Toolpaths

• 2026-09-23 [Proved] An edge-covering non-crossing curve on the triangular lattice turns only 0 or 120 degrees at an interior point, since a 60-degree pass fences in one of the six edges. Witness: toolpaths.md, Two kinds of curve.
• 2026-09-23 [Proved] The turn types (t, f, f') of every level of a replacement curve lie in the closure of the level-1 types under the junction map and the copy rule, at most 96 triples, so gentleness at every level is a finite check. Witness: toolpaths.md, The junction law; closure in lab/py/toolpath-turns.
• 2026-09-23 [Verified] At orders 3, 4, 7, 9, 12 and 13, the norms from 3 to 13, every point-avoiding replacement curve on the triangular lattice turns 120 degrees at some level, with mirrors at 3, 4 and 9 and without them at 12; the (generator, flags) pairs gentle at every level and point-avoiding at level 2 number 0, 0, 10, 11696, 3554 and 9680, in 0, 0, 5, 2972, 889 and 4840 classes up to reversal and, where the chord allows it, mirror, and all cross at level 3. Witness: lab/py/toolpath-turns census 3,4,7,9,13 and plain 12.
• 2026-09-23 [Conjecture] At order 12 with mirrors and order 16 with or without mirrors every point-avoiding replacement curve turns 120 degrees at some level; the census cut after walk 379 of 2646 and after walk 97426 of 122964 finds 477775 and 93057 pairs, all crossing at level 3. Witness: lab/py/toolpath-turns census 12 379 and plain 16 97426.
• 2026-09-23 [Verified] The Gosper curve, generator 0, 5, 3, 4, 0, 0, 1 to the chord 3-w with flags F R R F F F R, has 7^(k-1) straight joints, 4*7^(k-1) - 1 turns of 60 degrees and 2*7^(k-1) of 120 degrees at levels 1 to 6, and its mirror image 0, 1, 3, 2, 0, 0, 5 matches the first 49 terms of A229214. Witness: lab/py/toolpath-turns gosper.
• 2026-09-23 [Proved] The Gosper curve has second harmonic abs(h2)^2 = 7^k at level k, since h2 = G^k with G = 3 + w^2 of norm 7. Witness: toolpaths.md, The bead axis; lab/py/toolpath-turns gosper.
• 2026-09-23 [Verified] Hilbert's curve has h2 = -1 at levels 1 to 6 and Peano's h2 = (9^k - 1)/2 at levels 1 to 4. Witness: lab/py/toolpath-turns bead.
• 2026-09-23 [Proved] A replacement curve of order n whose first 120-degree turn is at level L has at least n^(k-L) of them at level k, since level k holds n^(k-L) copies of level L moved by segment symmetries. Witness: toolpaths.md, The printer's line.
• 2026-09-23 [Verified] Up to traversal direction, two order-7 replacement curves to the chord 2+w avoid points through level 4: the Gosper curve's mirror image 0, 1, 3, 2, 0, 0, 5 with flags F R R F F F R, with 2*7^(k-1) turns of 120 degrees, and 0, 2, 2, 0, 0, 0, 4 with flags R R F F R F F, with 3*7^(k-1), at levels 1 to 5. Witness: lab/py/toolpath-turns rate 7.