Beal Conjecture, which asserts that for $a^k + b^m = c^n$ with $k, m, n > 2$, the bases $a, b,$ and $c$ must share a common prime factor. We prove it to be true with an approach that utilizes a sequence of rational perturbations $delta={delta_i}:{delta}_{i in mathbb{N}} subset mathbb{Q}$ , $delta_i > 0$ and $lim_{i to infty} delta_i = 0$ to treat such Diophantine equation as the critical limit-state of a geometrically constrained configuration. By defining a sequence of non-degenerate triangles $mathcal{T}_delta$ with rational side lengths ${a^k, b^m, c^n - delta_i}$, we establish a continuous mapping to the moduli stack of elliptic curves $mathcal{M}_{1,1}$.We demonstrate that the requirement for {rationality of the configuration} (the existence of a rational altitude $h_delta$) induces a sequence of Frey-Hellegouarch curves $E_delta$ that converge algebraically to the limit-state $E_{Beal}$. For signatures where $min(k,m,n) geq 3$, we invoke Ribet’s Level-Lowering Theorem to show that the associated Galois representation $ho_{E,n}$ is necessitated to reside within the {em empty space} of weight-2 cuspidal modular forms $S_2(Gamma_0(2))$.Simply speaking, our proof follows the often anticipated path of reasoning by which if Beal Conjecture were trueit must ultimately stand on the foundationthat underpins the validity of Fermat's Last Theorem.Furthermore, we provide a formal textit{Parity Lemma} to delineate the bifurcation at $n=2$, explaining why the modular sieve permits coprime solutions in Fermat-Catalan and Pythagorean signatures. This topological and arithmetic framework confirms that for strictly hyperbolic signatures, a solution exists if and only if $gcd(a, b, c) > 1$.