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 → Rezs, 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 − v²/c²) 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 3c²/L³ is consistent with ϕ ̸= 0 — this is the remaining internal mathematical task.