Fermat's Last Theorem: Linear Space Analysis

<jats:p>Abstract This paper presents a definitive algebraic proof of Fermat’s Last Theorem strictly within a one-dimensional linear space where the components do not form a geometric triangle. When bases are restricted to a flat line, the system is governed by the perimeter constraint a + b = c. By applying Taha’s Coefficient Fact 1 (TCF1), the higher-power expression \(a^n + b^n\) is factored into lower-power base segments and their corresponding exponential scaling multipliers. Because individual components grow at a strictly slower rate than the unified target threshold (\(a^{n-1} &lt; c^{n-1}\) and \(b^{n-1} &lt; c^{n-1}\)), strict inequality bounds are established. Substituting the linear baseline directly into the bounding function mathematically demonstrates that \(a^n + b^n &lt; c^n\) for all exponents n &gt; 2. This structural divergence proves that power scales are permanently blocked from reaching equality on a flat line.</jats:p>