Taha's Coefficient Facts TCF1 & TCF2

This paper introduces Taha’s Coefficient Fact1 (TCF1), a proposed axiom in number theory. TCF1 asserts that for natural numbers ( a < b < c ), if ( a^n + b^n = c^n ), then the scaled coefficients ( \frac{a^n}{a^{n-r}} + \frac{b^n}{b^{n-r}} ) cannot equal ( \frac{c^n}{c^{n-r}} ), provided ( a^{n-r} \ne b^{n-r} \ne c^{n-r} ). While axioms traditionally require no proof, the author offers a rigorous argument to validate TCF1 in response to institutional skepticism. This work contributes a novel perspective to the study of exponential equations and invites reevaluation of foundational assumptions in mathematics.