The Invisible Structure of Time: Hidden Dimensions as Reality Support

1. Introduction

The traditional understanding of the universe is anchored in a three-dimensional model of space coupled to time. However, this model shows limitations when confronted with phenomena such as galactic rotation curves, which suggest the presence of invisible matter. This theory presents an alternative: that such effects are products of vector stresses accumulated in folded and hidden dimensional layers, whose presence stabilizes the visible space-time.

Contemporary physics, even in its most advanced forms, remains anchored in a reality model structured by three spatial dimensions and a temporal 3+1 model proposed by classical physics. This framework has been sufficient to describe the main phenomena of gravity, light and cosmic expansion. However, anomalies that challenge its completeness persist: the acceleration of the expansion of the universe, the rotational stability of galaxies and the absence of ontological explanations for what we call "vacuum". In this scenario, there is a need to reformulate not only the constituent elements of matter and energy, but the very structure of space-time.

The Theory of the Invisible Structure of Time and the Hidden Dimensions as Support of Reality arises as a structural and ontological response to these failures. Its fundamental principle is that the universe is not only what you see or measure directly, but what sustains - even invisibly. Thus, the visible space-time is treated as an apparent mesh, supported internally by folded vector layers, organized tension structures that remain hidden to the sensory and instrumental apparatus, but which can be simulated mathematically.

This theory is based on the postulate:

ω ε = 1

where ω represents the local vector angular frequency of a dimensional cell and ε_{-}ε represents its vector resistance to collapse. The product of an inverse vector constant symbolizes a dynamic balance between curvature and restraint. When one of the variables increases (as occurs at the edge of a galaxy or in an accelerated expansion), the other reacts compensatively to preserve dimensional integrity. This behavior can be interpreted as a vector mechanism of self-regulation of reality.

The universe, then, is seen as a system tensed by internal vector folds that contract, resist and expand not by chance, but by invisible structural laws. This folded structure not only supports the fabric of space-time - it is the fabric itself . What we see as continuous and flat is actually the average expression of a folded and compressed system.

To broaden the understanding of these concepts, we use physical and geometric analogies that make it possible to visualize the logic of TEITDO as Support of Reality:

• The corrugated paper, whose strength is not in the thickness of the paper, but in the trapped air and internal folds. Just like the vacuum of TEITDO as Reality Support, corrugated paper is seemingly fragile but structurally stable due to its invisible architecture.
• The pomegranate, fruit that expresses a fractal, radial and densified internal organization. Its geometry evidences a natural distribution of internal chambers and an organizing redundancy topology, reflecting how the hidden dimensions of TEITDOse project themselves into the universe, not out.
• The cube, symbol of the classical limitation of Euclidean space, but here reinterpreted as an internal vector tension box. In its appearance, it is finite and flat. But when treated vectorally, it is profound, multiple and dynamic.
• The vector mesh, metaphor of the autoregulated totality. Each node of the mesh represents a tensioned dimensional cell. The mesh vibrates, pulsates, collapses and rebuilds itself in dynamic equilibrium with the vector forces that compose it - ω and ε varepsilon_{-}ε .

The structure of the universe, in this model, does not need the insertion of dark matter or dark energy as explanatory mechanisms. What is observed in the galactic rotation curves, for example, can be perfectly reproduced by the accumulation of folded vector stresses on the periphery of simulated dimensional meshes . The space itself ceases to be an inert stage to become a living, vibratory, reactive agent.

Based on this principle, the TEITDO model as Reality Support was submitted to numerical simulations. First in 2D, showing the local variations of the vector mesh under gravitational compression. Then in 3D, with 30 dimensions overlaid - each modulating the angular tension of a specific layer. And finally, in 4D, simulating time as a continuous pulse of deformation and self-regulation. In these representations, we visualize the galactic curve emerging as a direct response of the internal vector mesh - without the need for exotic entities.

Such results support that the observable reality itself is only a superficial response of a deeper, dynamic, bent structural background. And this structure is falsifiable: if the predictions about vector tension can be measured - for example, in gravitational lensing without visible mass - then its postulate can be confronted directly with the universe.

Finally, this theory is not alone. It is integrated into a larger theoretical body, composed of previous contributions that dialogue in a profound way:

• The Octagonal Mesh acts as a geometric organizing structure, where each vector cell is positioned in harmonic intersections of a mirrored grid, which allows both expansion and vector containment.
• The Infinite Vectors function as continuous flows of structural stability, defining the routes by which vector stresses dissipate or accumulate, like breath fields in the universe.
• Emergent Dimension Mirroring proposes the natural duplication of dimensional domains from the superposition of harmonic layers. This explains phenomena such as duality, symmetry and topological bifurcations observed at macro and microscale.
• And the Pomegranate, as a fractal model of distribution and internal condensation, provides the symbol of the internal organization of reality as something alive, self-structured and dense, although invisible at first sight.

These theories, together, make up a new dimensional ontology, where reality is simultaneously visible and hidden, flat and bent, stable and pulsating. The TEITDO as Reality Support is its stabilizing core.

2. Ontological and Structural Reasoning

The visible reality is not only supported by its three traditional dimensions. Analogous to a corrugated paper, whose internal structure and its inner tubes provide invisible resistance, the universe is composed of an internal vector mesh, folded, invisible but present. These hidden dimensions are the structural beams of reality and guarantee full coherence, elasticity and contained expansion.

3. The Vector Postulate: ω ε = -1

This postulate describes the fundamental relationship between local angular frequency (ω) and vector vacuum resistance (ε ). When this equation is satisfied, the dimensional point is in tensional equilibrium. Minimal variations generate local instabilities, which can be interpreted as folds, compressions or structural fluctuations in space-time.

**Relationship to Traditional Models**

Although the formalism ω ε = -1 does not derive directly from Einstein’s equations, it can be interpreted as a dual model to the metric tensor g_{μν}, acting as a **network of compensatory stresses** (analogous to the content of the energy-momentum tensorT_{μν}).

Approximate matching proposal:

- ω ⇔ curvatura espacial local (análogo a componentes de Ricci ou Riemann)

- ε Effective energy density distributed vector (similar to T_{00})

- Differential equation:$d\omega /dt·\epsilon ₋+\omega ·d\epsilon ₋/dt=0$ local conservation equation$\nabla ^\mu T\_\left\{\mu \nu \right\}=0$

It is open the bridge for generalized Lagrangian-type formulations or approximations via effective vector field theory.

5. Simulation Methodology

The numerical simulation was performed in a two-dimensional (100x100) and three-dimensional (50x50x50) vector mesh, with gaussian parameters centered on ω = 1.0 and ε = -1.0. A central mass function was used to simulate gravitational compression and the temporal evolution was modeled with oscillatory fluctuations of ω. The 30 dimensions of the original model were superimposed as layers of accumulated angular modulation .

**Reproducibility**

All the source code used in vector simulations (2D/3D/4D mesh, projections and visualizations) is available in repository:

Github: https://github.com/CharlesEugeniodr/teitdo-vetorial-simulator

Includes:

- Scripts to generate the vector mesh with ω and ε

- Boundary conditions and parameters adjustable

- Dataset of simulated stresses

- Animations and graphics generated

2D vector mesh reproducibility code with central gravitational compression, based on the postulate ω ε = 1:

Code

• Grade 2D: 100x100 células vetoriais
• Fixed parameters:

$$\omega (x,y)=1+variacão gaussiana central$$

$$\epsilon -\left(x,y\right)=-1{\mathsf{\epsilon }}_{-}\left(x,y\right)=-1\mathit{\epsilon }-\left(\mathit{x},\mathit{y}\right)=-1$$

Vector voltage heat map $\mathit{\tau }=\mathit{\omega }\cdot \mathit{\epsilon }-$

6. Computational Results

A) In the vector 2D mesh, it was observed that the local gravitational compression caused deformations in the vector balance with formation of high-voltage zones around the mass.

B) The overlapping of the 30 dimensions resulted in a field of complex stability with zones of greater support concentrated around the center.

C) The galactic rotation curve simulated from the vector mean of the stresses showed a stable behavior over long distances without the need for dark matter.

) The simulation of temporal evolution demonstrated a harmonic pulsation in the vector stresses of the mesh, reinforcing the hypothesis of an active universe, coherent and self-regulated.

2D vector mesh with gravitational compression of the vector tension based on the postulate ω ε = 1 in Py:

"""

2D Vector Mesh Simulation with Central Gravitational Compression

Baseado no postulado vetorial: ω(x, y) · ε₋(x, y) = -1

Author: Charles de Paula Eugênio

"""

import numpy as np

import matplotlib.pyplot as plt

import pandas as pd

# Parameters of the mesh

grid_size = 100

omega = np.ones((grid_size, grid_size)) # Frequência angular ω(x, y)

epsilon = -1 * np.ones((grid_size, grid_size)) # Resistência vetorial ε₋(x, y)

# Centered gravitational compression function (gaussian distribution)

def compressao_gravitacional(x, y, sigma=10):

 cx, cy = grid_size // 2, grid_size // 2

 return np.exp(-((x - cx) ** 2 + (y - cy) ** 2)/ (2 * sigma ** 2))

# Apply gaussian compression variation to angular frequency ω

for i in range(grid_size):

 for j in range(grid_size):

  omega[i, j] += 0.5 * compressao_gravitationally(i, j)

# Calculate vector voltage τ(x, y) = ω(x, y) * ε (x, y)

tension = omega * epsilon

# Voltage field visualization (heat map)

plt.figure(figsize=(8, 6))

plt.imshow(tensao, cmap='viridis')

plt.colorbar(label='Vector Voltage τ(x, y)')

plt.title('Simulation of 2D Vector Mesh with Central Compression')

plt.xlabel('x')

plt.ylabel('y')

plt.tight_layout()

plt.savefig('malha_vetorial_2d.png')

plt.close()

# Export stress matrix to CSV

df = pd.DataFrame(tension)

df.to_csv('malha_vetorial_2d.csv', index=False)

print("Simulation completed.")

print("Generated files: malha_vetorial_2d.csv and malha_vetorial_2d.png")

7. Discussion

The results suggest that hidden dimensions, represented by folded vector layers, may provide a natural and elegant explanation for the stability of galactic curvature and universal expansion. The internal structure of space-time, according to the TEITDO model as Reality Support, acts as an active support mesh, capable of absorbing and redistributing vector stresses without resorting to invisible entities such as dark matter.

8. Cosmological Implications

The Theory of Hidden Dimensional Folding proposes a new reading of the universe as a tensioned vector entity, where visible reality is the result of a folded structure, pulsating and in constant self-regulation. The absence of dark matter in the model and the ability to simulate coherent galactic curves indicate that space-time may be deeper and more vivid than assumed.

10.2. Pomegranate - Fractal and Symmetric Distribution

The structure of the pomegranate is symmetrical and fractal, with repetitive chambers. This can be modeled as regular angular divisions with radial fluctuation.

Fractal polar model:

$$r\left(\theta \right)=r^0·\left(1+\alpha ·sin\left(n\theta \right)\right)$$

Where:

• r = mean radius;
• α = vector amplitude of variation;
• n = number of internal divisions.

This pattern describes the dense and symmetrical organization of the inner folded dimensions.

10.3. Cube - External Limitation with Internal Depth

The cube represents the classical three-dimensional boundary. However, its real depth is given by invisible internal tensional vectors.

Internal vector model:

$$C\left(x,y,z\right)=\left[V^1\left(x\right),V^2\left(y\right),V^3\left(z\right)\right]$$

Where:

• V , V , V = hidden voltage vectors;
• The edge represents the observable dimension;
• The interior contains folded vector layers.

10.4. Mesh - Self-Regulated Continuous Network

The dimensional mesh of TEITDOis an evolutionary vector matrix. Each node is a vector cell with equilibrium or tension.

Model of the mesh:

$$M\left[i,j,k\right]\left(t\right)=\left|\omega \left(i,j,k,t\right)·{\epsilon }^{-}\left(i,j,k,t\right)+1\right|$$

Stability depends on the preservation of the fundamental postulate. This network lives and pulsates as the basis of observed reality.

9. Mathematical Structure and Projections of Dimensions

The TEITDO model as Reality Support integrates two fundamental forms of vector projection to model dimensional support: exponential projection and quadratic projection. These curves describe the stress and growth behavior of each vector layer along the x coordinate, representing the internal progression of the hidden dimensions.

The functions used are:

• Exponential projection:$f\_exp\left(x\right)=ae^\left\{bx\right\}$

• Quadratic projection:$f\_quad\left(x\right)=ax^2+bx+c$

Each vector dimension manifests its specific curve, being interpreted as a support fold that directly influences the stability of space-time. The exponential curve represents nonlinear vector acceleration, while the quadratic one reflects local symmetric stabilization.

Vector Projections by Dimension (Sample of 50 Lines).

Dimension	X	F_EXP	F_QUAD
1	0.0	1e+17	1.1000000000000002e+17
2	0.0200400801603206	1.003010534608705e+17	1.1201003208822456e+17
3	0.0400801603206412	1.0060301325360402e+17	1.1403211232083405e+17
4	0.0601202404809619	1.00905882106744e+17	1.1606624069782854e+17
5	0.0801603206412825	1.0120966275704826e+17	1.1811241721920797e+17
6	0.1002004008016032	1.0151435794951373e+17	1.2017064188497235e+17
7	0.1202404809619238	1.0181997043740122e+17	1.2224091469512173e+17
8	0.1402805611222445	1.0212650298226032e+17	1.2432323564965603e+17
9	0.1603206412825651	1.0243395835395443e+17	1.2641760474857533e+17
10	0.1803607214428857	1.0274233933068566e+17	1.2852402199187955e+17
11	0.2004008016032064	1.0305164869902e+17	1.3064248737956877e+17
12	0.220440881763527	1.033618892539125e+17	1.3277300091164291e+17
13	0.2404809619238476	1.0367306379873251e+17	1.3491556258810206e+17
14	0.2605210420841683	1.039851751452891e+17	1.3707017240894616e+17
15	0.280561122244489	1.0429822611385624e+17	1.392368303741752e+17
16	0.3006012024048096	1.0461221953319854e+17	1.4141553648378925e+17
17	0.3206412825651302	1.0492715824059667e+17	1.436062907377882e+17
18	0.3406813627254509	1.0524304508187306e+17	1.4580909313617213e+17
19	0.3607214428857715	1.0555988291141754e+17	1.4802394367894106e+17
20	0.3807615230460922	1.0587767459221322e+17	1.502508423660949e+17
21	0.4008016032064128	1.0619642299586227e+17	1.5248978919763373e+17
22	0.4208416833667334	1.0651613100261198e+17	1.5474078417355754e+17
23	0.4408817635270541	1.0683680150138074e+17	1.570038272938663e+17
24	0.4609218436873747	1.0715843738978397e+17	1.5927891855856e+17
25	0.4809619238476953	1.0748104157416066e+17	1.615660579676387e+17
26	0.501002004008016	1.0780461696959933e+17	1.6386524552110234e+17
27	0.5210420841683366	1.081291664999645e+17	1.661764812189509e+17
28	0.5410821643286573	1.0845469309792306e+17	1.6849976506118448e+17
29	0.561122244488978	1.0878119970497085e+17	1.70835097047803e+17
30	0.5811623246492986	1.0910868927145912e+17	1.7318247717880653e+17
30	0.6012024048096192	1.0943716475662128e+17	1.7554190545419498e+17
30	0.6212424849699398	1.0976662912859965e+17	1.779133818739684e+17
30	0.6412825651302605	1.1009708536447218e+17	1.8029690643812675e+17
30	0.6613226452905812	1.1042853645027949e+17	1.826924791466701e+17
30	0.6813627254509018	1.1076098538105168e+17	1.851000999995984e+17
30	0.7014028056112224	1.1109443516083562e+17	1.8751976899691165e+17
30	0.721442885771543	1.1142888880272184e+17	1.899514861386099e+17
30	0.7414829659318637	1.1176434932887198e+17	1.9239525142469306e+17
30	0.7615230460921844	1.1210081977054594e+17	1.948510648551612e+17
30	0.781563126252505	1.1243830316812938e+17	1.9731892643001437e+17
30	0.8016032064128256	1.1277680257116109e+17	1.997988361492524e+17
30	0.8216432865731462	1.1311632103836067e+17	2.0229079401287546e+17
30	0.8416833667334669	1.1345686163765603e+17	2.0479480002088346e+17
30	0.8617234468937875	1.1379842744621123e+17	2.073108541732764e+17
30	0.8817635270541082	1.1414102155045424e+17	2.0983895647005437e+17
30	0.9018036072144288	1.1448464704610483e+17	2.1237910691121725e+17
30	0.9218436873747494	1.1482930703820253e+17	2.1493130549676506e+17
30	0.94188376753507	1.1517500464113464e+17	2.174955522266979e+17
30	0.9619238476953909	1.1552174297866451e+17	2.2007184710101565e+17
30	0.9819639278557114	1.1586952518395971e+17	2.226601901197184e+17

Vector Projections by Dimension continuous time applied on the three-dimensional vector volume with 30 dimensions folded.

The 4D simulation - representing the continuous time applied on the three-dimensional vector volume with the 30 dimensions folded.

The following are comparative graphs between exponential and quadratic functions for different dimensions:

The vector simulations were generated based on your CSV of projections by dimension:

Graph 2. - Vector Quadratic Projection by DimensionesRepresents the function , showing a less explosive progression $y=ax2+bx+cy=ax^2+bx+cy=ax2+bx+c$, but with predictable and symmetrical structure, as the support of a spatial mesh with internal reinforcements.

Graph 2. - Vector Quadratic Projection by DimensionesRepresents the function , showing a less explosive progression $y=ax2+bx+cy=ax^2+bx+cy=ax2+bx+c$, but with predictable and symmetrical structure, as the support of a spatial mesh with internal reinforcements.

It is observed that the vector curves follow distinct growth patterns, demonstrating how the composition between accelerated and symmetrical vector folds contributes to the formation of the dimensional structure. The overlap of these projections defines the internal behavior of the three-dimensional vector mesh, directly influencing the galactic curvature, local time and reality’s resistance to fragmentation.

The dynamic simulation of the temporal evolution of the vector mesh with folded layers.

The simulation shows how the 2D vector mesh responds to the postulate ω ε = 1, both before and after the simulated gravitational compression in the center of the structure.

Interpretation:

• Left: the vector field with small Gaussian fluctuations - most of the points are close to stability (darker colors).
• Right:When simulating the central gravitational compression, there is a significant increase in deviation in the regions closest to the center (lighter tones), indicating local dimensional folds, stresses or collapses .

This pattern reinforces the logical and empirical concept: the hidden folded dimensions (represented by the vector variations of ω and ε ) support structure to a certain point. When the pressure exceeds this threshold, the field enters into deformation - the fabric of space-time bends, not by dark energy, but by internal vector redistribution.

Vector projection of the 30 hidden dimensions on the mesh 2Dwith folded layers . What you are seeing is the result of the overlap of each dimension as a stabilizing or deforming vector stress field, modulated by the deviation of the postulate ω ε = 1 and the distance factor to the central mass.

Here is the simulated galactic rotation curve offset in the three-dimensional distribution of the 30 hidden dimensions doubled according to the postulate ω ε = 1.

Interpretation of the chart:

• The curve shows a stabilising behaviour of orbital velocity in large radii - no need for dark matter;
• This is because the dimensionally folded layers act as a vector source of gravitational support, projecting mass/influence not directly visible, but distributed by internal folds;
• The pattern consistently simulates what we observe in real galaxies: the extremities do not collapse, even with insufficient visible mass- exactly as this theory proposes.

These illustrations bring a conceptual structure , hierarchical and directly applicable to scientific formalization - uniting:

• The wavy mesh and reinterpreter of vacuum;
• The sectioned cube to express the distinction between visible and folded layers;
• The vector projective functions (exponential and quadratic) as engines of coherent expansion;
• The ω ε wave that symbolizes the hidden pulse of dimensional stability.

11. Expansion of Mathematical Formalism - Private Solutions

In addition to the general differential equations that describe the vector structure of TEITDO, we propose here particular solutions applicable to concrete astrophysical contexts.

Example 1: Static Voltage Field in EquibrioSupose a static system with vector disturbance located in the center (x = 0), modeled by:

$\delta \left(x\right)=A e^(-\alpha x²)$

$\mathrm{A} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{a} \mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{s}\mathrm{e}\mathrm{r}\mathrm{á}:$

$T\left(x\right)=-\nabla \delta \left(x\right)=2\alpha Ax e^(-\alpha x²)$

$\mathrm{E} \mathrm{o} \mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{l} \mathrm{d}\mathrm{e} \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{a} \mathrm{a}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{d}\mathrm{o}:$

${\nabla }^2\varphi \left(x\right)=\left|\left|T\left(x\right)\right|\right|=\left|2\alpha Ax{e}^{\left(-\alpha x^2\right)}\right|$

$\mathrm{E}\mathrm{s}\mathrm{s}\mathrm{a} \mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{ç}\mathrm{ã}\mathrm{o} \mathrm{m}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a} \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{o} \mathrm{o} \mathrm{c}\mathrm{a}\mathrm{m}\mathrm{p}\mathrm{o} \mathrm{s}\mathrm{e} \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a} \mathrm{s}\mathrm{o}\mathrm{b} \mathrm{f}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{a}\mathrm{ç}\mathrm{ã}\mathrm{o} \mathrm{g}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{a} \mathrm{e} \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{o} \mathrm{o} \mathrm{s}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{a}$

$$\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e} \mathrm{à} \mathrm{a}\mathrm{u}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{ç}\mathrm{ã}\mathrm{o}.$$

Corrugated Structure (Vector Longitudinal)

The geometry is described by a sinusoidal function, modeling vector voltage fluctuation:

$$y\left(x\right)=A\cdot sin(kx+\varphi )$$

Where:

• AAA represents the voltage range;
• $\mathit{k}={\displaystyle \frac{2\mathsf{\pi }}{\mathsf{\lambda }}}$ the angular frequency (linked to the density of folds);
• The local vector voltage is defined as:

$$\tau \left(x\right)=\omega \left(x\right)\cdot \epsilon -\mathrm{}\left(x\right)\approx -1$$

This structure is compatible with a continuous 1D or 2D mesh.

Hexagonal Mesh (Radial Vector)

The most direct mathematical model is the regular spatial repetition of hexagonal cells. To simulate a discrete vector field:

Each cell has center (xi,yj)(x_i, y_j)(xi ,yj ), and we can define a vector field associated with each cell as:

$$\mathbf{V}⃗ij\mathrm{}=[\omega ij\mathrm{},\epsilon -ij\mathrm{}$$

Where:

• ω ij omega_{ij}ω ij represents rotation or local angular frequency
• ε ij varepsilon_{-ij}ε ij is the resistance of the cell to deformation;
• The local stability condition is the same:

$$\omega ij\mathrm{}\cdot \epsilon -ij\mathrm{}=-1$$

3. Comparative Geometric-Mathematical Coherence
Property	Corrugated Structure	Hexagonal Mesh
Type of mesh	Continuous wavy (1D/2D)	Celular discreta (2D radial)
Forma base	Periodic senoid	Regular hexagons
Vectorization	Longitudinal continuous	Discrete vector cells
Field of voltage	τ(x)=ω(x) ε (x) tau(x) = omega(x) cdot varepsilon_{-}(x)τ(x)=ω(x) ε (x)	τij=ωij⋅ε−ij\tau_{ij} = \omega_{ij} \cdot \varepsilon_{-ij}τij​=ωij​⋅ε−ij​
Structural regulation	By wave frequency	By cell density and symmetry
Natural/analog application	Waves, vibrations, materials engineering	Hives, cellular structures, biomimetics

12. Proposals for Observational Testing

The ITDO is falsifiable . We suggest the following tests:

- Deep vacuum interferometry: search for micro-variations in areas of apparent void. - Gravitational lenseswithout visible mass: analysis of distortions where there is not enough matter to justify the optical effects. - Computational simulations with variation of ε : to mapthe vector response to pulses of ω(freq. angular).

13. Complementary Views

Heat map of the 2D vector mesh before/after compression. - 3D vector fieldwith colored directional arrows. - δ(x,t) gradient animation over time. Note: images available in . TIFF and . MP4 format in the article supplement.

14. Vector Projection Comparison Tables

Below, we present an isolated table with the vector projections (exp and quad) for the first 10 dimensions:

Dimension	X	F_EXP(X)	F_QUAD(X)
--------------------------	--------------------------	--------------------------	--------------------------
1	0.00	1,000e+17	1.100e+17
2	0.02	1.003e+17	1.120e+17
3	0.04	1.006e+17	1.140e+17
4	0.06	1,009e+17	1.160e+17
5	0.08	1.012e+17	1.181e+17
6	0.10	1.015e+17	1.202e+17
7	0.12	1.018e+17	1.222e+17
8	0.14	1.021e+17	1.243e+17
9	0.16	1.024e+17	1.264e+17
10	0.18	1.027e+17	1.285e+17

15. Critical Discussion: Comparison with Competing Theories

TEITDOoffers a distinct ontological approach. Important comparisons:

16. Automotive Radiators as Functional Corrugated Structures

In addition to the geometric analogies with corrugated paper and honeycomb, automotive radiators provide a technical example of simultaneous application of structural and functional vector function . Its construction is composed of thin parallel galleries , often made of aluminum or light metal alloys, interspersed by internal corrugated folds that enlarge the thermal exchange area and maintain the mechanical integrity of the assembly.

Although the primary purpose of a radiator is thermal dissipation - through the circulation of cooling liquid inside and forced ventilation outside -, the geometric structure of its galleries also fulfills a critical structural function: prevent deformation by pressure and vibration, distributing mechanical stresses through its corrugated and segmented architecture.

This organization allows the radiator, even being built with thin and potentially fragile materials, support extreme thermal variations, motor vibrations and torsion forces. This behavior is directly analogous to the TEITDO model:

• Internal galleries correspond to the folded vector dimensions that contain and redistribute internal energy of the system;
• The forcing air layers represent the external interaction with the field, which in the dimensional model is equivalent to the presence of curvature or gravitational deformation;
• Structural strengthening by geometry is a practical example of how form can compensate for the fragility of material - as proposed by TEITDOfor space -time: vacuum, although "fragile", is structurally robust by its internal vector organization.

Thus, the automotive radiator is not only an illustration of functional engineering, but an empirical example of the fundamental principle of TEITDO: stress containment and stabilization by self-regulated geometric structure, with exchange capacity, absorption and resilience.

17. Ontological Reasoning: The Reality Sustained by the Invisible

The Theory of the Invisible Structure of Time and the Hidden Dimensions proposes, at its core, that the visible three-dimensional reality - what is measured, observed and experienced - is not self-sufficient ontologically. It is sustained by a structural base not evident, whose existence is only revealed through its effects: stability, containment, regularity and resistance to fragmentation.

This proposition is not only physical, but ontological: it affirms that the being is sustained by layers of non-perceivable being; that what is manifested does not stand for itself, but for structures that operate silently in hidden layers of the dimensional mesh.

17.1. The Hidden Function of Form

Many systems of the physical world illustrate this principle by apparent purpose deviation. The automotive radiator, for example, is designed to exchange heat, but its shape - composed of thin, segmented and wavy galleries - also gives structural strength against torsions and pressures. Its geometry therefore performs a silent ontological function, parallel to the explicit technical function.

This structural- functional duplicity is the mirror of what occurs in the cosmos: the visible dimensions operate as the thermal surface of the radiator, while the hidden vector layers are its internal structural galleries, through which the "fluid" circulates of space-time - the field of invisible stresses that regulates its shape.

17.2. Perception as Dimensional Reduction

The non-perception of true structure does not deny its existence, but only attests to the limits of sensoriality and human instrumentation. The mind, conditioned to three-dimensionality, takes what is functional as fundamental. However, what is structural - what allows something to "be" with stability - is often hidden by definition.

This split between manifestation and support forces us to rethink the ontology of reality: the visible being is the emergence of an invisible background, and its coherence is only possible if we recognize a mathematical and vector structural basis that precedes and regulates it.

17.3. Matter as a Consequence, not an Origin

The stability of a galaxy, the strength of a pomegranate or the integrity of a radiator do not come exclusively from the matter that makes them up, but from the vector shape that organizes them-whether it be symmetries, internal stresses, or repetitive geometric patterns. Matter is thus the effect of a structure, not its cause.

In this context, the dimensional mesh proposed by TEITDOis not a metaphor but a radical ontological statement : reality is a system organized by hidden vector layers whose balanced tension keeps the universe in a state of cohesion. The equation ω ε = -1 is not just an algebraic relation - it is a postulate of stability of being, a statement that reality is supported by an internal vector equilibrium, even when this does not manifest directly to observation.

"Every observable three-dimensional structure - be it an organism, a technical artifact or the very fabric of space - reveals, under in-depth analysis, that its stability depends on invisible or not evident organizational elements in their primary manifestation.

In some cases, as in the automotive radiator, the structural function is hidden under the thermal function. In others, as in the mesh, the vector structure remains completely hidden from the senses. But in both cases, what is 'real' is sustained by that which at first sight is not seen."