Scale Graphs III: Onsager–Cheeger Observables as Order Parameters in a Navier–Stokes Scale Model

This paper is the numerical paper in the Scale Graphs sequence. The first two papers introduced the algebraic and analytic mechanisms: nonnegative Onsager edge production, Prigogine defect, Cheeger scale geometry, and two-channel tail control on dissipative attractors. Here we test the corresponding graph-thermodynamic observables in an archived Navier-Stokes scale model. The aim is deliberately narrower than a proof theorem: to determine whether the observables act as operational order parameters for the active/relaminarized transition in the tested data. The campaign is treated as a preregistered numerical dossier. Frozen constants are kept fixed before evaluation, and all headline quantities are read from shipped certificates or regenerated from shipped tables by the included script. The central result is a run-safe classifier using four Onsager-Cheeger features: minimum mobility, total Prigogine defect, bridge error, and no-pocket geometry. On the finite row set used by the certificate, the classifier separates ACTIVE from non-ACTIVE regimes with AUROC = 0.9848 on a direction holdout (d42 to d44) and AUROC = 0.9859 on a Reynolds holdout (Rc1500 to Rc2500). A raw-table cross-check recomputes the same claim independently from the shipped feature CSV, giving AUROC = 0.9847 and 0.9865. Group-k-fold by run is reported as degenerate because the archived runs are essentially mono-class. The release adds a fail-closed master reproduction check and a frozen protocol for future true out-of-sample v34 DNS features; no v34 classifier outcome is used in the present claims. A companion v34 closure package is used only to update numerical-health and bridge-context statements: the timestep check is mixed, the N=128 wall shows a qualitative large upward shift, and the bridge context is deliberately balanced. The original frozen P3F gate remains formally NO_PASS, while a separately frozen v34 addendum gives DEFECT_CONTROLLED_MULTI_EDGE on new active data; the bridge interpretation is deferred to the later bridge paper. Additional checks are reported with explicit epistemic weight: the weak bridge passes at run level with median Spearman correlation approximately one; terminal RELAM and ACTIVE channels have sharply different scale-graph signatures; the Rc2500,d44 wall is bracketed with a 1.45 percent gap; the N=128 anchor preserves active/relaminarized separation; the observed PDE RG sector is weakly unstable, with spectral radius 1.0332 and confidence interval [1.0086, 1.0562], paired leading moduli, and spectral gap one. Numerical reproduction dataset (v1.7): https://doi.org/10.6084/m9.figshare.33067970Paper I of the series: https://doi.org/10.6084/m9.figshare.32942279Paper II of the series: https://doi.org/10.6084/m9.figshare.33002558