Chapter 4: sweep the initial guess, and find the residual behind it

[qmohsu]

When an iterative positioning solve fails, the reflex is to blame the starting point. This adds ch4_rf_point_positioning/example_initial_guess_basin.py, which holds the geometry at the well-behaved square array, fixes one target, sets the measurement noise to zero, and sweeps the initial guess over 1681 seeds — twice, changing nothing but the space the residual is formed in.

	residual="tan"	residual="angle"
seeds that fail	785 / 1681	341 / 1681
quiet: stalled at the seed, or stopped somewhere plausible	263	0
loud: walked off past 100 m	522	341
failures that still reported converged=True	305	196

The example was written expecting the wrapped-angle form to remove the basin outright, and the sweep said otherwise. The docstring, the README section and the tests all state the narrower claim it can actually support: 2.3× fewer failures, and underneath that, the quiet class goes to zero — the failures that look like answers. A seed far outside the room still walks off under either parameterisation, and 196 of those still set the convergence flag.

So fixing the residual makes the solver honest, not safe, which is why core.rf.solve_batch's four conditions are not optional either way. One traced seed at (−4.5, −2.0) walked to 9.4 × 10¹¹ m in 18 iterations with converged=True; that run is the fourth panel of the figure.

Two details worth keeping

• The target sits off the seed lattice, at (3.2, 6.8). On a lattice point one of the 1681 seeds is the answer, so the solver cannot move and solve_batch scores it — correctly, by its own definition — as a stall. That is a classification artifact, and it is also the rule against seeding a solver with the ground truth, arrived at by accident. An assert in seed_grid() now makes the collision impossible rather than subtracting it afterwards.
• Classification goes through core.rf.solve_batch rather than being re-derived, and the failure modes are drawn as their own bar panel, per 030-figures-and-claims.mdc.

Also in this PR

• tests/ch4_rf_point_positioning/test_initial_guess_basin.py — pins both halves of the claim, so the caption cannot drift to "fixed", plus a guard that fails if the two parameterisations ever perform the same, at which point the example has stopped demonstrating anything. Companion to the existing test_aoa_initialisation_basin.py, which pins the same behaviour from a single cold start; this one sweeps the floor.
• Notebook: new Part 7, "When the solver lies." Parts 1–6 all report the accuracy of the solves that worked. Part 7 is the other half — first the collinear geometry from compare_geometries(), where TOA and TDOA fail 100/100 while GDOP calls the array fine at 1.43 against 1.02 for the square; then this sweep. Three "Try it" edits at the end.
• README: a new section, the Quick Start line, the File Structure entry, and a regenerated Architecture block (python -m tools.chapter_dependencies).

Checks

• tests/ch4_rf_point_positioning and the example-level meta tests: 368 passed (one genuine failure found and fixed — the example imported core without putting the repo root on sys.path first, per issue Examples do not put the repo root on sys.path, so a fresh clone fails on 'import core' #86).
• tests/docs: 507 passed, after regenerating the Architecture section and listing the new example in File Structure.
• ruff check: clean.
• The rendered PNG was opened and read before committing; the two panel titles and the y-label were re-laid-out after the first look, and the bar series were given their own palette because they had inherited the outcome colours.