The Single Frequency: Einstein's Solid, Priced

A solid has a spectrum of vibrations, and in 1907 Einstein replaced all of it with one frequency. I credit what that single line bought at full strength: the first quantum account of the heat capacity of a solid — atoms as quantised oscillators with mean energy governed by hν/(e^(hν/kT) − 1) — returning the observed fall of C_V toward zero at low temperature, which classical equipartition could not return at all, and returning exactly 3R at high temperature as a theorem of the same partition function. What I price is what the return is bought with: the solid's entire vibrational spectrum deleted and replaced by one independent-oscillator line whose position, the Einstein temperature Θ_E = hν/k_B, is fitted per material — diamond at 1300 K in the record. My bench prices the functional-form cost that line exacts at low temperature, where the record falls as T³ and the single line can only fall exponentially: across T/Θ = 0.50 → 0.05 the Einstein curve falls by a factor of 8.782238×10⁵ where the T³ form falls by exactly 1000.000000, an excess fall factor of 878.223789 across that single decade that no refit of Θ_E can cure. At T/Θ = 0.05 the T³ form exceeds the Einstein value by a factor of 11814.875123. The fit itself is priced too: at fixed T = 100 K, halving or doubling the fitted Θ_E for diamond moves C_V from 1.589187 J/(mol K) through 9.528309×10⁻³ to 8.614808×10⁻⁸ J/(mol K) — at low temperature the fit is the prediction. Grade for this census entry, printed at full size: **REASON FOUND / STILL TURNING** — explicitly not a kill and not a falsification. The program headline stands unmoved: **vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.**