The Count in the Crystal: Debye's Wall Read as a Saturation Count

The Debye model is usually taught as an approximation. Read structurally, it is something this frame recognizes on sight: a saturation count. A mole of atoms owns exactly 3N_A = 1.806642×10²⁴ vibrational modes — finite, exact, material-independent — and Debye's construction is the rule that the count stops there: modes are filled with a ω² density up to a wall ω_D, and there are no more. This paper carries that reading to numbers across six declared materials. The wall computed from declared densities and sound speeds returns measured Debye temperatures with ratios 0.955 (lead) to 1.362 (copper); silicon and diamond land at 0.996 and 0.995. The spread is the honest price of the model's one assumed step — a linear dispersion driven through a real band structure — and the low-temperature T³ law is shown to be the count itself speaking: at T = θ_D/10 exactly 0.0010 of the modes are thermally reachable, and the molar heat 1.9438 J/mol/K at that reduced temperature is universal — the count, not the material.