OMEGA Marked Reference Geometry: Construction-to-Curve Theorem for a One-Arcsecond Circular Seed with Bilateral Golden Extensions

Let φ = (1 + √5)/2, ε = 1″ = π/648000, and let a > 0. In an oriented Euclidean plane, choose a center α and an ordered pair d, b on the radius-a circle such that ∠dαb = ε. On the chord line, extend outward from each seed endpoint first by a and then by aφ. This paper proves that these complete construction data determine a unique marked curve, the OMEGA kernel. Since 1 + φ = φ², the six baseline coordinates are forced to be ±s, ±(1 + s), and ±(φ² + s) in units of a, where s = sin(ε/2). An existence, uniqueness, and exact-characterization theorem establishes the equivalence between the stated construction and the resulting marked OMEGA kernel. From the resulting coordinate formula, the paper derives all center-to-mark radii, all pairwise mark distances, all fifteen pairwise center angles, triangle areas, and the normalized Gram matrix. The six baseline marks are collinear and have affine dimension one; after the center is included, the configuration has affine dimension two, and its center-relative landmark and Gram matrices have rank two. In an oriented realization in ℝ³, the ordered seed determines an orthonormal frame (u, v, n), and every mark has the exact form α + a(cu + σxⱼv). The planar kernel therefore admits exact placement at arbitrary position and orientation in ℝ³, while genuinely nonplanar completion requires independent depth, incidence, or signed-orientation data. The paper also establishes the positive-homothety scaling family of the construction and gives a separately defined metric transcription of the outer archival notebook scaffold, with all associated identities stated explicitly as conditional on that definition. The accompanying reproducibility package contains the manuscript source, computational implementation, canonical numerical exports, verification code, and cryptographic release metadata. Numerical coordinate exports are finite-precision realizations; the exact mathematical layer is given by the symbolic formulas and proofs in the paper.