Paper 16: Near-Term Tests of the Complex Scale Coordinate Conjecture: Sub-leading Structure, Native vs Projected Treatment, and the Phase-Shift Prediction

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. The
Born-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.