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 aclassical 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 observationalprogramme (quantum interference near compact objects; Paper 16).

The mathematical development in this paper was produced in dialogue with Claude.ai (Anthropic) in Spring 2026, directed by the author. Use of AI assistance is acknowledged in accordance with standard scholarly practice.