Unit cubes in a cube: 60 new certified upper bounds for n < 1000 (three-band 3D L-frame and 45-degree lattice packings)

60 new upper bounds for packing n unit cubes in the smallest cube (s3(n), rotations allowed), all for n < 1000 and all verified EXACTLY (no floating point): the 45-degree packings in Z[sqrt2], the three-band packings in exact rational arithmetic. MATLAB verification scripts and certificates are included (13 files). A. Three-band 3D L-frame (side just below an integer K). A core of axis-parallel cubes, three tilted bands along three walls and a corner box. Each band is the prism of a 2D tilted band (L-frame of S. Ryu; end filling of E. Daniel as written by K. Fang; independent band parameters by M. Karimi), extruded along an integer length; the bands are arranged cyclically so that no layer is lost. The three-band rule is due to M. Karimi. - n = 34: s3(n) < 4 (side 3.998996662055) - n = 386-390: s3(n) < 8 (side 7.998998171459) - n = 569-584: s3(n) < 9 (side 8.998996662055) - n = 802-819: s3(n) < 10 (side 9.998981592910) For these n the best previously known bound was the trivial side K; no prism (stacked 2D packing) can reach them, since a prism loses a whole layer. B. 45-degree lattice packings (side a + sqrt(2)/2), found by exact integer programming on a structured lattice with M. Karimi's "CT-scan" rule (keep the axis-parallel cubes, re-place the tilted cubes in 3D). - n = 73-76: s3(n) <= 4 + sqrt(2)/2 - n = 137-143: s3(n) <= 5 + sqrt(2)/2 - n = 231-239: s3(n) <= 6 + sqrt(2)/2 C. Reconstruction: explicit coordinates for Friedman's 1998 bound s3(33) <= 3 + sqrt(2)/2 (hence also n = 31, 32), for which no reconstructible arrangement had been published. Computational setting: all results were obtained without GPUs, on MATLAB Online, under a slow and intermittent internet connection in Iran, using geometric ideas that shrink the search to laptop size. The constructions are deliberately simple and should be seen as seeds; GPU-based optimisation with free angles can very likely lower these sides further. Collaboration is welcome. Computations were designed and run with the assistance of an AI assistant (Claude, Anthropic); all ideas and rules credited to M. Karimi are his. Full details, certificate formats and references are in README.md.