Recovering Electron-Distribution Information from the Quiet-Sun Temperature Discrepancy

Temperature diagnostics compress an electron distribution into a scalar. If two diagnostics weight different velocity ranges, their disagreement can retain information either discards. We develop this measurement for the quiet Sun, where radio brightness and scale-height/ionization diagnostics read about 0.6 and 1.5 MK, a ratio of 2.4 plus or minus 0.3 stable across eight years. For specified projections, an exact relative-entropy identity partitions the discrepancy: the ratio fixes a family-independent temperature component; residual shape requires a family. Under the kappa family and stated projection assignments, the ratio gives kappa about 2.5 and a free-energy equivalent of 10--20% of the electron thermal energy. An independent EIS within-ion Fe IX test is consistent with kappa = 2.5 --3, not confirmed; its confirm condition fired under neither calibration treatment. Under narrow-DEM conditioning, the Maxwellian residual is 2.8 times the conservative systematic floor. A published broad Maxwellian DEM restores spectroscopic consistency, but its material gives a class-level radio-to-EUV cap of 1.21 against the measured class value 2.4 plus or minus 0.3. Across all stated treatments, the Maxwellian fails at least one constraint in this class-level joint comparison; the records are neither co-temporal nor co-spatial. Conditional tests, not further evidence, find that the local Coulomb/runaway channel falls 39--56 times short and that a 1.8--3.5 keV stopping-column scale overlaps the inferred 1.7--3 keV sharp-edge bracket. Termination there remains a working hypothesis. The central result is the measurement construction: information lost to either temperature alone becomes recoverable from their disagreement. Direct shape confirmation requires a cross-class or distribution-resolving measurement.