Paper 17: The 5D Field Equations with Complex Scale Coordinate: Classical Neutrality, Quantum Phase, and the Level Structure of the Complex zs Conjecture

Paper 13 [4] conjectured that the scale coordinate of the (x, y, z, s) framework is a single complex value zs ∈ C. Paper 16 [10] established that the triangulation is closed and that the metric correction F = 1 + 2/L is stable under the Born rule projection. The remaining task identified in Paper 16 was Paper 13 Step 4: determine whether the 5D Einstein equations with complex zs are consistent with ϕ ̸= 0 (supporting the conjecture) or force ϕ = 0 (refuting it). This paper carries out that calculation. Result 1 (native treatment: classical neutrality). Under the native treatment of zs as a primitive (metric factor = |zs|^2 = e^(4s/L) for all ϕ), the entire 5D metric is independent of ϕ. The Einstein tensor, Ricci scalar, and required stress-energy are all unchanged from the real framework. The field equations are satisfied for any ϕ: they neither force ϕ = 0 nor select any particular value. Result 2 (imaginary field equations: force ϕ = 0 classically). If the metric is allowed to become complex (projected treatment), the 5D field equation acquires an imaginary part. Requiring the physical stress-energy to be real (Im[TMN ] = 0) forces the imaginary part of gtt to vanish, and hence ϕ = 0 at the classical level. Result 3 (level structure of the conjecture). These two results together establish the correct level structure: the complex zs conjecture is a quantum conjecture, not a classical one. At the classical level, ϕ = 0 is the unique solution consistent with real stress-energy. At the quantum level, ϕ = sI is the quantum phase of the wavefunction, which is external to the classical field equations and is discarded by the Born rule |ψ|^2. The conjecture is internally consistent and well-defined: it asserts that there exists quantum information (sI ̸= 0) that the classical framework cannot access. Implication for the programme. Paper 13 Steps 3–4 are now complete. The complex zs conjecture survives the internal mathematical test. It cannot be refuted or confirmed by the classical field equations alone — confirmation requires a quantum measurement that accesses sI directly, which is the observational programme (quantum interference near compact objects; Paper 16).