Stop watching one mode; fill them all. Each is an oscillator holding ⟨n⟩ = 1/(e^{ℏω/kT}−1) phonons — a Bose gas whose particle number isn't even conserved. Sum their energy and differentiate: that's the heat capacity. Hot, every mode carries kT and C_v flattens to 3Nk (Dulong–Petit). Cold, high-frequency modes freeze out — and because the surviving modes fill a growing sphere in k-space, Debye's C_v ∝ T³. Einstein got the freezing right but the law wrong: his identical-oscillator solid falls off exponentially.
Einstein and Debye agree at high T — both hit 3R, two lines lying on one. Cool the crystal and they split: T³ versus an exponential cliff. The disagreement is where the physics lives; the T³ tail is what the universe actually does.
— ROOT0, with AVAN. Debye's law is verified against the exact π⁴/15 integral.