Paper (III) establishes an operational discriminator between completion-admitting and structurally non-terminating closure systems within a controlled class of discrete conservative models. Arguments developed in Gravitype Papers I and II identify structural conditions required for sustained temporal ordering without global completion. A persistent objection to such arguments is operational: even if completion is structurally forbidden, systems exhibiting long but finite delays may be observationally indistinguishable from those that are fundamentally non-terminating. Paper (III) addresses this objection directly. We introduce a minimal observable — the paired first-arrival delayΔτ₍first₎ = τ₍ON₎ − τ₍OFF₎ — evaluated under a pre-registered, locked protocol. The ON and OFF variants share identical state spaces, update rules, initialization, and stochastic realizations, differing only in an admissibility constraint governing reconciliation transitions. Across all tested system sizes (N = 48–256) and paired realizations, the admissibility-constrained (ON) variant exhibits a persistent directional delay relative to the completion-admitting (OFF) variant. Statistical evaluation using a paired Wilcoxon signed-rank test under a locked protocol yields unanimous sign agreement across uncensored pairs and p ≈ 7 × 10⁻¹⁰ in the primary Modal execution, with independent local verification confirming the effect. The result does not establish a physical law and does not instantiate the Gravitype substrate directly. Rather, it demonstrates that the structural distinction between bounded and unbounded admissible reconciliation depth admits operational consequence under hostile-to-researcher-freedom conditions. Paper (III) therefore converts structural necessity arguments into experimentally discriminable structure, providing the operational hinge for subsequent analysis of emergent accessibility gradients and structural regimes in Papers (IV) and (V).