Omega Phase Energy Node:Definition, Existence, Uniqueness, and the Twelve-Layer Phase Closure

Omega Phase Energy Node (open), with the fixed symbol δopen: δopen = 6Φ² - 5π = 9 + 3√5 - 5π ≈ 2.4066 × 10⁻⁴. This paper does exactly one thing: to make this quantity itself fully clear — its existence, its numerical value, its uniqueness, its phase realization, and its structural role. Existence is guaranteed by the domain separation between Φ² (a quadratic algebraic number) and π (a transcendental number): for every integer x ≠ 0 and every y, xΦ² - yπ ≠ 0; the residual never closes to zero. The numerical value and the ratio are uniquely locked by the iterated expansion of the golden sequence: the structural inheritance condition m₀ = x requires the expansion depth on the algebraic side to equal the iteration depth on the phase side; the unique pair satisfying this ratio (6, 5), with flip layer m₀ = 6 = x, forming a finite phase closure of twelve layers — six positive and six negative. The precision wall theorem elevates this uniqueness from "numerically closest" to "structurally unique." The structural role of δopen is that of a cross-domain interface and dimensional yardstick: it is the unique quantity that simultaneously touches the closed form 6Φ² on the Φ side and the closed form 5π on the π side; it measures cross-domain distance, but does not enter the endpoint formulas. Section 0 gives the naming rationale: Ω takes the meaning "two dimensions is the terminus of downward projection," phase takes "twelve-layer phase locking," and energy node takes "the minimal resolvable node between the two domains."