🧠 Zero as a Dimension Switch

Overview

In standard mathematics, the number line is a single continuous axis stretching from negative infinity to positive infinity. Zero sits in the middle.

This project proposes an alternative framework:

Treat 0 not as a number within a line, but as a "Switch" or "Mirror" between two separate positive spaces.

Instead of passing through zero into negative numbers, values reflect off zero and enter a Mirror Space. This creates a V-shaped or folded number system, conceptually similar to 1D polar coordinates or signed magnitude representation.

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Core Concept: The "Bounce" Mechanism

In this framework, there are no "negative numbers" in the traditional sense. Instead, there are Positive Numbers and Mirror Numbers.

• Positive Space ($\mathbb{R}^+$): Standard values (e.g., $1, 2, 3...$)
• Mirror Space ($\mathbb{R}^-$): Reflected values (e.g., $1^-, 2^-, 3^-...$)
• Zero (0): The Boundary or Gateway connecting the two spaces.

Visualizing the Difference

Standard Number Line: Values pass through 0 linearly.

... <--- (-2) --- (-1) --- 0 --- (1) --- (2) ---> ...

Dimension Switch Model (V-Shape): Values bounce at 0.

   Mirror Space (R⁻)          Positive Space (R⁺)
         2⁻                         2
          \                        /
           1⁻                     1
             \                   /
              \                 /
               \               /
                -----> 0 <-----
                  (The Switch)

How Calculation Works

When an operation crosses 0, the "Dimension Switch" is triggered. The remaining magnitude is conserved but transferred to the opposite space.

Example: Subtraction ($1 - 3$)

In standard math, $1 - 3 = -2$. In this framework, we interpret subtraction as movement towards the origin.

1. Start: At 1 in Positive Space.
2. Move: Subtract 1 to reach 0 (The Switch).
3. Remaining Force: We still have 2 units of subtraction left.
4. Switch: Since we hit the boundary, we deflect into the Mirror Space.
5. Result: $2^-$ (A magnitude of 2 in the Mirror Space).

Formalizing the Rule

If we define a value as a pair $(magnitude, space)$, where space is $+1$ or $-1$:

$$ x \ominus y = \begin{cases} (x - y, +1) & \text{if } x \ge y \quad (\text{Stay in Positive Space}) \\ (y - x, -1) & \text{if } x < y \quad (\text{Switch to Mirror Space}) \end{cases} $$

This ensures that magnitude is conserved across the boundary, behaving like a physical reflection or a billiard ball bouncing off a wall.

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Why This Matters

• Physical Intuition: Fits physical quantities that cannot be negative (like mass or energy) but can have "opposite direction" or "anti-matter" states.
• Eliminating "Negative" Ambiguity: Clearly separates the magnitude of a value from its state/orientation.
• Educational Value: Provides a tangible model for understanding vectors, absolute values, and polar coordinates without assuming a linear number line.

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Project Structure

• docs/ — Formal definitions, axioms, and proofs (LaTeX/PDF).
• README.md — Concept overview.

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Status

This is an experimental mathematical framework. The goal is to construct a consistent axiomatic system where 0 acts as a non-numeric boundary.

Feedback and contributions are welcome!