FCP v6

v6 as committed. (1) Point clouds drop their short, small tracks (at most 3 layers, under 5 % of the largest slice): the slicers' blobs on noisy and sparse clouds; a mesh keeps them. (2) encode(atoms="auto") stops at twice the noise level read at the top height degrees of the long parts (the atoms are orthonormal: noise spreads evenly over them; median |score| / 0.6745); a prefix like any other. The format and the decoder are v5.31's; meshes encode to the same bytes. 287 tests pass (6 new).

One rule instead of tolerances. With orthonormal atoms, keeping those above a level lam is the best descriptor of its size; its error plus lam² per atom kept is the sum over atoms of min(score², lam²). The partition is chosen by the same rule: each track cut into the pieces of least total cost (every piece's rings projected on its own height basis, plus the energy the basis cannot hold, plus lam² × 0.5 for a part's header), by dynamic programming over cut positions. lam is the level of the T0-th atom at the reference budget T0 (iterated with the partition), so the partition depends on the data and one constant, and prefixes stay exact. It replaces FIT_TOL, FIT_GAIN, STEP_SLACK, FIT_MIN and the clouds' spacing tolerance; the same rule on meshes and clouds. A level from the noise alone (2 σ or the universal threshold) is optimal for large budgets and cuts 2-3 times more parts, worse at 256 atoms. scripts/rd_split.py (swapped in for the package's cut rule); CPU NumPy, 0.2-1 s per object, not optimized. Tracks are not dropped by it (a ring's energy includes its position, which is not the geometric cost of leaving it out): v6's short-track filter stays.

The medians fall with every step for both rules, on meshes, clean and noisy clouds; per object, rate-distortion has fewer or as many increases (meshes, Chamfer: none against one). The increases are at small budgets and on a few objects: the open box, the GSO bowl, the battery, all with cavities or thin walls. Two causes, neither the cut rule: (1) the angle error is one-sided (decoded surface to the nearest true normal), so a decode that draws less of the object can score better normals; the open box at 128 atoms has good normals (11°) and poor Chamfer (4.2 %), at 256 it is drawn more fully (Chamfer 2.1 %) with noisy walls (35°). (2) On the noisy clouds the slicer leaves thin walls as one ring around both faces (no hole part for the cavity over most of the height); its first harmonics are a filled ellipse, so the cavity fills as atoms arrive before later atoms carve it out (the bowl from 1.6 to 4.0 % at 32 → 64 atoms; no single part's atoms cause it). That is the point slicers' thin-wall problem (OPEN.md 9). Tested and rejected: parts drawn without both mean coordinates (dropping them is worse everywhere).

What joins a long part on one side (a handle's root, a limb) is peeled into a part that overlaps the body, so the body's part goes on through it; the decoder takes the union (the distance at a layer to the region's boundary only: the same bits on NumPy, numba and CUDA). fcp/_lobes.py: a robust fit along the height locates lobes (each round keeps at least the better half of the layers); each ring is cut on its own outline, walked off the lobe's tip and down to the fillets; the body is closed across the gap by its own wall (the ring either side of the gap in polar form, a cubic in angle, matched to both ends); a peeled run is extended along the moving cut until nothing sticks out; a run goes to the track it touches, and is not peeled when it touches none (free lobe parts added 6 parts on the elephant and the horse for nothing). Results (ten meshes, on against off): the mug's body is one part through the lower handle root, and the mug is better at every budget (256 atoms: Chamfer 0.264 against 0.282, normals over 30° 7.7 % against 12.7 %; with its upper root a part of its own, 0.246 and 4.0 %); the dino slightly better; the dress, shoe and horse worse at 256 / 1024 atoms; overall Chamfer 1.019 / 1.014 / 0.998 times, normals 0.97 / 1.02 / 1.00 at 256 / 1024 / 6144. The boundary flags cost 100-500 ms per decode on the host (the CUDA decode is 30-40 ms without them). Not in v6. 287 tests pass on the branch. Why the dress, shoe and horse lose. Their lobes are where a leg or an arm joins the body (the dress at 41-49: a leg's track ends at 40 and its remnant is peeled and given back to it); the body is cut at those joints anyway, so the peel only adds an overlapping part. The decisive test (second commit on the branch): keep a run only when the cut rule then makes fewer parts of the body and of the track its lobe joins. No run on the ten meshes passes: mug 6 → 7, dress 11 → 13 and 10 → 10, horse 8 → 9 and 4 → 4, shoe 12 → 12, dino 3 → 3, 6 → 6, 7 → 7 (the other runs touch no track). Peeling a lobe never makes the body's parts longer; the mug's gain at small budgets came from where the cuts fall and one more part. Conclusion: lobe parts do not make parts bigger; left on the branch as a record. Speed (1024 atoms, best of 3): v6 encodes the mug / dino / shoe in 62 / 75 / 91 ms on CUDA (118 / 129 / 168 on the CPU) and decodes in 13 / 16 / 17 ms (45 / 56 / 58); with lobes, encode about 1 s and decode 252 / 71 / 174 ms (host NumPy: the robust fit, the cuts, the boundary flags on every decode).

Offset patches over an angle range (an offset along a smooth base's normal over a height and an angle range, Chebyshev × the height basis): a protrusion's sides run along the normal, so the offset is a step in angle and a polynomial rings (the mug's handle root: 0.3-2.6 layer heights off, against 0.15-0.38 for the parts). A cut tolerance raised with the noise: no fewer parts and a worse Chamfer (the clouds' extra parts are tracks, not cuts). Spike cut placement: meshes 0.96 times the Chamfer at 256 atoms, not validated on clouds. The partition rules of the study before (higher degree, cut tests in shape, minimal partitions): fcp_2810_partition.