Toolpath Turns

• Replacement curves on the triangular lattice: a generator of n unit steps from 0 to a chord c with N(c) = n, and a flag F, R, M or MR per step; M and MR only when c/conj(c) is a unit.
• census [orders] [stop]: every gentle generator walk (turns at most 60 degrees, point-avoiding), every flag assignment gentle by the junction closure and point-avoiding at level 2, expanded to the first crossing, up to level 4; mirrors where the chord allows them; prints the (generator, flags) pairs and, when the order is exhausted, their classes up to reversal and mirror; stop cuts after that many walks.
• plain [orders] [stop]: the same with F, R only.
• gosper: the Gosper curve's turn counts, axis counts and abs(h2)^2 at levels 1 to 6, asserted against 7^(k-1), 4*7^(k-1) - 1, 2*7^(k-1) and 7^k; its mirror image at level 2 asserted against the first 49 terms of A229214.
• bead: the second harmonic h2 of Hilbert's curve at levels 1 to 6 and Peano's at levels 1 to 4.
• rate [orders]: every curve avoiding points through level 4 at an order, one per class up to reversal and mirror, with its 120-degree turn counts at levels 1 to 5.
• Asserts exit nonzero; nothing is written.

RUN

• uv run python research/lab/py/toolpath-turns/toolpath.py census 3,4,7,9,13: 4 s.
• uv run python research/lab/py/toolpath-turns/toolpath.py plain 12: 1 s.
• uv run python research/lab/py/toolpath-turns/toolpath.py census 12 379: 84 s, order 12 with mirrors cut after walk 379 of 2646.
• uv run python research/lab/py/toolpath-turns/toolpath.py plain 16 97426: 60 s, order 16 with F, R cut after walk 97426 of 122964.
• uv run python research/lab/py/toolpath-turns/toolpath.py gosper: under 1 s.
• uv run python research/lab/py/toolpath-turns/toolpath.py bead: under 1 s.
• uv run python research/lab/py/toolpath-turns/toolpath.py rate 7: under 1 s; order 9 with mirrors passes 300 s.

WITNESSES

• toolpaths.md, The census: orders 3, 4, 7, 9 (mirrors), 12 (F, R) and 13 exhausted, pairs and classes, no gentle curve avoids points at level 3 (census 3,4,7,9,13, plain 12, Verified); the cuts at order 12 with mirrors and order 16 (census 12 379, plain 16 97426, Conjecture).
• toolpaths.md, The Gosper count: the turn triple at levels 1 to 6 and the mirror image against A229214 (gosper, Verified).
• toolpaths.md, The bead axis: abs(h2)^2 = 7^k for Gosper, h2 = -1 for Hilbert, h2 = (9^k - 1)/2 for Peano (gosper, bead, Verified).
• toolpaths.md, The printer's line: the two order-7 classes and their sharp counts 2*7^(k-1) and 3*7^(k-1) (rate 7, Verified).