This paper establishes a comprehensive mathematical program investigating the struc tural evolution of integer sequences under the iterated Euler’s totient function ϕk using absolute difference triangles. We demonstrate a universal phenomenon of boundary decay, where initially chaotic sequences—including linear and quadratic polynomials, prime num bers, prime powers, and products of consecutive primes—inevitably collapse into strictly periodic geometric boundaries. The core of this work is the formulation of the Grand Con jecture, which asserts the existence of a minimum iteration depth N for any such sequence to reach a stable periodic state. A significant result of this program is the derivation of the long-standing Gilbreath’s Conjecture (1958) as a localized corollary of the broader boundary decay mechanism applied to prime sequences. Our findings suggest that the iterated totient function acts as a universal filter, uncovering an infinite reservoir of undiscovered periodic structures within number theory.