[qmohsu]

Chapter 4's inline comparison had no two-way row, so "what does estimating the receiver clock actually cost?" was answerable only by intuition. example_toa_positioning does run an RTT chain, but on different anchors, a different target and a different error budget — its 0.704 m and this table's numbers were never comparable, and a deck built from the two would have compared them anyway.

The design: one varied factor

The new RTT (two-way TOA) series shares the anchors, the 50 test points, the seed, the sigma ladder and the realisation drawn from it — draw_range_noise is called once per level and handed to both range series. Only the state vector differs:

	measurement	state	clock bias
TOA	one-way pseudorange	(x, y, c·δt), Eqs. (4.24)-(4.26)	1.5 m injected
RTT	Eq. (4.7) converted range	(x, y), TOAPositioner	none — the round trip cancels it

Sharing the draw rather than redrawing is the point: aoa_bearings exists in this same file because the weighted and unweighted AOA columns used to draw their own noise, so their ratio carried a full realisation on each side. Same defect, avoided before it was written.

RTT is deliberately not noised through processing time and clock drift, though simulate_rtt_measurement offers exactly that. Doing so would vary the error budget as well as the state vector, and the row could attribute the difference to neither. The standalone RTT section keeps teaching that cost and is untouched. Nothing here charges RTT for the transponder either, so the result is the best case for two-way ranging, not a net verdict on it.

Measured

Median over all 50 fixes, python -m ch4_rf_point_positioning.example_comparison:

Level	σ TOA/RTT (m)	AOA (deg)	RSS (dB)	TOA	RTT	TDOA	AOA	AOA unw	RSS	RTT fail	AOA fail	RSS fail
1	0.00	0.0	0.0	0.000	0.000	0.000	0.000	0.000	1.652	0	0	21
2	0.05	1.0	2.0	0.044	0.044	0.047	0.121	0.355	2.219	0	0	19
3	0.10	3.0	4.0	0.088	0.083	0.091	0.384	1.430	3.246	0	0	26
4	0.20	5.0	6.0	0.162	0.152	0.175	0.519	2.433	4.029	0	0	34
5	0.50	10.0	8.0	0.507	0.431	0.567	1.129	7.026	8.045	1	4	38

TOA/RTT ratio: — / 1.00× / 1.06× / 1.07× / 1.18×.

Read that ratio as one number, not a trend

Adding the clock column to H inflates the position covariance by a factor set by the geometry alone — it does not grow with σ. Analytically this array gives 1.026 (median of per-point DOP ratios; 1.033 as a ratio of medians). The 1.00→1.18 spread is a 50-point median resolving a 2.6% effect through roughly ±7% of sampling noise: over 40 repeats per level the ratio of medians ranges 0.87 to 1.27, so a single level cannot even establish the sign.

So the tests pin the claim where it is real, and pin the shipped ordering separately as a property of seed 42:

• exactly, at every one of the 50 points, via the Schur complement of the clock column;
• statistically, over a 1200-point sample where 2.6% is resolvable — mean ratio bounded in (1.010, 1.045), a floor measured at ~4 SE below the observed mean of 1.0257 (10 seeds, std 0.0070);
• as a regression pin, RTT ≤ TOA at every noisy level of the shipped sweep, docstringed as a realisation pin rather than as physics.

The four pre-existing series are byte-identical

Verified at full precision, not at three decimals: running main's copy of the module and this one in a single process, every error value, solved mask and n_failed of TOA, TDOA, AOA, AOA_unw and RSS agree with max|difference| = 0.0. The mechanism is that draw_range_noise returns None at zero σ instead of a zero array, so the noiseless level still consumes nothing from the global stream, and the new series consumes nothing at any level.

Two defects found in this change's own output

• The ratio line first guarded on median > 0. The noiseless medians are 3.5e-10 m and 9.9e-10 m — both zero to a positioning system, both nonzero to float division — so it printed "0.35×" as the cost of the clock state. The floor is now STALL_M, the solver's own 1e-6 m step tolerance.
• A first test asserted the recovered bias was 1.5 m on noisy data; it reads 1.4642, because the solve absorbs the injected bias plus whatever common part the noise happens to have. What is exact there is the difference between the two clock estimates, and that is what is asserted now.

A log y-axis was rendered and rejected, not argued about

RSS at 8 m does crush the range series near zero, and the sibling geometry figure fixes exactly that with set_yscale("log"). It does not work here: level 1 is noiseless, so those medians are roundoff and a log axis spends ten of twelve decades on them, compressing every informative level into the top fifth of the panel. Recorded in the panel comment and the README so the next reader does not repeat it. Colours and dashes are now keyed by method rather than positional, so inserting a series cannot silently recolour the four already there; RTT is purple with its own dash pattern, needed because four of five methods coincide at 100 % in the solved-fixes panel.

Verified

• tests/ch4_rf_point_positioning — 78 passed (11 new).
• Docs guards — 751 passed / 1 skipped: test_chapter_architecture_sections, test_readme_file_structure, test_chapter_readme_code_blocks, test_readme_example_output (the transcript pin), test_documented_flags_exist, test_documented_paths_exist, test_documented_commands_use_module_form.
• test_repo_conventions + test_lint_debt_only_shrinks + test_examples_answer_help — 522 passed.
• test_every_figure_has_a_demo_behind_it — 203 passed, 8 skipped.
• ruff check on both changed files reports the same 3 pre-existing typing.Dict findings as main, and nothing new.
• python -m tools.chapter_dependencies — "already up to date" (no new core dependency).
• python -m ch4_rf_point_positioning.example_comparison exits 0; the regenerated PNG was opened and read — legend legible, five series present, nothing clipped.
• The notebook is untouched and needs no change: it imports only compare_geometries, which this PR does not modify.

Three mutations were run to check the new guards can fail: giving RTT the clock state too makes the ratio exactly 1.0000 (the position estimates become bit-identical, since a common bias is absorbed whole into c·δt); letting RTT redraw its own noise breaks the shared-draw test; injecting the bias into RTT takes its median to ~1.2 m and its failures to 47–48 of 50.

---

🤖 Generated with Claude Code