Victor Edmonds

For a plasma whose electrons carry a $κ\approx 2.5$ suprathermal tail, the Spitzer-Harm conductive closure does not exist: the conductive flux is the tail-carried third velocity moment, and the local conductivity integral diverges across the entire $κ\in [2,3]$ range -- the finite value the closed-form $κ$-conductivity returns at $κ= 2.5$ is an analytic continuation of a divergent integral, not a physical conductivity. Edmonds (2026a) places the quiet solar corona (QS) in this regime. Taking that as premise, two failures follow for any plasma in the class: the standard EUV-DEM diagnostic cannot resolve such a plasma, and the conductive term of the standard QS energy budget has no valid form. The diagnostic failure is shown end-to-end. A single-T $κ= 2.5$ probe, a multi-T $κ= 2.5$ source, and a multi-T Maxwellian source, all run through the regularized DEM inversion of Hannah & Kontar (2012), recover $\log T$ widths inside the FWHM distribution the same pipeline returns from 80 real quiet-Sun AIA patches; the pipeline cannot distinguish them. Two structural features also emerge: a Fe XI charge-state crossover and an EUV continuum reversal. The ionization-gated diagnostic structurally returns the tail-weighted effective temperature $T_{\mathrm{eff}}$, while Spitzer-Harm takes the bulk-core $T_{\mathrm{core}} = (κ- 3/2)/κ\cdot T_{\mathrm{eff}}$ as input. The mismatch invites a temperature substitution yielding a budget reduction -- mechanically correct and physically empty, because the coefficient it corrects has no convergent form: it is the Fourier-law closure itself that fails, not its temperature input. Two QS pillars for impulsive heating -- DEM-width multi-thermality and the conductive-budget gap -- lose their structural assumptions, and the budget question shifts to non-local kinetic transport outside any fluid closure.