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DEFINITION OF SPHERION

1️⃣ Main Concepts

Spherion Metrics is an innovative computational system designed to replace traditional Cartesian coordinates (x, y, z) with a new spherical-based approach for measuring and calculating positions, uncertainty, and spatial relations. Spherion’s nature, rooted in spherical abstraction, enables it to transcend not only the three dimensions of the plane but also the temporal dimension. This is because the sphere is fundamentally relativistic, both in form and in its movements.

2️⃣ Symbol and Meaning

The symbol 𝕊 (double-struck capital “S”) represents the Spherion Metrics computational system, clearly denoting its spherical foundation and mathematical orientation.

3️⃣ General Definition

Spherion Metrics (𝕊) is an innovative spherical coordinate system designed to represent spatial information with explicit uncertainty encoding in a hierarchical manner. Unlike traditional Cartesian or spherical systems, Spherion recursively subdivides a sphere into overlapping hemispheres and circles, creating zones of ambiguity that naturally model probabilistic or fuzzy spatial relationships.

4️⃣ Structural Summary

Instead of flat, three-dimensional grids, Spherion Metrics organizes information into spheres divided into overlapping hemispheres.

It features horizontal, lateral, and vertical diameters between the hemispheres, along with radial circles (twelve per diametral hemisphere) that are recursively subdivided until the required precision is achieved.

5️⃣ Core Definition in Quantum Mechanics Context

Spherion Metrics (𝕊) offers a natural, hierarchical, spherical coordinate system explicitly designed to handle positional uncertainty.

It matches quantum mechanical realities intuitively by encoding positional ambiguities directly into its coordinate structure.

6️⃣ Formal Mathematical Definition

The hierarchical structure uses recursive subdivision of hemispheres:

P𝕊 = (H, C₀, C₁, ... , Cₙ)

Where:

H : One or more overlapping hemispheres

C₀, C₁, ... , Cₙ : Subdivisions, with six circles per level

This structure allows positions to be encoded at any desired resolution and includes overlapping zones for ambiguity.

Probabilistic or Fuzzy Uncertainty Encoding: overlaps represent spatial uncertainty explicitly.

7️⃣ Key Characteristics

Hierarchical Spherical Subdivisions: space is recursively divided into structured spherical regions.

Intentional Overlapping Partitions: hemispheres and circular subdivisions overlap by design, creating zones of ambiguity.

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Spherion Metrics: Compiled Equations and Explanations

Spherion Metrics (𝕊) is a hierarchical spherical coordinate system designed to represent spatial information with explicit uncertainty and probabilistic encoding. Below are the compiled equations and their explanations from our previous discussions:

1. Base Coordinate Definition

A point on the unit sphere in Cartesian coordinates $(x, y, z)$ can also be represented in spherical coordinates $(\theta, \phi)$, with ranges: $$ \theta \in [0, \pi], \quad \phi \in [0, 2\pi) $$

where: * $\theta$ is the polar angle measured from the positive z-axis. * $\phi$ is the azimuthal angle measured from the positive x-axis.

For beginners, click here.

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What this means:
$$θ∈[0,π], ϕ∈[0,2π)$$
These two expressions tell us the range of two angles, $θ$ (theta) and $φ$ (phi), which are used to describe points on the surface of a sphere (like Earth or a globe).
$θ$ (theta):
This angle tells you how far up or down you are on the sphere.
It starts at the top (North Pole) with 0 and goes to the bottom (South Pole) with π (pi).
In degrees, that’s from 0° to 180°.
Example:
If $θ$ = π/2 (or 90°), you’re at the equator.
$φ$ (phi):
This angle tells you how far around the sphere you are—like turning around the globe left or right.
It starts at 0 (think: facing forward) and goes up to just below 2π (which is one full turn around the circle).
In degrees, that’s 0° to almost 360°.
Example:

If $φ$ = π (or 180°), you’re halfway around the sphere from where you started.
Together:
With $θ$ and $φ$, you can pinpoint any location on the surface of a sphere—just like using latitude and longitude on Earth!
Let a point on the unit sphere be given by Cartesian coordinates p=(x,y,z) and (equivalently) by spherical coordinates (θ,φ), where θ ranges from 0 to π and φ ranges from 0 to 2π.”
In other words, any spot on the ball’s surface can be named either by its x-y-z position or by the pair of angles showing its direction from the centre.

2. Hemisphere Definitions

The Spherion is initially divided into six overlapping hemispheres: $$ H_i = \{(x,y,z) | \text{specific inequalities define each hemisphere}\} $$
where $i \in \{1, 2, 3, 4, 5, 6\}$, representing pairs such as top/bottom, front/back, left/right.
For beginners, click here.

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What are Hemispheres in Spherion?

In the Spherion system, we split a sphere into six big sections called hemispheres. You can think of hemispheres like slicing an orange into large curved wedges, but instead of doing it just once, we do it three times in different directions:

Top and Bottom halves (like North and South on Earth)

Front and Back halves

Left and Right halves

What Does the Equation Mean?

$H_i~=~ \{{ (x,y,z) ~specific~ inequalities~ define~ each~ hemisphere\}}$

$H_i$: This stands for the i-th hemisphere, where $i$ is a number from 1 to 6.
The curly braces {} mean: “a group of points.”
$(x,y,z)$ are the 3D coordinates of a point on the sphere.
The part inside means: “All the points where a certain condition is true”—like “all points where $x ≥0$” to define the right hemisphere.

So, this just says:
But here’s the twist: these hemispheres overlap. That means some areas are part of more than one hemisphere—this is important because it helps the system represent uncertainty or fuzziness about where something is.

So, this just says:

Each hemisphere is defined by a mathematical rule that tells us which points (x, y, z) are inside it.
Why Use This?

Because the Spherion model is meant to handle fuzzy or uncertain locations, overlapping hemispheres give us more flexible and realistic ways to describe where something might be—not just in one exact place, but in several places at once.

3. Spherion Hierarchical Coordinates

The hierarchical encoding of a point $P_𝕊$ is given by: $$ P_𝕊 = (H, C_0, C_1, C_2, \dots) $$
where:
$H$ is the hemisphere label.

$C_i$ are subsequent subdivision circles at each hierarchical level, refining positional accuracy.

For example: $$ P_𝕊 = (\text{Top Hemisphere}, C_0 = 2, C_1 = 4, C_2 = \{3,4,5\}) $$
For beginners, click here.

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What does this mean?

$$ P_S~ =~(H, C_0, C_1, C_2, …)$$

This is how we describe where a point is located on a special kind of sphere used in the Spherion system.

Let’s break it down:

$P_S$

This just means “a point in the Spherion system.”

Think of it like an address for a place on a globe..

$H$

This is the hemisphere (or big half of the sphere) the point is in.
There are six overlapping hemispheres: Top/Bottom, Front/Back, Left/Right.
So $H$ tells you the general region on the sphere.
$C_0, C_1, C_2 ,…$

These are like zoomed-in layers that help pinpoint the location more precisely:
$C_0$ is a big circle dividing the hemisphere.
$C_1$ divides $𝐶_0$ into smaller parts.
$C_2$ divides again, and so on.
Each level gives you more detail—kind of like zooming in on Google Maps.
Example:
$$ P_𝕊 = (\text{Top Hemisphere}, C_0 = 2, C_1 = 4, C_2 = \{3,4,5\}) $$
…it’s like saying:

“The point is in the top half of the sphere, inside the second region, zoomed into the fourth subdivision, and even deeper into the third sub-subdivision.”

4. Distance Function $d_𝕊$

The distance between two points $P, Q$ in the Spherion Metrics considers both angular separation and hierarchical depth: $$ d_𝕊(P,Q) = f(\Delta \theta, \Delta \phi, \text{hierarchical similarity}) $$
This function captures:
Angular differences between points.

Depth of shared hierarchy to measure similarity or positional uncertainty.

For beginners click here.

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What does this equation mean?
$$d_𝕊(P,Q) = f(\Delta \theta, \Delta \phi, \text{hierarchical similarity})$$

This equation tells us how to calculate the distance between two points, P and Q, on a special kind of sphere called the Spherion Sphere.

But this isn't just any distance—it includes not only how far the points are physically, but also how similar their locations are in terms of structure. Let’s look at each part:

Term-by-term:

$d_𝕊(P,Q)$

This means “the distance between point P and point Q in the Spherion system.”

$f(...)$

This just means “a function of” — it’s like a machine that takes in some information and gives back a number.

$Δθ$

This is the difference in vertical angle (like north–south direction on a globe).

$Δϕ$

This is the difference in horizontal angle (like east–west direction).

Hierarchical similarity

This checks how deep into the same zone structure the two points are.

If P and Q are in the same hemisphere and also share the same small subdivisions, they are considered closer in this system—even if they’re physically farther away.

Example:

Let’s say two points are:

Both in the top hemisphere (✅ similar)

In the same Level 0 and Level 1 zones (✅ very similar)

Slightly different angles (◻️ small difference in 𝜃θ and 𝜙ϕ)

Even though their angles are a little different, their shared structure means they’re very close in the Spherion sense.

Why is this important?

In real-world uses like quantum mechanics or AI, you often want to measure how similar things are, not just how far apart they are. This equation lets you do both at once.
Even though their angles are a little different, their shared structure means they’re very close in the Spherion sense.

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Explicação da Equação 3

Este é o 3o. pop-up, e ele também aparecerá corretamente centralizado na tela.

5. Zone Assignment Function $h(p)$

Each point $p$ on the sphere is assigned a primary hemisphere by maximizing its proximity: $$ h(p) = \arg\max_{i \in \{1,\dots,6\}} d(p, H_i^c) $$
Here, $H_i^c$ denotes the complement of hemisphere $H_i$, and the assignment favors the hemisphere closest to the point.
For beginners, click here.

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Equation
$$ h(p) = \arg\max_{i \in \{1,\dots,6\}} d(p, H_i^c) $$
What does it do?

This equation helps decide which of the 6 hemispheres a point belongs to in the Spherion system.

Step-by-step Explanation:

$p$

This is just a point somewhere on the surface of the sphere (like a dot on a globe).
$H_i^c$

This means the complement of the i-th hemisphere — in simpler terms, it’s the area outside of one specific hemisphere.

For example, if $H_1$ is the top half of the sphere, $H_i^c$ is the bottom half.

$d(p,H_i^c$\)

This tells us how far the point $𝑝$ is from the outside of each hemisphere.

It’s like asking, “How far is this dot from being outside that hemisphere?”

$ arg⁡~max_{⁡𝑖∈}~{1,…,6}$

This part means: “Find the value of 𝑖 i (from 1 to 6) that gives the biggest result.”

In our case, it’s choosing the hemisphere where the point is most deeply inside (furthest from the outside).

ℎ(𝑝)

The result of all this: which hemisphere should the point be assigned to?

It’s the one where it fits best—where it’s farthest from the edge.

Example:

Imagine a point is deep in the “Top” hemisphere. The equation checks:

“How far is this point from being outside each hemisphere?”

Then it picks the hemisphere where the point is most securely inside—and assigns that hemisphere as the point’s main label.

6. Probabilistic Representation

Quantum mechanical applications represent wavefunctions probabilistically across multiple overlapping zones: $$ \Psi(p) = \sum_{i,j,k} a_{ijk} \psi_{H_i, C_j, C_k}(p) $$
where:
$a_{ijk}$ are complex amplitudes.
$\psi_{H_i, C_j, C_k}(p)$ denotes wavefunctions localized within hierarchical subdivisions.
For beginners, click here.

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The Equation:
$$ \Psi(p) = \sum_{i,j,k} a_{ijk} \psi_{H_i, C_j, C_k}(p) $$
What It Means:

This equation is used in quantum physics to describe where a particle (like an electron) might be on the surface of a special kind of sphere called a Spherion Sphere.

But instead of saying “the particle is exactly here,” it says something more realistic:

“The particle could be in several different places at once, each with a different chance.”

Breaking It Down:

$Ψ(𝑝)$: This is the overall "wave function" at a point 𝑝. Think of it like a map that tells you how likely it is to find the particle at that point.

$∑_{𝑖,𝑗,𝑘}$: This symbol means “add up a bunch of pieces.” Each piece is like one possible location.

𝑎_𝑖𝑗𝑘: These are the weights or probabilities for each possible location. If 𝑎_𝑖𝑗𝑘 is large, it means the particle is more likely to be found there.

$𝜓 𝐻_𝑖 , 𝐶𝑗, 𝐶𝑘 (𝑝)$

$(p)$: This is a mini-wave function for one specific part of the sphere. The labels 𝐻𝑖, 𝐶𝑗C j, and 𝐶𝑘 tell you:

Which hemisphere the region is in.

Which Level 0 circle (a big slice of that hemisphere).

Which Level 1 circle (a more zoomed-in slice)

So, you’re combining many small “waves” from different parts of the sphere to build one big picture of where the particle might be.

Think of it Like:

You’re trying to find where someone is hiding in a big park divided into areas and sub-areas.

Each small area has a different chance of hiding the person.

This equation adds up all those chances to give you the full map of where they’re most likely hiding.

7. Quantum Collapse Representation

In quantum measurements, collapse refines probabilistic representations:
* **Before Measurement:** Multiple overlapping zones encode superposition states.
* **After Measurement:** Single refined zone indicates collapsed definite state.
For beginners, click here.

8. Uncertainty Zones

Overlap regions (intersections of hemispheres and circles) explicitly encode uncertainty: $$ U = \bigcap_{i,j} (H_i \cap C_j) $$
These zones are crucial for applications involving fuzzy logic, robotics, AI, and quantum computing, naturally reflecting spatial ambiguities.
For beginners click here.

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Explicação da Equação 8

Este é o 8o. pop-up, e ele também aparecerá corretamente centralizado na tela.

Conclusion

These compiled equations from Spherion Metrics illustrate its ability to encode spatial information hierarchically, probabilistically, and with explicit uncertainty. Its adaptability makes it highly suitable for advanced applications ranging from quantum mechanics to spatial computing.

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Spherion Metrics: Compiled Equations and Explanations

1. Base Coordinate Definition

2. Hemisphere Definitions

3. Spherion Hierarchical Coordinates

4. Distance Function \(d_𝕊\)

5. Zone Assignment Function \(h(p)\)

6. Probabilistic Representation

7. Quantum Collapse Representation

8. Uncertainty Zones

Conclusion

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