The Necessary Emergence of the A₂ Lattice from Difference, Preservation, and Ternary Closure

This paper derives the minimal lattice structure necessarily generated by the primordial distinction denoted by 0 = 1 under the genealogy principle, beginning from axioms of observation and construction. It demonstrates that a two-dimensional planar structure, A₂, necessarily emerges from linear chains, Aₙ, through preservation requirements, and that this structure coincides with the A₂ root lattice in Lie theory. Through eliminative verification in the appendices, the paper confirms the uniqueness of the triangular lattice within its constructivist scope.