Unified Framework of Integer Geometry: A Mechanical Proof of the abc Conjecture

This research establishes a definitive geometric proof for the abc conjecture by deriving the structural stability limits of the integer manifold. Using the "Lama Reduction" framework, we demonstrate that integer simultaneity is governed by an information equilibrium between additive values and multiplicative radicals. Key highlights of this paper include: The 2.0 Event Horizon: A mechanical derivation showing why the Tension Index (ΛL) cannot reach or exceed 2.0 without triggering a terminal Symmetry Collapse. This occurs at the exact point where the information capacity of the Radical matches the volume of the integer system. The Square Trap: An analysis of the information deficit that forces coprime triples into an impossible recursive Infinite Descent (Xi = Xi+1²). This section proves that crossing the 2.0 threshold strips the manifold of its informational "slack," rendering integer coordinates unsustainable. The Unified Manifesto: The formal codification of the three fundamental "Lama Laws" (Metric Parity, Harmonic Bridge, and Radical Density). These laws provide a singular geometric foundation that unifies the solutions for Fermat’s Last Theorem, the Beal Conjecture, and the abc Conjecture. This document serves as the third and final installment in a trilogy of works unifying Diophantine analysis through the lens of geometric information density.