The Algebraic Vacuum: Zero-Ramification Conductor Model for the Goldbach Conjecture at N = 2ᵏ

When the even integer N is a power of 2, the Goldbach–Frey curve enters a zero-ramification regime: the static conduit factor radₒdd (N/2) = 1, so the conductor is governed entirely by the boundary summands p and q. We exploit this algebraic vacuum to isolate the pure effect of radical rigidity on the conductor distribution. Scanning N = 2ᵏ for k = 7, …, 14, we establish three structural phenomena: (1) Ground state locking: the minimum Chen's ratio satisfies ρₘin → 2 from above, locked by rad (p) = p for primes (proved unconditionally) ; (2) Bandwidth rigidity: the composite-to-Goldbach bandwidth ratio is consistently ≥ 2. 3 across all tested k; (3) Conductor floor: Goldbach pairs satisfy ρ > 2, while composite decompositions can reach ρ = 0. These results provide the sharpest available experimental evidence that the conductor geometry of the Goldbach family is fundamentally constrained by the algebraic rigidity of prime numbers.