OMEGA NAUGHT Ω₀

Let φ = (1+√5)/2 and Ω₀ = φ + π, and define τ = Ω₀⁻¹, λ = φ/Ω₀ = φτ, and μ = π/Ω₀ = πτ = 1 - λ. This reference note records exact identities and algebraic relations for Ω₀ and the associated constants τ, λ, and μ. It determines the generated field, its relative algebraic closure and quadratic automorphism, complete polynomial-relation ideals, traces, norms, normal forms, and Fibonacci and Lucas identities. It also presents the associated prime-cyclotomic intersection and finite-Fourier kernel criterion, unit-sum normalization of positive vectors, positive-ray classification, and exact symbolic arithmetic in the basis (1, λ) over ℚ(τ).

A finite rational specialization is defined by S = 10¹⁵ and Q = ⌊Sλ⌋ = 339949771344778. An exact rational enclosure proves agreement of the irrational and rational floor maps on every unsigned 15-bit input. Ordered Euclidean remainders and validated 15-bit byte folding provide exact finite reconstruction. The final section records the direct relation between the normalized distinguished ray and the frozen Scale-Resolved State Dynamics encoder.

The accompanying reproducibility archive contains the 27-page publication PDF, complete LaTeX source, three-pass build instructions, exact symbolic arithmetic implementation, exact 15-bit bridge verifier, Euclidean-ledger and byte-folding tests, generated JSON reports, and a SHA-256 manifest.