed, primarily applies to entities characterized by a non-zero radius \(R_u\). However, exploring the limit case where \(R_u \to 0\) is fundamental to understanding massless entities such as photons.

If a Sealed Universe (SU) were characterized by a radius \(R_u = 0\), the fundamental relationships of TUS take on special significance. According to the relation \(c = R_u F_u\), for the speed of light \(c\) to retain a finite and constant value, the frequency \(F_u\) of this SU must tend to infinity (\(F_u \to \infty\)). Consequently, the proper time \(T_u = 1/F_u\) of this entity must tend to zero (\(T_u \to 0\)). In this scenario, the absence of spatial (\(R_u = 0\)) and temporal (\(T_u = 0\)) confinement reflects a state of pure energy, devoid of rest mass. This limiting case is intrinsically linked to the description of photons in TUS.

When an electron emits a photon, its radius \(R_u\) decreases by an amount \(\Delta R_u\) equal to the product of the number of energy steps exchanged and the space step \(S_u\). This \(\Delta R_u\) represents the "loss of space" from the electron, manifesting as photonic energy. The wavelength \(\lambda_{\text{photon}}\) of the emitted photon is directly related to the expelled energy.

The ontology of TUS reveals a profound connection by considering an extreme theoretical scenario: one in which the electron transfers its entire rest mass energy (\(E_u = m_e c^2\)) into a single photon. In this conceptual limit, the emitted photon—though intrinsically a "Sealed Universe with \(R_u \approx 0\)"—would have energy \(E_{\text{photon}} = m_e c^2\). Its wavelength \(\lambda_{\text{photon}}\) would then be expressed as: \[\lambda_{\text{photon}} = \frac{hc}{m_e c^2} = \frac{h}{m_e c}\] The **reduced Compton radius** of the electron, \(R_{\text{Compton}}\), is defined by \(R_{\text{Compton}} = \hbar / (m_e c)\). Using the fundamental relationship between Planck’s constant and the reduced Planck constant (\(h = 2\pi\hbar\)), we can rewrite the photon’s wavelength as: \[\lambda_{\text{photon}} = \frac{2\pi\hbar}{m_e c} = 2\pi \cdot R_{\text{Compton}}\] Within TUS, the "phase radius" of a photon can be conceptualized as its reduced wavelength, \(\lambda_{\text{reduced}} = \lambda / (2\pi)\). For this hypothetical photon carrying the electron’s full energy, its phase radius becomes: \[R_{\text{photon,phase}} = \frac{\lambda_{\text{photon}}}{2\pi} = \frac{2\pi \cdot R_{\text{Compton}}}{2\pi} = R_{\text{Compton}}\] This remarkable coincidence suggests that even though the photon itself is an SU with negligible intrinsic spatial radius (\(R_u \approx 0\)) and vanishing proper time (\(T_u \approx 0\)), the spatial manifestation of its most fundamental interaction (when it results from the total energy conversion of a massive particle like the electron) is directly tied to the characteristic dimension of the **reduced Compton radius** of the electron. TUS thus establishes an ontological continuity between matter and pure energy, where the spatial properties of a massive entity are reflected in the wave properties of a massless entity under extreme conditions.

Toward an Internal Dynamics of the Electron: The Toric Model

Previous analyses, particularly the study of the photon emission limit case where its wavelength is directly linked to the electron’s reduced Compton radius, raise a fundamental question: What internal dynamics of the electron can account for such interactions while ensuring its own stability and intrinsic properties (mass, charge, spin)?

The fact that a photon can inherit spatial characteristics linked to the electron from which it originates (via \(R_{\text{Compton}}\)) suggests that the emission process is not a simple "loss" of energy but a "projection" of part of the electron’s spatiotemporal structure. Similarly, the absorption of a photon modifies the electron’s radius, indicating that its "sealed space" is responsive and reconfigures itself based on energy exchanges.

To explain this ability to emit photons with specific characteristics (wavelength, helicity, spin, and invariant speed) while maintaining its own energy, charge, and spin, the Theory of Sealed Universes proposes an internal statistical dynamics model: the **toric structure**.

In this model, the fine-structure constant \(\alpha\) is not merely an electromagnetic coupling factor; it emerges as a **fundamental geometric parameter of the torus**. It is defined by the ratio of the classical electron radius (\(r_e\)) to its reduced quantum radius (\(R_u\)): \(\alpha = r_e / R_u\). This relationship is critical for understanding the "degree of winding" or "compression" of the toric structure and how it governs its dynamics and interactions.

A torus provides a geometry capable of reconciling multiple requirements:

- Energy Confinement: The closed and wound topology of the torus, with compression governed by \(\alpha\), stably confines the electron’s energy, explaining its resilience and conservation of rest mass.

- Coherent Emission and Absorption: Variations in the SU radius (\(\Delta R_u\)) during interactions can be interpreted as changes in the "tension" or "compression" of the torus. Released or absorbed energy manifests as a geometric reconfiguration, generating the "space steps" and "energy steps."

- Transmission of Photon Characteristics: The wave nature of emitted photons—including their spin, helicity, and speed \(c\)—is directly tied to rotational and winding dynamics within the electron’s torus. A specific oscillation or perturbation of the toroidal surface can "project" a helical wave (the photon), whose properties (wavelength, helicity, spin) reflect the characteristics of this perturbation.

- Stability of the Electron’s Spin: The electron’s \(1/2\) spin can be naturally interpreted as an intrinsic topological property of the torus, requiring a double rotation (\(4\pi\)) to return to its initial state, consistent with the spinorial nature of fermions.

This toric model transcends a mere description of interactions to propose an underlying geometric mechanism for particle stability and energy exchanges. The geometry, intrinsically linked to \(\alpha\), generates the "space steps" and "energy steps" through deformation, ultimately driving emission and absorption phenomena.

Geometry of the Torus and the Role of \(\alpha\)

The toric model, proposed to explain the internal dynamics of the electron, is not a mere analogy. Its geometry is strictly constrained by fundamental constants, providing a mathematical framework to derive the particle’s properties. This section formalizes this geometry and reveals the crucial role of the fine-structure constant \(\alpha\), not as a simple coupling strength, but as a **fundamental geometric parameter** governing the discretization of the Sealed Universe (SU).

Parameters of the Quantum Torus

Within the TUS framework, the electron’s torus is defined by two fundamental radii:

- Major Radius (Principal Radius): Corresponds to the characteristic radius of the Sealed Universe, the reduced Compton radius \(R_u = \hbar / (m_e c)\). It defines the global scale of the particle’s energy confinement.

- Minor Radius (Cross-Sectional Radius): Identified with the classical electron radius, \(r_e = e^2 / (4\pi\epsilon_0 m_e c^2)\). It represents the scale of intrinsic electromagnetic interaction.

From the relationship between these scales emerges a profound geometric interpretation of the fine-structure constant, \(\alpha\): \[\alpha = \frac{r_e}{R_u} = \frac{e^2}{4\pi\epsilon_0 \hbar c}\] Thus, \(\alpha\) is no longer merely a measure of electromagnetic interaction strength but a **universal geometric ratio**. It quantifies the degree of compression of the electromagnetic scale (\(r_e\)) within the quantum confinement scale (\(R_u\)).

\(\alpha\) as a Principle of Discretization and Confinement

The role of \(\alpha\) as a geometric parameter becomes explicit in describing the path followed by energy within the torus. This path requires a double rotation (\(\theta\) varying from \(0\) to \(4\pi\)) to return to its initial state, consistent with the electron’s spinorial nature (spin 1/2). The parametric equations for this toroidal path are: \[x(\theta, \phi) = (R_u + r_e \cos\phi) \cos(\alpha\theta) \\ y(\theta, \phi) = (R_u + r_e \cos\phi) \sin(\alpha\theta) \\ z(\theta, \phi) = r_e \sin\phi\] The term \(\alpha\theta\) in the trigonometric functions indicates that \(\alpha\) directly governs the "winding" of the path around the torus.

This geometric formalism links the torus structure to the fundamental quanta of TUS. The **space step** (\(S_u\)), representing the elementary spatial quantum of the electron, can be derived directly from this geometry. By defining a "fundamental compression length" of the torus, \(\lambda_{\text{comp}} = 2\pi R_u / \alpha\), we recover the value of \(S_u\): \[S_u = \frac{\alpha^2 \lambda_{\text{comp}}^2}{4\pi^2 c} = \frac{\alpha^2 (2\pi R_u / \alpha)^2}{4\pi^2 c} = \frac{R_u^2}{c}\] This derivation is a major corroboration: it shows that the space step \(S_u\), initially defined by the base formalism (\(S_u = R_u / F_u\)), is not an arbitrary postulate but an **emergent property of the toroidal geometry**, itself dictated by \(\alpha\). The fine-structure constant is thus the organizing principle of the internal discretization of the Sealed Universe.

Creation Time and Topological Stabilization

High-energy photon collision experiments producing electron-positron pairs measure an initial creation time of \(6.44 \times 10^{-22}\) s, precisely half the electron’s proper time according to TUS (\(T_u = 1.288 \times 10^{-21}\) s).

The path requires a double rotation (\(\theta\) varying from \(0\) to \(4\pi\)) to restore the initial state, consistent with the electron’s spinorial nature (spin 1/2).

The experimentally observed time (\(6.44 \times 10^{-22}\) s) corresponds to the first phase of pair creation—the initial emergence of structure—while the full proper time (\(T_u\)) includes the additional period required for complete topological stabilization of the toroidal structure: - The first half-cycle (\(6.44 \times 10^{-22}\) s) represents the initial creation of the electron-positron pair. - The second half-cycle corresponds to the full stabilization of the internal toroidal structure.

Without this double rotation, the electron would lack its spin.

Geometric Confinement Energy

The electron’s stability and resistance to decay arise from a colossal energy barrier originating from the torus’s geometric "tension." The energy is confined by an effective curvature with radius \(R_{\text{eff}} = \alpha R_u\). The resulting confinement energy, \(E_{\text{conf}}\), is purely geometric: \[E_{\text{conf}} = \frac{\hbar^2}{2m_e (R_{\text{eff}})^2} = \frac{\hbar^2}{2m_e (\alpha R_u)^2} = \frac{m_e c^2}{2\alpha^2}\] This relation defines the energy ensuring the particle’s integrity solely from its rest mass energy and the geometric constant \(\alpha\).

Numerical calculation of this energy reveals an immense protective barrier: \[E_{\text{conf}} = \frac{0.511 \text{ MeV}}{2 \times (1/137.036)^2} \approx \frac{0.511 \text{ MeV}}{2 \times 5.325 \times 10^{-5}} \approx 4.79 \text{ GeV}\] This confinement energy of nearly **4.8 GeV** is approximately 10,000 times greater than the electron’s rest mass energy. It is non-releasable and represents the torus’s topological "rigidity," making the electron exceptionally stable and resilient.

Modeling the Toric Projection During Photon Emission

The toric model of the electron, where the fine-structure constant \(\alpha\) acts as a fundamental geometric parameter, describes photon emission not as a simple energy loss but as a **geometric projection**. This process is viewed as a dynamic reconfiguration of the torus, which "projects" a helical wave (the photon) whose properties directly reflect modifications to the electron’s internal structure.

The characteristics of this projection are calculated using the example of a green photon (540 nm) emitted by an excited electron, as previously introduced.

State of the Electron and Torus Reconfiguration

According to the formalism, photon emission is the process by which an electron in an excited state (characterized by a reduced radius \(R_{u,\text{adapted}}\)) returns to its rest state (radius \(R_u\)).

- Excited State (pre-emission): The electron has absorbed the energy of a green photon (\(E_{\text{photon}} \approx 3.679 \times 10^{-19}\ \text{J}\)). Its torus is in a "compressed" state, characterized by a reduced major radius: \[R_{u,\text{adapted}} \approx 3.86157533 \times 10^{-13}\ \text{m}.\]

- Rest State (post-emission): The electron returns to its stable ground state with a major radius: \[R_u = \frac{\hbar}{m_e c} \approx 3.86159268 \times 10^{-13}\ \text{m}.\]

- Geometric Variation (\(\Delta R_u\)): The transition between these states corresponds to a "relaxation" of the torus. The variation in its major radius is: \[\Delta R_u = R_u - R_{u,\text{adapted}} \approx 1.735 \times 10^{-18}\ \text{m}.\] This variation is, as previously calculated, the product of the number of "steps" exchanged and the "space step" \(S_u\).

Characteristics of the Geometric Projection (the Photon)

TUS postulates that this geometric variation \(\Delta R_u\) is not abstract; it materializes in space as the emitted photon. The photon’s characteristics are thus a direct projection of the torus reconfiguration.

- Energy of the Projection: The projected energy is the energy restituted by the electron, corresponding to the green photon’s energy: \[E_{\text{restituted}} = \text{Number of steps} \times N_u \approx 3.679 \times 10^{-19}\ \text{J}.\]

- Spatial Extension of the Projection (Wavelength): The photon’s wavelength is the spatial manifestation of this energy: \[\lambda_{\text{photon}} = \frac{hc}{E_{\text{restituted}}} = 540\ \text{nm}.\]

- Phase Radius of the Projection: Following the limit case analysis, we define a "phase radius" for the photon. This is not a material radius but the characteristic radius of its wave structure: \[R_{\text{photon,phase}} = \frac{\lambda_{\text{photon}}}{2\pi} = \frac{540 \times 10^{-9}}{2\pi} \approx 8.59 \times 10^{-8}\ \text{m}.\]

Synthesis: The Torus Projection as a Unified Geometric Event

Photon emission by the electron is modeled as a unique, coherent event:

1. Cause: A **topological relaxation** of the electron’s torus, transitioning from a "compressed" major radius \(R_{u,\text{adapted}}\) to its stable radius \(R_u\). The quantized variation is \(\Delta R_u\).

2. Effect: The **projection of a helical perturbation** arising from the torus’s winding dynamics (governed by \(\alpha\)). This projection is the photon.

3. Correspondence: A direct correspondence exists between the electron’s geometry and the photon’s properties: the torus radius variation (\(\Delta R_u\)) is the energy source, while the photon’s wave properties (\(\lambda\), \(R_{\text{photon,phase}}\)) manifest this variation. The photon’s helicity and spin inherit the torus’s intrinsic rotational dynamics.

In conclusion, the calculation confirms the model’s coherence. The "torus projection" is not a mere metaphor but a quantifiable mechanism where modification of an internal geometric parameter (\(\Delta R_u\)) deterministically generates the emitted photon’s physical and wave properties.

Electron-Positron Pair Annihilation

The annihilation process of an electron-positron pair, within the framework of TUS, is a topological collapse resulting from the interaction of two mirror Sealed Universes (SUs). This mechanism unfolds through several geometric stages that lead to the conversion of mass energy and confinement energy into distinct products.

Approach and Synchronization

The electron and positron are identical toric structures but with opposite internal dynamics. The attractive interaction due to their opposite charges brings them closer together. When their mutual distance reaches the order of magnitude of the reduced Compton radius (\(R_u\)), their internal characteristic frequencies (\(F_u = c/R_u\)) synchronize through geometric resonance. The phases of the two tori then align in perfect opposition (\(\theta_{\text{electron}} = -\theta_{\text{positron}}\)).

Torus Interlocking and Topological Collapse

Once synchronized, the two tori interlock. Their opposite helicities, corresponding to their respective spins, enable a superposition that neutralizes their internal dynamics. This interlocking cancels the effective curvature of the tori (\(R_{\text{eff}} = \alpha R_u\)). The cancellation of this curvature triggers the collapse of the topological structure that ensured the stability and confinement of each particle.

Energy Release and Annihilation Products

The topological collapse releases the entire energy contained in the two particles in two distinct forms.

Photon Emission (Conversion of Mass Energy)

The rest mass energy of each particle (\(E_u = m_e c^2 \approx 511 \text{ keV}\)) is converted into a photon. Annihilation thus produces two 511 keV photons, emitted in opposite directions to conserve the system’s momentum. The helicity and spin of these photons are a direct inheritance of the internal dynamics of the tori from which they originate.

Space-Time Wave (Conversion of Confinement Energy)

The geometric confinement energy, which is significantly greater than the mass energy, is also released. For each particle, this energy is given by the relation: \[E_{\text{conf}} = \frac{m_e c^2}{2\alpha^2} \approx 4.78 \text{ GeV}\] Annihilation therefore releases a total confinement energy of \(2 \times 4.78 = 9.56 \text{ GeV}\). This energy is not converted into particles but is returned to the quantum vacuum in the form of a space-time wave. The characteristic wavelength of this wave is: \[\lambda_{\text{conf}} = \frac{hc}{E_{\text{conf}}} = \frac{1240 \text{ MeV·fm}}{4780 \text{ MeV}} \approx 0.26 \text{ fm}\] This process describes annihilation as a complete topological transition, where the confined structure of the torus converts into a free wave (photon) and a vacuum perturbation (space-time wave).

Corroboration by Detailed Experimental Analogies

The postulate of confinement energy conversion into a space-time wave is corroborated by quantitative and mechanistic analogies from several experimental domains.

Resonance in Particle Physics (LEP/CERN)

The observed phenomenon is the existence of a resonance in photon-photon collisions at a center-of-mass energy on the order of 10 GeV. The TUS model proposes a geometric interpretation where this energy corresponds to the threshold required to excite and collapse a pair of electron-positron tori. The calculation of this total energy is: \[E_{\text{total}} = 2 \times (m_e c^2 + E_{\text{conf}}) = 2 \times (0.000511 \text{ GeV} + 4.78 \text{ GeV}) \approx 9.58 \text{ GeV}\] This value calculated by TUS is in direct quantitative agreement with the energy scale of the observed phenomenon, supporting the hypothesis of a topological resonance.

Dissolution of Knots in Condensed Matter (BEC)

The observation is the dissolution of topological knots in a Bose-Einstein condensate (BEC), which releases its structural energy in the form of matter waves. This process is a direct analog of topological collapse in TUS. The comparison of wavelength ratios is particularly relevant:

- TUS: The ratio between the confinement wave wavelength and the Compton wavelength is \(\lambda_{\text{conf}} / \lambda_{\text{Compton}} \approx 6.6 \times 10^{-4}\).

- BEC: The ratio between the emitted matter wave wavelength and the characteristic sound wavelength in the condensate is \(\lambda_{\text{obs}} / \lambda_{\text{sound}} \sim 10^{-3}\).

The fact that these two dimensionless ratios are of the same order of magnitude indicates that the underlying physical mechanism is the same: the conversion of topological confinement energy into a wave.

Annihilation in Materials Science (Graphene)

The observed phenomenon is the annihilation of an electron-hole pair in graphene, which releases its energy simultaneously in the form of a photon and a phonon (acoustic wave). This two-channel mirror mechanism corroborates the TUS annihilation model:

- The photon emitted in graphene is analogous to the conversion of mass energy (\(m_e c^2\)) into a photon in TUS.

- The phonon (vibrational wave of the lattice) is analogous to the conversion of confinement energy (\(E_{\text{conf}}\)) into a space-time wave.

By adapting the TUS confinement energy formula to graphene with its effective parameters (fine-structure constant \(\alpha_g \approx 0.3\)), we obtain a confinement energy for the quasiparticle on the order of \(E_{\text{conf,g}} \approx 0.42 \text{ eV}\). This value is consistent with the energy scale of high-energy phonons measured in these processes.

In conclusion, these three detailed and quantified analogies, covering systems ranging from elementary particles to quasiparticles in materials, demonstrate that the mechanism of converting topological confinement energy into a wave is a universal and measurable physical principle. This provides indirect but strong experimental support for the central postulate of TUS regarding the nature of annihilation.

The Electron’s Charge

Geometric Origin of Charge

In the Theory of Sealed Universes (TUS), the electron’s charge is not a fundamental constant but an emergent property of the toric structure. This structure is defined by two fundamental scales:

- Reduced Compton Radius: \[R_u = \dfrac{\hbar}{m_e c} \approx 3.86 \times 10^{-13}~\text{m}\]

- Classical Electron Radius: \[r_e = \dfrac{e^2}{4\pi\epsilon_0 m_e c^2} \approx 2.82 \times 10^{-15}~\text{m}\]

These quantities are linked by the fine-structure constant: \[\alpha = \frac{e^2}{4\pi\epsilon_0 \hbar c} = \frac{r_e}{R_u} \approx \frac{1}{137.036}\]

This geometric ratio \(\alpha\) plays a central role: it represents the degree of compression of the electron’s toroidal energy structure. Thus, the charge \(e\) is not a primary entity but a derived quantity from the internal geometry of the Sealed Universe (SU): \[e = \sqrt{4\pi\epsilon_0 \hbar c \, \alpha}\]

Manifestation of a Dynamic

Charge emerges from the torus’s internal dynamics. A closed helical energy flux generates a stable spatial asymmetry. This perpetual motion creates a structured pressure on the quantum vacuum, perceived externally as a field.

The charge’s sign is determined by the torus’s chirality: - Electron: Left-handed winding \(\Rightarrow\) charge \(-e\) - Positron: Right-handed winding \(\Rightarrow\) charge \(+e\)

This topological duality naturally explains matter-antimatter symmetry and the discrete quantization of charge.

Geometric Confinement Energy

The electron’s stability is ensured by an energy barrier linked to the torus’s geometry. The confinement energy is given by: \[E_{\text{conf}} = \frac{m_e c^2}{2\alpha^2} \approx \frac{0.511~\text{MeV}}{2 \times (1/137)^2} \approx 4.78~\text{GeV}\]

This energy, inaccessible to conventional collisions, corresponds to the torus’s topological rigidity. It is released only during electron-positron annihilation, where the two structures interlock and collapse.

Attraction and Repulsion

Electrostatic interaction is not a primitive force but results from geometric compatibility between two tori: - Electron + Positron: Opposite chiralities \(\Rightarrow\) complementary vacuum curvatures \(\Rightarrow\) interlocking \(\Rightarrow\) attraction. - Electron + Electron: Identical chiralities \(\Rightarrow\) curvature conflict \(\Rightarrow\) repulsion.

This mechanism is regulated by \(\alpha\), which defines the torus’s effective curvature scale. The geometric confinement radius is: \[r_{\text{eff}} = \alpha R_u = \frac{e^2}{4\pi\epsilon_0 m_e c^2} = r_e \approx 2.82 \times 10^{-15}~\text{m}.\]

At this scale, quantum vacuum deformation is maximal. The effective interaction range emerges at larger distances, where Coulomb’s law \(F \propto 1/r^2\) approximates this geometric interaction.

Experimental Corroboration

The geometric charge model finds indirect support in observed phenomena:

Annihilation in Graphene

In graphene, electron-hole pair annihilation simultaneously releases a photon (light) and a phonon (acoustic wave). This dual emission mirrors the TUS mechanism: conversion of topological confinement energy into two waves, one being a spacetime wave (phonon analog).

10 GeV Resonance at LEP

The LEP collider observed a resonance in \(\gamma\gamma \to e^+e^-\) collisions near 10 GeV. The TUS model predicts: \[E_{\text{total}} = 2 \times (m_e c^2 + E_{\text{conf}}) = 2 \times (0.000511 + 4.78)~\text{GeV} \approx 9.58~\text{GeV}\] This aligns remarkably with data, corresponding to the geometric excitation threshold of \(e^\pm\) tori.

Bose-Einstein Condensates (BEC)

In some BECs, stable quantized vortices emerge with circulation \(\kappa = h/m\). This analogous behavior—closed wave, chirality, topological stability—supports toric dynamics as the source of quantized properties.

Conclusion

Charge is a direct consequence of the Sealed Universe’s geometry, dynamics, and causal discretization. It illustrates the principle: All fundamental physical properties are projections of self-coherent geometric structures. The constant \(\alpha\) reveals the deep granularity of spacetime itself.

Antimatter

Matter and antimatter are two manifestations of fundamental entities whose internal structure is defined by a closed toroidal geometry. Each particle possesses a characteristic radius \(R_u = \hbar/(m_e c)\), corresponding to the reduced Compton radius, and an internal dynamics governed by a proper frequency \(F_u = c / R_u\). The fine-structure constant \(\alpha = r_e / R_u\), where \(r_e\) is the classical electron radius, determines the degree of winding of this structure.

The electron and positron are tori with identical geometry but opposite chirality. One exhibits left-handed winding, the other right-handed winding. This chirality difference manifests as opposite electric charges. When an electron and positron interact, their internal dynamics (of opposite direction) can align through geometric resonance. Their tori interlock, their effective curvatures (\(R_{\text{eff}} = \alpha R_u\)) cancel out, and the topological confinement structure disappears.

This collapse process releases two distinct forms of energy.

The first is the rest mass energy of each particle, \(E_u = m_e c^2 \approx 511\,\text{keV}\). It is converted into two 511 keV gamma photons, emitted in opposite directions to conserve momentum. These photons inherit the helicity and spin of the initial tori, their internal dynamics projecting as heliocentric waves.

The second is the geometric confinement energy, \(E_{\text{conf}} = m_e c^2 / (2 \alpha^2) \approx 4{,}78\,\text{GeV}\) per particle. This energy, related to the topological tension of the torus, is not converted into massive particles. It is returned to the quantum vacuum in the form of a spacetime wave, with characteristic wavelength \(\lambda_{\text{conf}} = h c / E_{\text{conf}} \approx 0{,}26\,\text{fm}\).

When a high-energy photon collides with an electron, it does not cause collapse. Even at energies exceeding 1 GeV, the interaction is limited to quantized energy exchange. The electron absorbs an integer number of quanta, modifying its characteristic radius by space steps \(S_u = R_u / F_u\). This modification is reversible: the electron restitutes the energy as a photon whose properties are determined by the geometric variation of the torus.

In contrast, high-energy photon-photon collisions reveal a critical threshold. In experiments conducted at LEP, a resonance is observed when the center-of-mass energy reaches approximately 10 GeV. At this threshold, electron-positron pair production becomes significant. The available total energy corresponds to \(2 \times (m_e c^2 + E_{\text{conf}}) \approx 9{,}58\,\text{GeV}\), in direct agreement with the predicted sum of mass energy and confinement energy.

This threshold is not the mass creation threshold (which occurs at 1.022 MeV), but the geometric excitation threshold of the system. It marks the necessary condition for topological collapse of a forming pair, where the two tori—though separated—reach a critical curvature configuration.

In other systems, analogous phenomena are observed. In condensed matter physics, the dissolution of quantum vortices in a Bose-Einstein condensate releases a matter wave whose wavelength is proportionally related to the system’s coherence wavelength, similar to the \(\lambda_{\text{conf}} / \lambda_{\text{Compton}}\) ratio. In graphene, electron-hole pair annihilation simultaneously produces a photon and a phonon, reflecting a dual emission channel where structural energy is converted into a collective lattice wave.

These observations, though made in different contexts, share a common structure: the release of topological confinement energy as a wave when a closed structure collapses.

The collapse of an electron-positron pair is not merely the conversion of mass to radiation. It involves the disappearance of a self-coherent spatiotemporal structure, whose geometric signature persists in the emitted spacetime wave. This wave, though not directly measurable, manifests through its global effect: an injection of energy into the vacuum.

If such annihilations occur on a large scale, or continuously through virtual pairs, the accumulation of these spacetime waves could contribute to the vacuum energy density. This contribution, homogeneous and non-local, exhibits characteristics similar to those attributed to dark energy. The release of confinement energy thus appears as a microscopic process that could have macroscopic effects on the evolution of the universe.

The distinction between matter and antimatter is not a fundamental asymmetry, but a topological duality necessary for causal stability. Annihilation is not destruction, but a transition where the closed structure yields to a free wave and a vacuum perturbation. Chirality, confinement, and energy release are linked by the same geometry, whose fundamental constants—\(c\), \(\hbar\), \(\alpha\)—express invariant ratios.

Electron-Positron Pair Generation

Creation Conditions: The Breit-Wheeler Process

The generation of an electron-positron pair (\(e^-e^+\)) from light, known as the Breit-Wheeler process, describes the collision of two energetic photons (\(\gamma\)).

Opposite Polarization Condition

The interaction is strongly dependent on the photons’ helicity (\(\lambda\)), requiring opposite polarizations to be effective: \[\gamma_1(\lambda = +1) + \gamma_2(\lambda = -1) \rightarrow e^- + e^+\] This configuration is necessary for the conservation of angular momentum and to allow for the topological entanglement of the fluxes.

Experimental Corroboration

This process has been experimentally confirmed.

• Direct Observation (STAR, 2021): The STAR experiment at RHIC performed the first clear observation of the Breit-Wheeler process by using the photon fields surrounding heavy ions.

• Polarization Dependence: The experiment also validated the strong dependence of the interaction on photon polarization, thereby corroborating a fundamental premise of the geometric model.

Toroidal Stabilization Mechanism

The geometric model explains how the particles acquire their stability.

Formation of the Double Torus

The two photons, modeled as open helical waves, entangle to form two conjugate tori of opposite chirality: \(\chi = -1\) for the electron and \(\chi = +1\) for the positron.

Parametric Formalization of the Torus

Modeling the energy path within the electron’s torus with the following parametric equations: \[\begin{aligned} x(\theta, \phi) &= (R_u + r_e \cos\phi) \cos(\alpha\theta) \\ y(\theta, \phi) &= (R_u + r_e \cos\phi) \sin(\alpha\theta) \\ z(\theta, \phi) &= r_e \sin\phi \end{aligned}\]

• \(R_u\) is the major radius of the torus (reduced Compton radius).

• \(r_e\) is the minor radius (classical electron radius).

• The factor \(\alpha\) in the cosine and sine functions directly governs the degree of winding of the path around the torus.

• The spinorial nature of the electron (spin 1/2) is explained by the necessity of a double rotation (\(\theta\) varying from 0 to \(4\pi\)) for the structure to return to its initial state.

Geometric Stability and Uniqueness of \(\alpha\)

The stability of the tori is constrained by the fine-structure constant, \(\alpha\), interpreted as a geometric ratio. The parameters of the torus are: \[\begin{aligned} R_u &= \frac{\hbar}{m_e c} \quad \text{(Major radius)} \\ r_e &= \frac{e^2}{4\pi\varepsilon_0 m_e c^2} \quad \text{(Minor radius)} \\ \alpha &= \frac{r_e}{R_u} \end{aligned}\] The internal coherence of the torus, via its fundamental space step \(S_u\), leads to an identity that is only satisfied for the observed value of \(\alpha\). This implies that \(\alpha\) is a necessary and non-arbitrary geometric constant.

Specific Predictions

The geometric model leads to precise quantitative predictions.

The Topological Energy Threshold

An energy threshold required for the complete creation of the topological structure, including the geometric confinement energy (\(E_{\text{conf}}\)). \[E_{\text{total}} = 2 \times (m_ec^2 + E_{\text{conf}}) = 2 \times \left(m_ec^2 + \frac{m_ec^2}{2\alpha^2}\right) \approx 9.58 \ \text{GeV}\] This threshold corresponds to resonances observed at LEP (CERN).

The Characteristic Creation Time

The topological stabilization time of the pair is half the "proper time" of the electron, which is: \[\Delta t_{\text{creation}} = \frac{T_u}{2} = \frac{1}{2} \frac{R_u}{c} \approx 6.44 \times 10^{-22} \ \text{s}\]

Observability of the Toric Model

The Theory of Sealed Universes (TUS) postulates that the electron’s fundamental properties emerge from the dynamics of an internal toric structure. However, high-precision experiments depict the electron as a point-like entity with no measurable spatial structure. This section demonstrates that this apparent contradiction is not a refutation of the model, but a predictable consequence of the nature of measurement at the quantum scale. The model’s validity does not rely on direct observation, but on a constellation of indirect evidence and testable predictions.

Fundamental Limits to Direct Observation

Three primary reasons explain why the toric structure is not "visible" in the classical sense.

- Scale and Resolution: The postulated geometry is defined by extraordinarily small dimensions: a major radius \(R_u \approx 3.86 \times 10^{-13}\) m and a minor radius \(r_e \approx 2.82 \times 10^{-15}\) m. Any probe capable of resolving such scales (e.g., a photon) would need to possess such high energy (\(E \gg 1\) GeV) that it would so profoundly alter the electron’s dynamics that discerning its internal structure would become impossible. The interaction itself, rather than the particle’s rest geometry, would become the sole object of measurement.

- Statistical Averaging and Ultra-fast Dynamics: The torus is not a static structure but hosts dynamics oscillating at the characteristic frequency \(F_u \approx 7.76 \times 10^{20}\) Hz. Any experimental interaction, occurring over a duration much longer than the electron’s proper time (\(T_u \approx 10^{-21}\) s), performs an average over billions of cycles. The resulting observable is therefore not the torus itself, but the averaged effect of its dynamics, contained within a statistical sphere that exhibits apparent spherical symmetry.

- Unified Geometric Structure: Unlike a composite proton, the electron’s torus is a unified and indivisible geometric structure. During a high-energy collision, one interacts with the entirety of this dynamic structure at once, without "breaking" it into subcomponents, which reinforces the interpretation of a point-like signature.

Indirect Evidence and Experimental Signatures

If the torus is invisible, its existence is supported by its ability to unify several observed phenomena that would otherwise remain disjointed postulates.

- Spin 1/2 as a Topological Property: The electron’s intrinsic spin is one of the strongest indicators. The toric model provides a geometric origin: it is a direct consequence of the torus topology, which requires a \(4\pi\) rotation (a double turn) to return to its initial state—precisely the mathematical definition of a spinor.

- Fine-Structure Constant \(\alpha\) as a Geometric Ratio: TUS confers physical meaning to the ratio between the classical radius (\(r_e\)) and the reduced Compton radius (\(R_u\)) of the electron, postulating \(\alpha = r_e/R_u\). Thus, \(\alpha\) is no longer merely a coupling constant but the geometric imprint of the torus’s "compression."

- Confinement Energy and Annihilation: The model predicts the existence of a colossal geometric confinement energy, given by \(E_{\text{conf}} = \frac{m_e c^2}{2\alpha^2} \approx 4.79 \, \text{GeV}\). TUS postulates that this energy is released during annihilation. The total energy for a pair (\(2 \times (m_e c^2 + E_{\text{conf}})\)) is approximately \(9.58\) GeV, a value in quantitative agreement with the energy scale of resonances observed in photon-photon collisions at LEP/CERN.

Perspectives and Falsifiability of the Model

The strength of the toric model lies in its ability to generate specific, testable predictions, making it falsifiable. Among proposed experiments for validation:

- Annihilation Photon Interferometry: Analysis of phase correlations between photons from annihilation of electron-positron pairs with aligned spins could reveal signatures specific to torus synchronization.

- High-Field Spectroscopy: Extreme magnetic fields (\(B > 10^4\) T) should induce measurable torus distortion, causing predictable deviations in electron energy levels.

- Condensed Matter Analogies: Electron-hole annihilation in graphene, which produces both a photon and a phonon, serves as a testbed for the dual-channel mechanism (mass energy and confinement energy) predicted by TUS.

Conclusion: The non-observation of the toric structure is a prediction of the TUS model, not a refutation. The theory’s validity rests on the coherence of its explanations for already observed phenomena (spin, \(\alpha\)) and its capacity to be tested through future experiments.

C is Omnipresent

Attempt to Derive \(c\) from Observable Constants

The TUS formalism establishes a fundamental relationship between the electron’s parameters and universal constants. From this relationship emerges an expression for the speed of light: \[c = \frac{e^2}{4\pi\epsilon_0\alpha\hbar}\] At first glance, this equation appears to be a derivation of \(c\). Each constant on the right-hand side can be treated as a measurable quantity obtained through independent experiments:

- The elementary charge, \(e\).

- The vacuum permittivity, \(\epsilon_0\).

- The reduced Planck constant, \(\hbar\).

- The fine-structure constant, \(\alpha\), which is one of the most precisely measured dimensionless constants in physics.

Using the accepted experimental values for these four constants, the calculation yields the known value of \(c\) with extremely high precision. This numerical success creates the impression that \(c\) has been successfully derived from observables.

The Coherence Circularity

However, a deeper examination reveals a fundamental circularity. The question is not whether the calculation is correct, but *why* it is correct.

The reason the measured values of \(e\), \(\epsilon_0\), \(\hbar\), and \(\alpha\) combine to yield \(c\) is that these quantities are not physically independent. They are constrained by a physical law that connects them, and this law is precisely the definition of the fine-structure constant: \[\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c}\] This relationship is not merely a theoretical formula; it describes how our universe is structured. It holds true for both theoretical concepts and measured values.

The initial equation \(c = \frac{e^2}{4\pi\epsilon_0\alpha\hbar}\) is therefore merely an algebraic rearrangement of the definition of \(\alpha\). Using it to "derive" \(c\) amounts to demonstrating that \(c = c\).

Conclusion

The attempt to derive \(c\) from other observable constants does not produce an origin for \(c\). Rather, it demonstrates the extraordinary **internal coherence** of nature’s constants. This coherence is organized by fundamental relationships where \(c\) plays a central and inalienable role.

The impossibility of deriving \(c\) without presupposing it, even when using experimental values, is the conclusion of the demonstration and shows that \(c\) is not a consequence of other phenomena.

The Nature of \(c\)

The analysis of circularity in the derivation of \(c\) leads to a reinterpretation of its nature. TUS proposes that \(c\) is not an emergent constant, but a foundational principle that governs the very manifestation of energy.

The Fundamental Limitation and Discretization of Reality

TUS postulates that \(c\) is a fundamental and irreducible ratio of the universe. It is not a speed in the classical sense, but the value of an intrinsic "limitation" that conditions the existence of physical reality.

In this framework, **pure energy** is a potential that cannot exist or be expressed without manifestation. This manifestation occurs through a physical dimension—a length—which represents the discrete spatial expression of the cause-effect principle. This discretization is a direct consequence of the limitation of which \(c\) is the measure.

Thus, before being a speed, \(c\) is the fundamental ratio that establishes the relationship between length and time, enabling energy to unfold in an orderly manner.

\(c\) as the Organizing Principle of Reality

The constant \(c\) transcends its role as a physical constant to become the **organizing principle** that ensures the coherence of the universe.

- It is the **fundamental ratio** that guarantees a fixed and stable relationship between space and time, endowing the universe with its "orthonormal" structure and causality.

- It is the rule that allows pure energy to manifest in an orderly manner, as required by the universe’s "orthonormal arithmetic."

From this perspective, the fine-structure constant \(\alpha\) becomes the "sub-ratio" of \(c\)’s limitation. It reveals the **fundamental granularity** of energy’s deployment—the very translation of the universe’s discretization.

Emergence of Geometric and Causal Invariants

The Theory of Sealed Universes (TUS) proposes that fundamental constants such as \(\pi\) and \(c\) are not arbitrary parameters, but structural invariants that logically emerge from the necessity of constructing a coherent, orthonormal, and causal universe.

\(\pi\): The Invariant of Spatial Circularity

The emergence of \(\pi\) is illustrated by a geometric construction in an orthonormal Euclidean space, where proportions are preserved. The process consists of a sequence of fixed-length steps, each followed by a rotation of a constant angle \(\theta\). For the resulting trajectory to form a closed figure, such as a regular polygon, the sum of rotations must be a multiple of a complete turn.

The constant \(\pi\) thus appears as the unique structural ratio that makes circularity possible. It is not a postulated constant, but a factor of spatial circularity inherent to any geometric system with stable proportions.

\(c\): The Invariant of Temporal Causality

The emergence of \(c\) is demonstrated in a discrete space-time, composed of space steps (\(\Delta x\)) and time steps (\(\Delta t\)). The only rule imposed is that of local causality: information cannot traverse more than one unit of space for each unit of time.

This fundamental constraint defines an intrinsic maximum speed for the system: \[c = \frac{\Delta x}{\Delta t}\] This value is not the measured speed of an object, but the causality bound of space-time itself. It represents a homogeneity constant between space and time, the only one that guarantees a coherent causal sequence.

The Coherence of the Orthonormal Universe

The necessity of the value of \(c\) is reinforced by its unifying role between quantum physics and relativity. Planck’s relations (\(E=\hbar\omega\)) and De Broglie’s (\(p=\hbar k\)) are only mutually consistent with the relativistic relation \(E=pc\) if \(c = \omega/k = \Delta x / \Delta t\).

The appearance of the factor \(2\pi\) in the definition of frequencies (\(\omega = 2\pi/\Delta t\)) and wave numbers (\(k = 2\pi/\Delta x\)) is essential. It presupposes that cycles and waves are defined on an orthonormal geometric support where rotations are uniform.

In conclusion, the value of \(c\) is the unique ratio that simultaneously confers to the universe its properties:

- Causal, by setting the limit of information propagation.

- Quantified, by ensuring coherence with Planck’s constant \(\hbar\).

- Geometrically orthonormal, by being compatible with a geometry where cycles are defined by \(\pi\).

Genesis of the Universe

The cosmological progression of the universe’s genesis is the process of spacetime structuring through the establishment of the constant \(c\).

Before \(10^{-43}\) s: The Pre-Geometric Potential

At this primordial scale, the universe transitions from a pure energy potential where space and time are not yet distinct dimensions, to a first step toward causality. The fundamental relation \(c = \Delta x/\Delta t\), which had not yet been established, manifests within a set of one-dimensional lengths. The organizing principle of physical reality is in place.

From \(10^{-43}\) s to \(\sim 10^{-21}\) s: The Geometrization Phase and Emergence of \(c(t)\)

This period represents the crucial stage of geometrization. The universe does not immediately adopt its stable structure. The causal relation establishes progressively: \(c\) is not yet a constant, but a function of time, \(c(t)\), which converges toward its final stable value that we know (\(c_{ortho}\)). During this phase, spacetime "rigidifies." The physical laws as we conceive them are still forming, as their geometric foundation itself is evolving.

At \(6.44 \times 10^{-22}\) s: The Creation Threshold in a Non-Orthonormal Universe

This moment marks a critical stage. The value of \(c(t)\) enables the materialization of the first particle pairs. However, since the universe is not yet perfectly "orthonormal," the creation conditions for particles and antiparticles are not rigorously symmetric.

From \(6.44 \times 10^{-22}\) s to \(1.288 \times 10^{-21}\) s: Topological Stabilization and Convergence of \(c\)

During this brief window, two processes occur in parallel:

1. Existing particles complete their topological stabilization with the definitive acquisition of their spin characteristics and full confinement energy.

2. The constant \(c(t)\) completes its convergence toward its stable and definitive value, \(c_{ortho}\). The universe becomes orthonormal.

At the end of this period, all new pair creation will occur in a perfectly symmetric manner.

After \(1.288 \times 10^{-21}\) s: The Orthonormal Universe and its Inherited Imbalance

The universe enters its stable era, governed by fixed fundamental constants. Electrons have become complete and autonomous Sealed Universes. However, it inherits a particle imbalance forged during the geometrization phase. The observed matter-antimatter asymmetry does not originate from a violation of current physical laws, but is the relic of the formation of these laws themselves.

Causal Break and Domains with Distinct \(c\)

The model postulates that the value of the constant \(c\) observed in our universe is the necessary condition for its orthonormal and causal structure. From this perspective, a horizon (such as that of a black hole) is not a limit where physical laws cease to apply, but the boundary of a domain where the fundamental ratio \(c\) takes a different value from ours.

Incompatibility of Spacetime Frameworks

An observer, whose reality is structured by the value of \(c\) in our universe, cannot establish a coherent causal relationship with a region governed by a different \(c\). This incompatibility is fundamental:

- The succession rules of causes and effects are no longer shared.

- The spacetime structure is no longer mutually compatible.

Phenomena perceived from a distance, such as infinite time dilation or extreme spectral shift when approaching a horizon, are interpreted as manifestations of this structural incompatibility. They are symptoms of observing a causal system from another non-superimposable framework.

Loss of Relation and Causal Fragmentation

The direct consequence of this incompatibility is a complete break in the relationship. No information from the region with a different \(c\) can be coherently integrated by the observer. The causal connection is broken.

This implies that the universe can be conceived as a set of causally disjoint domains, each potentially defined by a distinct value of \(c\). Interaction or observation is only possible within the same domain, or between domains that strictly share the same ratio \(c\). Horizons mark the boundaries where this condition is no longer fulfilled.

Mathematics

The Logic of an "Orthonormal" Universe

The idea that our mathematics is intrinsically linked to an "orthonormal" universe, and that quantified observations are consequently irrefutable, stems from a central premise: the constant speed of light, \(c\), is not merely a measured speed but the fundamental principle that organizes reality itself.

\(c\) as the Organizing Principle

The theory posits that \(c\) establishes a fixed and stable relationship between space and time, which it describes as an "orthonormal" structure. This term is not used in the strict mathematical sense of perpendicular vectors, but rather as a conceptual term for a universe that is structured in a coherent, ordered, and causal manner. From this perspective, \(c\) is the fundamental rule, or "invariant of temporal causality," that allows energy to manifest in a discrete and ordered way. Without this principle, the universe would be an unstructured and acausal potential.

Mathematics as a Reflection of Reality

If our universe is fundamentally built upon this "orthonormal" and causal logic defined by \(c\), then any logical system we develop within it—namely mathematics and arithmetic—will inevitably reflect this underlying structure. Our mathematical systems work because they are "calibrated" to the very rules that govern the universe. The success and coherence of mathematics in describing the universe is not a coincidence; it is a consequence of the fact that mathematics is the product of a logically structured reality.

The Irrefutable Nature of Quantified Observation

This leads to the assertion that quantified physical observations are irrefutable. Given that:

1. The universe operates according to a coherent and "orthonormal" logic established by \(c\).

2. Our mathematics is a formal expression of this same logic.

3. A quantified observation is the result of applying our mathematical framework to a physical phenomenon.

Consequently, a precise measurement, such as the mass of an electron or the value of the fine-structure constant, represents a direct point of correspondence between the inherent logic of the universe and our description of it. These quantified values are considered "irrefutable" facts that any valid physical model must account for and explain. The only room for debate lies in the logical and mathematical construction that a model uses to arrive at and explain this observed value.

Description of a Universe with \(c = -c_{\text{ortho}}\)

The Central Argument

Our mathematics—arithmetic, geometry, and calculus—is not a universal abstraction but a product of the orthonormal universe defined by \(c > 0\). The causal structure of this universe (\(c = \Delta x / \Delta t\)) generates stable geometric invariants (for example, \(\pi\)) and quantization rules (for example, \(E = \hbar \omega\)). These invariants form the foundation of our mathematical-logical physics. Attempting to describe a universe with \(c = -c_{\text{ortho}}\) (where causality is reversed) forces our mathematics to operate outside its domain of validity. The result is not a coherent alternative physics, but a cascade of contradictions.

Mathematical Decomposition: Key Examples

a. The Schrödinger Equation (Acausal Collapse)

In our orthonormal framework, the Schrödinger equation describes causal evolution: \[i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi \quad (\text{where } t \text{ flows forward})\] For \(c < 0\), one might attempt to reverse time (\(t \to -t\)): \[i\hbar \frac{\partial \psi}{\partial (-t)} = \hat{H} \psi \quad \implies \quad -i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi\] This appears to rewrite physics for reversed causality. But the problem runs deeper:

- Contradiction: The complex unit \(i = \sqrt{-1}\) relies on our algebraic axioms (for example, \(i^2 = -1\)), which assume orthonormal spatial rotations. In a universe where \(c < 0\), rotations are acausal helices—no closed loop exists to define \(i\).

- Self-referential trap: Writing \(\partial / \partial t\) assumes a direction of time, contradicting the premise of reversed causality.

b. The Emergence of \(\pi\) (Geometric Invariant)

In our universe, \(\pi\) emerges from closed circular paths in orthonormal space: \[\text{Circumference} = 2\pi r \quad (\text{requires } c > 0 \text{ for spatial closure})\] For \(c < 0\):

- Paths are open helices: "circles" cannot form because spatial displacements correspond to negative time intervals.

- No \(\pi\): The circumference/radius ratio is undefined. Without \(\pi\), Fourier transforms (essential for quantization) and trigonometric functions collapse.

c. Planck-de Broglie Relations (Quantization Failure)

Our quantum relations assume \(c > 0\): \[E = \hbar \omega, \quad p = \hbar k, \quad \text{with } \frac{\omega}{k} = c > 0\] For \(c < 0\): \[\frac{\omega}{k} = -c_{\text{ortho}} \quad \implies \quad E = \hbar \omega, \quad p = -\hbar k\] This implies:

- Energy-momentum conflict: \(E\) and \(p\) have opposite signs, violating relativistic consistency \(E^2 = p^2c^2 + m^2c^4\).

- No wave-particle duality: Waves propagate backward in time, but "particles" cannot be localized without causal measurement.

The Self-Referential Trap

Any \(c < 0\) model constructed with our mathematics:

- Uses axioms derived from \(c > 0\) (for example, real numbers require causal ordering).

- Requires an "observer" whose cognition is tied to orthonormal logic.

- Paradox: Describing "retrocausality" (effect \(\rightarrow\) cause) in equations forces causes to precede effects in syntax. For example:

- Equation: "Effect at \(t_1\) = Cause at \(t_0\)" with \(t_1 < t_0\)

- Syntax: Symbols are written left to right (cause \(\rightarrow\) effect), incorporating our causality.

Conclusion: The Unknowable Universe

A universe with \(c = -c_{\text{ortho}}\) is mathematically unknowable from our framework. This is not a limitation of imagination, but a consequence of the TUS principle:

Mathematics is the formal language of orthonormality. It cannot transcend the causal structure that engendered it.

The irrefutability of quantified observations (for example, \(m_e = 9.109 \times 10^{-31} \, \text{kg}\)) stems from this alignment. Demanding that such values "make sense" in a universe where \(c < 0\) is to demand the impossible—like asking a computer program calibrated for integers to calculate with irrational numbers it cannot represent.

The only coherent conclusion is that universes with fundamentally distinct values of \(c\) are causally sealed domains. Our mathematics, confined by \(c_{\text{ortho}}\), cannot describe \(c < 0\) any more than a thermometer can measure darkness.

Physical reality and its mathematical description form a unified and sealed system—a coherent whole where orthonormality is both the rule and the proof.

Reality as Logical Projection

This geometric construction of reality’s nature leads to an extreme intellectual consequence that presents itself as logically coherent, provided its premises are accepted.

Content and Rule

The hypothesis rests on two minimal elements:

- The Content: Unstructured pure energy, existing as a potential of one-dimensional lengths.

- The Rule: A single causal relation, \(c = \Delta x/\Delta t\), applied to this content.

Geometrization and Emergence of Causality

The plausibility of the hypothesis rests on the transformation of 1D content into a structured and causal universe.

Geometric Transformation (1D \(\rightarrow\) 3D)

Consider a 1D length subjected to identical and opposing compression forces at its endpoints. Unable to shorten, it deforms in space to accommodate the energy. The most stable resulting form is a three-dimensional helix. The projection of this helix is a circle, thereby causing the constant \(\pi\) to emerge as a geometric ratio expressing an orthonormal space.

Emergence of Causality

Beyond geometry, this helical structure generates causality itself. The helix is not a set of random points, but an ordered sequence of elementary steps. The position of each step on the curve is rigorously determined by the previous step through a fixed angle that determines the next step.

This interdependence creates a necessary sequence; one turn of the helix can only be followed by the next, in an immutable order. This sequential order is the very manifestation of the causality principle. The direction of the helix (its winding sense) provides directionality to this sequence, which is interpreted as the emergence of an arrow of time.

The Nature of Reality

Pushed to its extreme, this hypothesis leads to considering that reality does not exist "in itself"—neither electron, nor living domain, nor universe.

The universe would then be merely the conceivability or projection of a logical sequence resulting from the application of a rule (\(c\)) to conceptual content.

This vision, though radical, presents itself as a coherent intellectual construction, whose non-refutability is conditioned upon accepting the transformation of a 1D potential into an ordered, causal 3D structure capable of producing a universe—real or conceptual—within the realm of possibilities.

Conclusion

The Theory of Sealed Universes has undertaken a deductive journey, starting from the postulate that a stable particle like the electron can be modeled as a coherent and autonomous entity: a Sealed Universe. The basic mathematical formalism, established from the measured properties of the electron, allowed for the quantification of this entity, but quickly raised the question of the internal dynamics necessary to explain its stability and interactions.

The answer to this question was the introduction of the **toric model**. This geometric structure, whose degree of winding is dictated by the fine-structure constant \(\alpha\), provided a mechanism for energy confinement, particle resilience, and photon emission. The torus is not a mere analogy, but a model whose geometric properties generate the quanta of space and energy that govern interactions.

The exploration of this model led to a deeper revelation regarding the nature of fundamental constants. The attempt to derive the speed of light \(c\) highlighted an inevitable circularity, demonstrating its status not as an emergent constant, but as a foundational **causal ratio**. Similarly, \(\pi\) emerged as a necessary **geometric invariant**. These constants are not arbitrary parameters of the universe; they are the logical rules that allow a causal and orthonormal universe to exist.

The culmination of this construction is the full meaning of the theory’s name. The Sealed Universe is the place where these abstract rules are embodied. The dynamics of the electron’s torus integrate and materialize the entire set of geometric and causal laws of the cosmos. Thus, each stable particle is not merely an object *in* the universe; it is a **microcosm** of this universe, a complete and autonomous system that contains within its own structure the totality of the principles that govern it.