The Diagnostic Disagreement and the Closure Failure of the Quiet-Sun Corona Are the Same Relative Entropy

In the weakly collisional quiet-Sun corona, the electron distribution departs from Maxwellian, and ``the electron temperature’’ is not single-valued: two standard diagnostics differ by a stable factor $R\approx 2.4$ across the solar cycle. We show this departure is a single relative entropy whose two orthogonal components are the EUV–radio diagnostic disagreement and the failure of the local conductive closure. By the generalized Pythagorean identity for Bregman divergences, the relative entropy of the $\kappa \approx 2.5$ distribution from its radio-diagnostic Maxwellian projection ($1.20$ nats) partitions with no cross-term into two components. The first is the energy-matched closure deficit ($0.32$ nats), the information discarded by the local Spitzer–Härm closure (Edmonds 2026c), which itself has no convergent form across $\kappa \in [2,3]$. The second is the inter-projection diagnostic gap ($0.876$ nats), what the EUV and radio diagnostics disagree by and a closed-form function of $R$ within the shape envelope. Closure failure and diagnostic disagreement are therefore the same relative entropy in two orthogonal coordinates: together they account for the corona’s full departure from Maxwellian. The physical heat flux that survives is non-local and tail-dominated. Any mechanism that sets this $\kappa$ must act on the suprathermal tail, operate non-locally, and fix $\kappa$ to the observed band; of the standalone candidates only velocity filtration (Scudder 1992) meets the first two, and whether it, or anything, fixes $\kappa$ is the open problem we leave posed.