The Amestoy Vazquez Theorem: Universal Equation for Orthogonal Invariance in N-Dimensional Systems.

This paper presents the formalization of a previously unrecorded mathematical invariant emergent in multidimensional arithmetic structures. The Amestoy Vazquez Theorem demonstrates that within any n-dimensional hypercube (D) constructed via a contiguous arithmetic progression starting at any real value (a), the sum of an orthogonal selection—defined as the selection of exactly one element per hyperplane—results in a structural constant (S).

The discovery is synthesized into the Universal Equation: S = n * a + (n * (n^D - 1)) / 2

Where: S = The Invariant Sum (Amestoy Vazquez Constant) n = The edge length or magnitude of the system a = The starting value or origin of the progression D = The number of dimensions (D >= 2)

Key contributions of this work include:

1. The Proof of Translation Invariance: Demonstrating that the constant S remains predictable regardless of the starting point (a) on the number line.

2. Dimensional Scaling: Extending the logic from 2D matrices to n-dimensional hypercubes.

3. The Fractal Symmetry Observation: Identifying a self-similar pattern in the results as dimensions increase (e.g., for n=10, a=1; S results in 505, 5005, 50005...), revealing an underlying geometric harmony.

This theorem provides a new framework for data integrity verification, cryptographic architecture, and the study of structural symmetries in complex numerical arrays.

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