The Euler-Mascheroni Constant from First Principles: 66-Digit Precision from Icosahedral Geometry

We present a closed-form expression for the Euler-Mascheroni constant γ achieving 66-digit precision using only e and integers derived from D=3 geometry. The base formula γ ≈ (e² + 5)/(9e − 3) with systematic corrections reveals that γ is not transcendental chaos but geometric necessity. The key insight: γ√3 ≈ 1 (0.02% accuracy), meaning γ is the Eisenstein projection of unity. The recurrence structure encodes twin prime signatures — the seed values (8, 11, 3) use 11 = V−1, the lower twin at the icosahedral vertex center. The number 19 = E−V+1 appears as the capacity threshold. v2.0 updates: Connection to twin prime structure (11, 19, 23 are twin-prime-adjacent), Completing Closure interpretation (the recurrence "tends without arriving"), explicit link to the fine structure constant α (same scaffold, α adds √3 for physical measurement), updated cross-references. The same geometric scaffold that generates γ also yields α to experimental precision, confirming that number theory and physics share a common foundation in D=3 icosahedral geometry.