Paper 13 [4] conjectured that the scale coordinate of the (x, y, z, s) framework is a single complex value zs ∈ C, and identified a programme of steps to test this. The triangulation unification [8] and tetrad sub-leading paper [9] completed significant groundwork, but left the nature of the discriminating test unclear. This paper carries out two calculations that resolve the situation. Result 1 (sub-leading test is closed). The area gap ∆(L) = e^(4/L) − 1 satisfies 1 + ∆(L) = e^(4/L) exactly, so the triangulation function f(∆(L)) = 1 + 1/2 ln(1 + ∆(L)) = 1 + 2/L exactly at all orders. There are no sub-leading corrections to F within the static diagonal ansatz. The sub-leading test is not a discriminating test for the conjecture. Result 2 (native vs projected treatment). If zs is treated as a primitive quantity, the metric correction is governed by the ansatz that the metric factor is |zs|^2 (the Hermitian norm), giving |zs| = e^(2s/L) independently of the imaginary part ϕ at all orders. The tetrad paper’s ϕ2/L2 correction arises from the real-axis projection zs → Re[zs], not from the native treatment. TheBorn-rule operation |ψ|^2 removes the phase from single-outcome probabilities but preserves it in amplitudes; it is distinct from the real-axis projection. The metric correction F = 1 + 2/L is stable under both operations. Result 3 (the genuine discriminating prediction). The conjecture does not predict a correction to F = 1 + 2/L. It predicts the existence of a scale-dependent quantum phase ϕs/L that is discarded by the Born rule but that affects interference patterns in quantum systems. Near compact objects (L ≈ 5 nats for a neutron star), this phase reaches ϕs/L ∼ 0.2 rad for ϕ ∼ 1 (a natural order-of-magnitude estimate; ϕ is dimensionless in this framework, measuring scale-imaginary displacement in logarithmic units), potentially observable through phase-sensitive spectroscopic or quantum phenomena near compact objects. What remains for Paper 13 Steps 3–4. Step 3 (proper time with zs primitive) is substantially addressed here: the proper time is dτ = sqrt(|zs|^2/|zs,ref|^2 − v^2/c^2) dt, which gives F = 1 + 2/L exactly and is independent of ϕ. The open question is: what determines ϕ from the field equations? Step 4 (Einstein tensor with zs primitive) requires showing the deficit 3c^2/L^3 is consistent with ϕ ̸= 0 — this is the remaining internal mathematical task.

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.