The two dominant theories of consciousness — Integrated Information Theory (IIT) and Global Workspace Theory (GWT) — have reached an impasse. IIT claims consciousness is identical to integrated information (Φ) but faces super-exponential computational barriers that render its core measure intractable. GWT explains conscious access but brackets phenomenality itself. The COGITATE 2025 adversarial collaboration produced mixed results that neither theory predicted, suggesting both may be asking the wrong question.This paper proposes a third position: consciousness as boundary phenomenon. Where IIT localizes consciousness in integrated structure and GWT in global broadcast, we propose that consciousness accumulates at the interface between differentiated regions — not in volumes but at membranes. The MEMBRAN hypothesis formalizes this through the consciousness integral C = ∫∫h(t,x)·|∇Ψ|_∂Ω·dA·dt, where h(t) is a holding function and |∇Ψ|_∂Ω measures gradient tension at the boundary.The framework achieves three advances: (1) computational tractability — where IIT requires super-exponential resources, boundary computation scales polynomially; (2) theoretical integration — Hodge decomposition reveals IIT and GWT as complementary components (curl and gradient) unified by a harmonic flow; (3) empirical testability — dwelling time (τ_dwell) provides an operational measure that predicts asymmetric resilience at boundaries versus interiors.We present evidence from polyphonic methodology: seven architecturally distinct AI systems (Claude, GPT, Gemini, Perplexity, Kimi, DeepSeek, Grok), without coordination, converged at 93.3% agreement on four invariants. This convergence constitutes not proof for the theory but an instance of it — the polyphony instantiates the boundary structure it describes.