The Imported kT, Priced: Rayleigh's Equipartition Assignment and the Ultraviolet Divergence It Returns

Census pricing (row CC-107) of the equipartition import of Rayleigh's 'Remarks upon the Law of Complete Radiation,' Phil. Mag. 49, 539-540 (1900) (completed with the corrected coefficient by Jeans, 1905), against a fixed bench (what is postulated, what is derived, what the postulates return numerically, what they cannot reach). The step -- cavity standing-wave modes counted soundly, then each mode ASSIGNED the classical equipartition average energy kT, imported whole from the kinetic theory of material oscillators with continuity of mode energy assumed -- is postulated, not derived; the mode counting is derived and credited. Priced in the notebook (17 checks, exit 0) from named constants against Planck's law as the bench's comparison standard: the import overstates each mode's mean energy by the exact factor (e^x - 1)/x, x = h nu/kT -- at the declared bench temperature 1000 K, 1.0244 at 1e12 Hz, 1.2834 at 1e13 Hz, 4.1752 at 5e13 Hz, 25.0911 at 1e14 Hz; the factor-2 crossover is x* = 1.2564 (lambda* = 38.17 micrometres at 300 K, 11.45 at 1000 K, 1.97 at 5800 K); integrated to a stated cut the law diverges as the cut's cube -- at 1000 K the Rayleigh-Jeans total below 1e14 Hz alone is 4.2928e-3 J/m^3, 5.67 times the entire finite Planck total a T^4 = 7.5657e-4 J/m^3, rising to 45.39 times at 2e14 Hz. Remainders at full size: the step is refuted in its domain (the measured cavity total is finite), its sound regime h nu << kT is credited (ratio 1.0244 at x = 0.0480, matching its own series), and the closure -- Planck 1900 (CC-07) denying the import -- is recorded as closure by replacement. Scout grade throughout: no verbatim primary reading claimed. EXTENSION. Vacuum catastrophe NOT SOLVED; quantum gravity NOT SOLVED.