A Dual-Vector Obstruction to Linear-Depth κ = 1 Lifts in the Collatz Parity-Word Framework

This paper analyzes κ=1 lifting in the affine parity-word representation of the accelerated Collatz map. Any κ=1 lift reduces to a 2-adic divisibility condition of the form v₂(C + tΔ) ≥ m, where Δ = 3ᵃ − 2ᵐ is odd, yielding a unique Hensel path for the lift parameter t modulo 2ʳ. We introduce a deterministic decoder defined as the inverse of a prefix-offset bijection and use it to reconstruct forced prefixes consistent with the lift constraint at each depth. Two complementary mechanisms obstruct linear-depth lifts in the near-critical regime a ≈ p* m, where p* = 1 / log₂ 3: • Vector A — deterministic feasibility failure for bounded lifts, where forced prefixes violate the prefix feasibility barrier before full depth is reached.• Vector B — a bridge inequality linking deep lifts to Diophantine approximation bounds on the critical defect εₘ = |3ᵃ − 2ᵐ| / 2ᵐ. Under explicit hypotheses isolating the remaining analytic tasks (Diophantine control and bridge inequality formalization), we obtain a conditional obstruction to linear-depth κ=1 lifts for sufficiently large m. Supporting computational experiments (Phase 5D, depths up to R = 4096, including dense lift sweeps) demonstrate strong empirical drift away from the critical density p*, with an observed uniform gap exceeding 0.10 across wide lift windows. This dual-vector architecture clarifies the structural obstruction mechanism and isolates the remaining analytic components required for an unconditional result.