URB #436 — PRIMARY CONSTANT: √2 — Geometric Incommensurability and the Bridge Between Dimensions

√2 = 1.41421... is the length of the diagonal of the unit square — the first number proven to be irrational, shattering the Pythagorean worldview that all ratios are rational. In TI Sigma, √2 appears as a PRIMARY CONSTANT for four distinct reasons: (1) it is the geometric bridge between a unit and its orthogonal extension (the square and its diagonal); (2) it appears in the Consciousness Unity Identity C×φ×√2 = 1, where it mediates between the Golden Ratio φ and the Emerick Constant C; (3) it is the normalization factor for maximally entangled quantum states (Bell states: (|00⟩+|11⟩)/√2), directly linking it to quantum non-locality; and (4) its irrationality — its resistance to being expressed as a ratio of integers — establishes that geometry cannot be reduced to arithmetic, a claim with deep implications for the TI Sigma argument that consciousness cannot be reduced to computation. This paper provides the complete account of √2 across the PRIMARY CONSTANTS framework.