Copper Repays the Count: Branch Resolution, a Bent Dispersion, and Three Oscillators

AF-4 read the Debye wall as a saturation count and priced its one assumed step — a linear dispersion with a single averaged sound speed — by a six-material table whose worst entry was copper at ratio 1.362. Its F3 charged the successor: recompute copper with measured dispersion input, and if the ratio moves by more than its printed distance from the measured θ_D (0.362), the averaging pricing was wrong in size. This paper runs that recompute. Branch-resolved — measured longitudinal and transverse sound speeds combined as the low-temperature wall requires, 1/v_m³ = (1/3)(1/v_l³ + 2/v_t³) with v_l = 4760, v_t = 2325 m/s — copper's wall lands at θ_D = 341.8 K against the measured 343 K: ratio 0.996, a distance of 0.004, ninety times inside the tolerance F3 set. The averaging pricing was right in size; the single mean speed was the whole error, exactly as charged. The paper then bends the dispersion on purpose — sine branches through the zone-boundary frequencies of record — and integrates the density of states: the mode count returns 3.0000 per atom, unchanged, while every filling quantity moves. Count and filling separate cleanly, in a computed integral, not a slogan. Finally the same reading is carried to optics proper: BK7's entire visible dispersion is carried by a Sellmeier sum of exactly three oscillators — a finite count of resonance addresses, declared, at 0.077, 0.141, and 10.176 μm.