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 phase-stable under the Hermitian-norm ansatz. 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 under two distinct treatments of zs — the native (Hermitian-norm) treatment and the projected (real-axis) treatment — which give complementary results that together fix the correct level structure of the conjecture.

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 (projected treatment: ϕ = 0 classically). This result applies to the projected treatment, not the native one. If the metric is allowed to become complex (real-axis projection applied before computing the metric), 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 under this treatment.

Result 3 (level structure of the conjecture). These two results apply to different treatments and are not contradictory: under native treatment ϕ is invisible to classical geometry; under projected treatment real stress-energy eliminates the imaginary metric component. Together they 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, we propose to interpret ϕ = sI as a quantum phase-like coordinate of the wavefunction — an identification that is plausible but not yet derived from the framework — which is discarded by the Born-rule operation |ψ|^2. The conjecture is internally consistent: 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 substantially addressed. The classical field-equation check does not refute the complex zs conjecture; it shows classical neutrality only. The conjecture cannot be confirmed or ruled out by 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).