From 55328c615a2d9c64c9baa2e7a38784650b327134 Mon Sep 17 00:00:00 2001
From: qmohsu <qmohsu@gmail.com>
Date: Wed, 2 Sep 2026 13:17:05 +0800
Subject: [PATCH 1/2] feat: Chapter 4 prices the clock state against a beacon
 that answers back

The 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 is one varied factor, and the discipline is the experiment.** 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: TOA solves `(x, y, c*dt)` per Eqs. (4.24)-(4.26) because a
one-way pseudorange carries an unknown receiver clock, RTT solves `(x, y)`
because Eq. (4.7)'s round trip is timed on one clock and cancels it. No bias is
injected into RTT.

Sharing the draw rather than redrawing is the point: `aoa_bearings` exists
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. It would vary the error
budget as well as the state vector and the row could attribute the difference
to neither. The standalone section keeps teaching that cost and is untouched.

Measured (median over all 50 fixes, sigma 0 -> 0.5 m):

    sigma  0.00   0.05   0.10   0.20   0.50
    TOA    0.000  0.044  0.088  0.162  0.507
    RTT    0.000  0.044  0.083  0.152  0.431
    ratio  --     1.00x  1.06x  1.07x  1.18x

**That column is one number, not a trend, and reading it as a trend was the
trap.** The clock column inflates the position covariance by a factor set by
the geometry alone -- it does not grow with sigma. Analytically this array
gives 1.026 (median of per-point DOP ratios). The 1.00-1.18 spread is a
50-point median resolving 2.6% through about +/-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. The tests therefore pin the claim where it is
real -- exactly, at every one of the 50 points, via the Schur complement, and
over a 1200-point sample where 2.6% is resolvable -- and pin the per-level
ordering separately, labelled as a property of the shipped seed.

**The four pre-existing series are byte-identical, verified at full precision
rather than 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 sigma 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 were found and fixed in this change's own output.** The ratio
line first guarded on `median > 0`, and 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 "0.35x" as the cost of the clock state. The floor is now
`STALL_M`, the solver's own 1e-6 m step tolerance. And 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 has. 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 rather than 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")` -- but 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. The reason
is recorded in the panel's comment and in the README so the next reader does
not repeat the twenty minutes.

Colours and dash patterns are keyed by method rather than positional, so
inserting a series cannot silently recolour the four that were 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: 78 passed in `tests/ch4_rf_point_positioning` (11 of them new); 751
passed / 1 skipped across the docs guards, including the README transcript pin;
`test_repo_conventions` and the lint ratchet green; ruff on the two changed
files reports the same 3 pre-existing `typing.Dict` findings as main and
nothing new; `python -m ch4_rf_point_positioning.example_comparison` exits 0
and the regenerated figure was opened and read.

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*dt`); 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.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
---
 ch4_rf_point_positioning/README.md            |  106 +-
 .../example_comparison.py                     |  284 ++-
 .../figs/ch4_rf_comparison.pdf                |  Bin 34734 -> 36310 bytes
 .../figs/ch4_rf_comparison.png                |  Bin 214642 -> 228286 bytes
 .../figs/ch4_rf_comparison.svg                | 1589 ++++++++++-------
 .../test_rtt_matched_comparison.py            |  396 ++++
 6 files changed, 1660 insertions(+), 715 deletions(-)
 create mode 100644 tests/ch4_rf_point_positioning/test_rtt_matched_comparison.py

diff --git a/ch4_rf_point_positioning/README.md b/ch4_rf_point_positioning/README.md
index 31304d0..e2bec3b 100644
--- a/ch4_rf_point_positioning/README.md
+++ b/ch4_rf_point_positioning/README.md
@@ -395,7 +395,7 @@ round trip measures distance better.
 Running `python -m ch4_rf_point_positioning.example_comparison` generates a comprehensive comparison:
 
 The left four columns are the noise injected at each level; the middle columns
-are the resulting median position error over all 50 fixes; the right two count
+are the resulting median position error over all 50 fixes; the right three count
 the fixes that failed. One draw is not a measurement, so this table reports
 medians rather than the single realisation it used to — see
 `.cursor/rules/030-figures-and-claims.mdc`.
@@ -404,25 +404,78 @@ medians rather than the single realisation it used to — see
 ```
 Results Summary (median error in metres)
 ======================================================================
-  Clock bias: 1.5 m (TOA only; cancels in TDOA)
+  Clock bias: 1.5 m (TOA only; cancels in TDOA and RTT)
   RSS config: Rayleigh short-term (sigma=0.5), 5 samples averaged
   AOA anchor 3 is 10x noisier than the others; 'AOA unw' solves the same bearings unweighted
-Level  TOA(m)    TDOA(m)   AOA(deg)  RSS(dB)   TOA       TDOA      AOA       AOA unw   RSS       AOA fail  RSS fail
---------------------------------------------------------------------------------------------------------------------
-1      0.00      0.00      0.0       0.0       0.000     0.000     0.000     0.000     1.652     0         ~
-2      0.05      0.05      1.0       2.0       0.044     0.047     0.121     0.355     2.219     0         ~
-3      0.10      0.10      3.0       4.0       0.088     0.091     0.384     1.430     3.246     0         ~
-4      0.20      0.20      5.0       6.0       0.162     0.175     0.519     2.433     4.029     0         ~
-5      0.50      0.50      10.0      8.0       0.507     0.567     1.129     7.026     8.045     ~         ~
+  'RTT' is two-way TOA on the identical range-noise draw as 'TOA', solved
+  for (x, y) alone: same anchors, points, seed and sigma, one fewer state
+Level  TOA(m)    TDOA(m)   AOA(deg)  RSS(dB)   TOA       RTT       TDOA      AOA       AOA unw   RSS       RTT fail  AOA fail  RSS fail
+----------------------------------------------------------------------------------------------------------------------------------------
+1      0.00      0.00      0.0       0.0       0.000     0.000     0.000     0.000     0.000     1.652     ~         0         ~
+2      0.05      0.05      1.0       2.0       0.044     0.044     0.047     0.121     0.355     2.219     ~         0         ~
+3      0.10      0.10      3.0       4.0       0.088     0.083     0.091     0.384     1.430     3.246     ~         0         ~
+4      0.20      0.20      5.0       6.0       0.162     0.152     0.175     0.519     2.433     4.029     ~         0         ~
+5      0.50      0.50      10.0      8.0       0.507     0.431     0.567     1.129     7.026     8.045     ~         ~         ~
 ```
 
-The two fail columns carry `~` because they are integer counts with no
+The three fail columns carry `~` because they are integer counts with no
 tolerance to give: a fix "fails" partly on whether an iterative solve reached a
 1e-6 m step inside its budget, and that is exactly the kind of thing a
 different LAPACK moves by one. The medians beside them are pinned. On this
-machine the RSS column reads 21, 19, 26, 34, 38 of 50 and AOA's level 5 reads
-4 — none of them the >100 m kind, which is `[0, 0, 0, 0, 0]` and printed under
-the table.
+machine the RSS column reads 21, 19, 26, 34, 38 of 50, AOA's level 5 reads
+4, and RTT reads 0, 0, 0, 0, 1 — none of them the >100 m kind, which is
+`[0, 0, 0, 0, 0]` and printed under the table.
+
+#### What estimating the clock costs (`TOA` against `RTT`)
+
+The two range columns are a **matched pair**, and the match is the experiment.
+`RTT` is two-way TOA — Eqs. (4.6)-(4.9) — and it shares everything with `TOA`
+except one thing: the state vector. Same anchors, same 50 test points, same
+seed, the same sigma ladder, and the *same realisation* drawn from it, because
+the example draws the range noise once per level and hands the array to both.
+What differs is that a one-way pseudorange contains an unknown receiver clock
+offset, so `TOA` solves for `(x, y, c*dt)` per Eqs. (4.24)-(4.26); a round trip
+is timed on one clock and cancels that offset by construction, so `RTT` solves
+for `(x, y)` and no bias is injected into it.
+
+So the gap between the two columns prices the third unknown, and it is
+measured rather than argued:
+
+| σ (m) | TOA (m) | RTT (m) | TOA / RTT |
+|---|---|---|---|
+| 0.00 | 0.000 | 0.000 | — both exact |
+| 0.05 | 0.044 | 0.044 | 1.00× |
+| 0.10 | 0.088 | 0.083 | 1.06× |
+| 0.20 | 0.162 | 0.152 | 1.07× |
+| 0.50 | 0.507 | 0.431 | 1.18× |
+
+**Read that column as one number and not as 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 — and for this square array the analytic value is
+**1.026** (median over per-point DOP ratios; 1.033 as a ratio of medians). The
+spread from 1.00 to 1.18 above is a 50-point median resolving a 2.6% effect
+through about ±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.
+`tests/ch4_rf_point_positioning/test_rtt_matched_comparison.py` therefore pins
+the claim twice: analytically, where the inequality is exact at every one of
+the 50 points, and over a 1200-point sample, where 1.026 is resolvable.
+
+Two things this row deliberately does **not** do. It does not model the RTT
+noise through processing time and clock drift, though `simulate_rtt_measurement`
+offers that and Example 5 of `example_toa_positioning` uses it — the σ here is
+charged on the *converted* range of Eq. (4.7). Varying the error budget as well
+as the state vector would leave the comparison unable to attribute the
+difference to either. And nothing here charges RTT for the transponder or for
+the processing-time calibration it needs, so the ~3% is the *best case* for
+two-way ranging, not a net verdict on it.
+
+The noiseless row is worth its own sentence: both range methods are exact
+there, at 3.5e-10 m and 9.9e-10 m, so the two-way series is in fact the larger
+of the two by 0.6 nanometres. Nothing is measured at σ = 0, which is why the
+example prints no ratio for level 1. An earlier draft guarded that print on
+`median > 0` and duly reported "0.35×" as the cost of the clock state — a
+ratio of two roundoff residues under a heading claiming to price a physical
+effect.
 
 **TOA reads 0.000 m at level 1 because it now estimates the clock.** The
 comparison injects a shared 1.5 m receiver bias into the TOA pseudoranges, and
@@ -470,18 +523,27 @@ read was a ratio between two different noise realisations.
 
 *This figure shows four subplots comparing RF positioning methods:*
 - **Top-Left:** median error over every fix, against the noise **level index**
-  1–5. The x-axis is an index and not a distance because the four methods have
-  four schedules in three different units; each series carries its own in the
+  1–5. The x-axis is an index and not a distance because the five methods have
+  five schedules in three different units; each series carries its own in the
   legend. They used to be drawn at TOA's metre positions under an axis labelled
   "Measurement Noise (m)", which put AOA's 10° at x = 0.5 m.
-- **Top-Right:** error CDF at level 3 — the panel title names all four methods'
-  noise there, rather than only TOA's. TOA/TDOA/AOA are sub-metre; RSS spreads
-  past 12 m.
+  **Do not look for the TOA/RTT gap here.** RSS at 8 m puts the three range
+  series in a band near zero, and a log axis — which is how the sibling
+  geometry figure solves exactly this — was rendered and rejected: level 1 is
+  noiseless, so those medians are roundoff at 1e-10 to 1e-8 m and a log axis
+  spends ten of twelve decades on them, compressing every informative level
+  into the top fifth of the panel. A 2.6% difference between two series that
+  also share a schedule is a table result, not a picture.
+- **Top-Right:** error CDF at level 3 — the panel title names all five methods'
+  noise there, rather than only TOA's. TOA/RTT/TDOA/AOA are sub-metre; RSS
+  spreads past 12 m.
 - **Bottom-Left:** error distribution at level 3 — RSS has far higher variance.
 - **Bottom-Right:** fixes solved, on the same four conditions as the table.
-  TOA and TDOA hold 100% across the sweep, AOA falls to 92% at level 5, and RSS
-  runs 58% down to 23%. This is where the failures live, so that the panel
-  above it can be an accuracy.
+  TOA, RTT and TDOA hold 100% across the sweep bar a single RTT fix at level 5,
+  AOA falls to 92% there, and RSS runs 58% down to 23%. This is where the
+  failures live, so that the panel above it can be an accuracy. Each method has
+  its own dash pattern, because four of the five coincide at 100% for most of
+  the sweep and solid lines would show only the last one drawn.
 
 ### Geometry is method-specific (`--compare-geometry`)
 
@@ -1118,7 +1180,7 @@ All figures are generated by the example scripts and stored in the `figs/` direc
 | Figure | Source Script | Description |
 |--------|--------------|-------------|
 | `toa_positioning_example.png` | `example_toa_positioning.py` | TOA positioning geometry showing anchors, range circles, true/estimated positions, and iterative solver convergence path |
-| `ch4_rf_comparison.png` | `example_comparison.py` | Comprehensive comparison of TOA, TDOA, AOA, and RSS methods: median error vs noise level, error CDF, box plots, and solved rates |
+| `ch4_rf_comparison.png` | `example_comparison.py` | Comprehensive comparison of TOA, RTT (two-way TOA), TDOA, AOA, and RSS methods: median error vs noise level, error CDF, box plots, and solved rates |
 | `ch4_aoa_geometry.png` | `example_aoa_positioning.py` | AOA positioning geometry demonstrating ENU coordinate convention with azimuth angles measured from North |
 | `ch4_dop_geometry.png` | `example_dop_geometry.py` | DOP field and walk-away curve showing how anchor geometry amplifies the same range noise into different position uncertainty |
 | `ch4_initial_guess_basin.png` | `example_initial_guess_basin.py` | Starting-point sweep showing where raw angle residuals and book sin/tan residuals converge or fail |
diff --git a/ch4_rf_point_positioning/example_comparison.py b/ch4_rf_point_positioning/example_comparison.py
index de64475..73a3dd4 100644
--- a/ch4_rf_point_positioning/example_comparison.py
+++ b/ch4_rf_point_positioning/example_comparison.py
@@ -1,8 +1,8 @@
 """
 Comparison of RF Positioning Methods.
 
-This script compares TOA, TDOA, AOA, and RSS positioning methods
-under various conditions using pre-generated datasets.
+This script compares TOA, RTT (two-way TOA), TDOA, AOA, and RSS positioning
+methods under various conditions using pre-generated datasets.
 
 Can run with:
     - Inline data (default):
@@ -13,7 +13,8 @@
         python -m ch4_rf_point_positioning.example_comparison --compare-geometry
 
 Implements:
-    - TOA positioning (Eqs. 4.14-4.23)
+    - TOA positioning (Eqs. 4.14-4.23), with the clock as a state (Eqs. 4.24-4.26)
+    - Two-way TOA / RTT ranging (Eqs. 4.6-4.9)
     - TDOA positioning (Eqs. 4.34-4.42)
     - AOA positioning (Eqs. 4.63-4.67)
     - DOP analysis (Section 4.5)
@@ -44,6 +45,7 @@
 from core.eval import save_figure, show_figures_if_requested
 from core.rf import (
     DIVERGENCE_M,
+    STALL_M,
     AOAPositioner,
     SolveOutcome,
     TDOAPositioner,
@@ -53,6 +55,7 @@
     solve_batch,
     toa_range,
     toa_solve_with_clock_bias,
+    two_way_toa_range,
 )
 from core.utils import resolve_data_path
 
@@ -389,11 +392,33 @@ def solve(self, ranges, initial_guess, **kwargs):
 INLINE_SEED = np.array([5.0, 5.0])
 
 
+def draw_range_noise(n_points, n_anchors, range_noise_std_m):
+    """One ``(N, K)`` draw of per-range gaussian noise, or ``None`` at zero.
+
+    Hoisted out of :func:`toa_positioning_test` so that the one-way and
+    two-way range series can be handed the **same realisation**. The two
+    columns exist to price one difference -- the estimator's state vector --
+    and drawing separately would put a full realisation of noise on each side,
+    which is the defect `aoa_bearings` was written to remove one series over
+    (and which `.cursor/rules/030-figures-and-claims.mdc` names: a paired
+    comparison on unpaired draws is two experiments reported as a ratio).
+
+    Returning ``None`` rather than a zero array at ``range_noise_std_m == 0``
+    is not tidiness: it keeps the global RNG untouched on the noiseless level,
+    exactly as the in-line ``if range_noise_std_m > 0`` it replaces did, so
+    every draw downstream in the sweep lands where it did before.
+    """
+    if range_noise_std_m <= 0:
+        return None
+    return np.random.randn(n_points, n_anchors) * range_noise_std_m
+
+
 def toa_positioning_test(
     anchors,
     true_positions,
     range_noise_std_m=0.0,
     clock_bias_m=0.0,
+    range_noise=None,
 ):
     """Test TOA positioning (inline mode). Returns a `SolveOutcome`.
 
@@ -404,6 +429,10 @@ def toa_positioning_test(
         clock_bias_m: Shared receiver clock bias in metres, added to
             every pseudorange.  TDOA differencing cancels this term; TOA
             has to estimate it, which is what the state below is for.
+        range_noise: A pre-drawn ``(N, K)`` noise array to add instead of
+            drawing one. `run_inline_comparison` passes the same array to
+            :func:`rtt_positioning_test`, so the two range series differ in
+            their estimator and in nothing else. ``None`` draws as before.
 
     **The state is (x, y, c*dt), not (x, y), and that is the whole point.**
     A shared bias is unobservable to a position-only solver: no (x, y)
@@ -430,8 +459,10 @@ def toa_positioning_test(
         ]
     )
     ranges += clock_bias_m
-    if range_noise_std_m > 0:
-        ranges += np.random.randn(*ranges.shape) * range_noise_std_m
+    if range_noise is None:
+        range_noise = draw_range_noise(*ranges.shape, range_noise_std_m)
+    if range_noise is not None:
+        ranges += range_noise
 
     return solve_batch(
         _ClockBiasSolver(anchors),
@@ -442,6 +473,67 @@ def toa_positioning_test(
     )
 
 
+def rtt_positioning_test(
+    anchors,
+    true_positions,
+    range_noise_std_m=0.0,
+    range_noise=None,
+):
+    """Test RTT (two-way TOA) positioning (inline mode). Returns a `SolveOutcome`.
+
+    Args:
+        anchors: Anchor positions, shape (K, 2).
+        true_positions: Ground-truth agent positions, shape (N, 2).
+        range_noise_std_m: Gaussian noise std in metres, on the **converted**
+            range -- the quantity Eq. (4.7) hands back after halving the round
+            trip -- not on the raw round-trip time.
+        range_noise: A pre-drawn ``(N, K)`` noise array; see
+            :func:`toa_positioning_test`.
+
+    **Exactly one thing differs from the TOA row above: the state vector.**
+    Same anchors, same 50 test points, same seed, the same sigma ladder and --
+    because `run_inline_comparison` draws once and passes the array to both --
+    the same realisation of it. What changes is that a round trip is timed on
+    one clock, so Eq. (4.7)'s ``d = c(t_rtt - t_proc)/2`` carries no receiver
+    clock offset to estimate: **no bias is injected here, and the solve is the
+    two-state `(x, y)` `TOAPositioner`** rather than the three-state
+    `(x, y, c*dt)` of Eqs. (4.24)-(4.26). The difference between the two
+    medians is therefore the price of the third unknown, measured rather than
+    argued.
+
+    **The noise is deliberately *not* modelled through processing time and
+    clock drift here**, though `simulate_rtt_measurement` offers exactly that
+    and `example_toa_positioning`'s Example 5 uses it. Doing so would change
+    two things at once -- the state vector *and* the error budget -- and the
+    row would no longer answer the question it is in the table to answer. The
+    cost of the RTT hardware's own uncertainty is taught in that standalone
+    section, on its own geometry; this one prices the clock state alone.
+
+    Note this makes the comparison favourable to RTT by construction at equal
+    sigma, which is the honest framing: two-way ranging buys you the missing
+    unknown, and pays for it in the transponder and the processing-time
+    calibration that this row charges nothing for.
+    """
+    ranges = np.array(
+        [
+            [two_way_toa_range(anchor, true_pos) for anchor in anchors]
+            for true_pos in true_positions
+        ]
+    )
+    if range_noise is None:
+        range_noise = draw_range_noise(*ranges.shape, range_noise_std_m)
+    if range_noise is not None:
+        ranges = ranges + range_noise
+
+    return solve_batch(
+        TOAPositioner(anchors, method="iterative_ls"),
+        ranges,
+        INLINE_SEED,
+        true_positions,
+        progress=partial(tqdm, desc="  RTT", leave=False, unit="pt"),
+    )
+
+
 def tdoa_positioning_test(
     anchors, true_positions, range_noise_std_m=0.0, clock_bias_m=0.0
 ):
@@ -640,6 +732,16 @@ def run_inline_comparison():
     schedule so that the comparison is apples-to-apples.  A shared
     receiver clock bias is injected into TOA pseudo-ranges to demonstrate
     that TDOA differencing cancels it.
+
+    **TOA and RTT are a matched pair, and the match is the experiment.** They
+    share 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. The only difference is the estimator's state vector:
+    TOA carries `(x, y, c*dt)` because a one-way pseudorange has an unknown
+    receiver clock in it, RTT carries `(x, y)` because a round trip is timed
+    on one clock and cancels it by construction. So the gap between the two
+    columns is the price of the third unknown, measured instead of intuited.
+    See :func:`rtt_positioning_test` for what is deliberately *not* varied.
     """
     print("=" * 70)
     print("RF Positioning Methods Comparison")
@@ -655,6 +757,10 @@ def run_inline_comparison():
 
     # ---- Independent noise schedules per method ----
     toa_range_noise_levels_m = [0.0, 0.05, 0.1, 0.2, 0.5]
+    # RTT shares TOA's ladder by design, not by coincidence: the two rows are a
+    # matched pair and the schedule is one of the things held equal. Kept as
+    # its own list so the figure's legend can carry it per series.
+    rtt_range_noise_levels_m = list(toa_range_noise_levels_m)
     tdoa_range_noise_levels_m = [0.0, 0.05, 0.1, 0.2, 0.5]
     aoa_noise_levels_deg = [0.0, 1.0, 3.0, 5.0, 10.0]  # degrees
     rss_fading_noise_levels_db = [0.0, 2.0, 4.0, 6.0, 8.0]
@@ -666,13 +772,24 @@ def run_inline_comparison():
 
     # "AOA_unw" is the unweighted control, reported in the table but not
     # plotted -- the figure compares measurement types, not solver options.
-    results = {"TOA": [], "TDOA": [], "AOA": [], "RSS": [], "AOA_unw": []}
+    results = {
+        "TOA": [],
+        "RTT": [],
+        "TDOA": [],
+        "AOA": [],
+        "RSS": [],
+        "AOA_unw": [],
+    }
 
     print("\nNoise configuration (independent per method):")
     print(
         f"  TOA : range noise {toa_range_noise_levels_m} m  "
         f"(+ clock bias {clock_bias_m} m)"
     )
+    print(
+        f"  RTT : range noise {rtt_range_noise_levels_m} m  "
+        "(no clock bias; the round trip cancels it)"
+    )
     print(f"  TDOA: range noise {tdoa_range_noise_levels_m} m  (clock bias cancels)")
     print(f"  AOA : angle noise {aoa_noise_levels_deg} deg")
     print(
@@ -695,17 +812,35 @@ def run_inline_comparison():
         print(
             f"\n[{i+1}/{n_levels}] TOA: {toa_range_noise_m:.2f}m "
             f"(+bias {clock_bias_m}m), "
+            f"RTT: {toa_range_noise_m:.2f}m (no bias), "
             f"TDOA: {tdoa_range_noise_m:.2f}m, "
             f"AOA: {aoa_noise_levels_deg[i]:.1f}deg, "
             f"RSS: {rss_fading_db:.1f}dB"
         )
 
+        # Drawn once and solved twice, which is what makes the TOA/RTT pair a
+        # measurement of the clock state rather than of two noise draws. It
+        # also keeps the sweep's global RNG stream where it was: this draw is
+        # the one `toa_positioning_test` used to make at exactly this point,
+        # and `rtt_positioning_test` consumes nothing.
+        shared_range_noise = draw_range_noise(
+            len(true_positions), len(anchors), toa_range_noise_m
+        )
         results["TOA"].append(
             toa_positioning_test(
                 anchors,
                 true_positions,
                 toa_range_noise_m,
                 clock_bias_m=clock_bias_m,
+                range_noise=shared_range_noise,
+            )
+        )
+        results["RTT"].append(
+            rtt_positioning_test(
+                anchors,
+                true_positions,
+                toa_range_noise_m,
+                range_noise=shared_range_noise,
             )
         )
         results["TDOA"].append(
@@ -745,7 +880,7 @@ def run_inline_comparison():
     print("\n" + "=" * 70)
     print("Results Summary (median error in metres)")
     print("=" * 70)
-    print(f"  Clock bias: {clock_bias_m} m (TOA only; cancels in TDOA)")
+    print(f"  Clock bias: {clock_bias_m} m (TOA only; cancels in TDOA and RTT)")
     print(
         f"  RSS config: Rayleigh short-term (sigma={sigma_short_linear}), "
         f"{n_samples_avg} samples averaged"
@@ -754,11 +889,16 @@ def run_inline_comparison():
         f"  AOA anchor {DEGRADED_ANCHOR} is {DEGRADED_ANCHOR_SCALE:.0f}x noisier "
         f"than the others; 'AOA unw' solves the same bearings unweighted"
     )
+    print(
+        "  'RTT' is two-way TOA on the identical range-noise draw as 'TOA', "
+        "solved\n  for (x, y) alone: same anchors, points, seed and sigma, one "
+        "fewer state"
+    )
     header = (
         f"{'Level':<6} {'TOA(m)':<9} {'TDOA(m)':<9} "
         f"{'AOA(deg)':<9} {'RSS(dB)':<9} "
-        f"{'TOA':<9} {'TDOA':<9} {'AOA':<9} {'AOA unw':<9} "
-        f"{'RSS':<9} {'AOA fail':<9} {'RSS fail':<9}"
+        f"{'TOA':<9} {'RTT':<9} {'TDOA':<9} {'AOA':<9} {'AOA unw':<9} "
+        f"{'RSS':<9} {'RTT fail':<9} {'AOA fail':<9} {'RSS fail':<9}"
     )
     print(header)
     print("-" * len(header))
@@ -777,10 +917,12 @@ def run_inline_comparison():
             f"{aoa_noise_levels_deg[i]:<9.1f} "
             f"{rss_fading_noise_levels_db[i]:<9.1f} "
             f"{results['TOA'][i].median_m:<9.3f} "
+            f"{results['RTT'][i].median_m:<9.3f} "
             f"{results['TDOA'][i].median_m:<9.3f} "
             f"{results['AOA'][i].median_m:<9.3f} "
             f"{results['AOA_unw'][i].median_m:<9.3f} "
             f"{results['RSS'][i].median_m:<9.3f} "
+            f"{results['RTT'][i].n_failed:<9d} "
             f"{results['AOA'][i].n_failed:<9d} "
             f"{results['RSS'][i].n_failed:<9d}"
         )
@@ -790,8 +932,43 @@ def run_inline_comparison():
         for i in range(n_levels)
     ]
 
+    # What one fewer state is worth, printed rather than asserted. Both
+    # columns solve the SAME range errors, so the ratio is not a comparison of
+    # two noise realisations -- it is the cost of carrying the clock.
+    #
+    # Level 1 is exact on both sides and must not print a ratio. `rtt > 0` is
+    # not the guard that gets this right: the noiseless medians are 3.5e-10 m
+    # and 9.9e-10 m -- both zero as far as a positioning system is concerned,
+    # both nonzero as far as float division is concerned -- and the first draft
+    # of this loop duly printed "0.35x", a ratio of two roundoff residues,
+    # under a heading claiming to price the clock state. The floor is
+    # `STALL_M`, the solver's own 1e-6 m step tolerance: nothing below it is a
+    # distance, which is exactly why `solve_batch` uses it to decide a fix
+    # never left its seed.
+    print()
+    print("  What the clock state costs, TOA / RTT on the identical draw:")
+    for i in range(n_levels):
+        toa_m = results["TOA"][i].median_m
+        rtt_m = results["RTT"][i].median_m
+        if max(toa_m, rtt_m) < STALL_M:
+            ratio = f"-- (both exact, under {STALL_M:g} m)"
+        else:
+            ratio = f"{toa_m / rtt_m:.2f}x"
+        print(
+            f"    level {i+1}  sigma {toa_range_noise_levels_m[i]:.2f} m   "
+            f"TOA {toa_m:.3f} m   RTT {rtt_m:.3f} m   {ratio}"
+        )
+    print("  Four ranges over three unknowns against four over two: one degree")
+    print("  of freedom instead of two, and the inflation is the price of not")
+    print("  having a beacon that answers back. RTT's noise is charged on the")
+    print("  post-conversion range (Eq. 4.7), NOT modelled through processing")
+    print("  time and clock drift -- that would change the error budget as well")
+    print("  as the state vector, and this row exists to change one thing.")
+    print("  example_toa_positioning's Example 5 prices the RTT hardware budget")
+    print("  on its own geometry; nothing here charges RTT for the transponder.")
+
     print()
-    print("  The two 'fail' columns count fixes that raised, reported")
+    print("  The three 'fail' columns count fixes that raised, reported")
     print("  converged=False, never left the seed, or landed over 100 m away")
     print("  -- core.rf.solve_batch's four conditions. Every median above is")
     print("  over all 50 fixes including those, which is the point: the RSS")
@@ -836,9 +1013,10 @@ def run_inline_comparison():
     print("  Only the spread between anchors carries information.")
 
     # The schedules travel with the results, because the figure needs them:
-    # four series in three different units cannot share one x-axis in metres.
+    # five series in three different units cannot share one x-axis in metres.
     level_schedules = (
         ("TOA", toa_range_noise_levels_m, "m"),
+        ("RTT", rtt_range_noise_levels_m, "m"),
         ("TDOA", tdoa_range_noise_levels_m, "m"),
         ("AOA", aoa_noise_levels_deg, "deg"),
         ("RSS", rss_fading_noise_levels_db, "dB"),
@@ -1076,9 +1254,9 @@ def plot_geometry_comparison(all_results: dict):
 def _schedule_label(method, schedule, unit):
     """A legend entry that carries the series' own noise units.
 
-    Panels 1 and 4 plot four methods against a level *index*, because the four
-    schedules are in three different units: metres for TOA and TDOA, degrees
-    for AOA, dB for RSS. They used to be drawn at TOA's metre positions under
+    Panels 1 and 4 plot five methods against a level *index*, because the five
+    schedules are in three different units: metres for TOA, RTT and TDOA,
+    degrees for AOA, dB for RSS. They used to be drawn at TOA's metre positions under
     an x-axis reading "Measurement Noise (m)", which puts AOA's 10 deg at
     x = 0.5 m and invites exactly one reading, the wrong one. The units now
     travel with the line that owns them.
@@ -1100,14 +1278,35 @@ def plot_inline_comparison(level_schedules, results):
     fig.suptitle("RF Positioning Methods Comparison", fontsize=16, fontweight="bold")
 
     methods = [name for name, _, _ in level_schedules]
-    colors = ["blue", "red", "green", "orange"]
+    # Keyed by method, not positional, so that inserting a series cannot
+    # silently recolour the four that were here before it -- a reader who
+    # learned "RSS is orange" from the committed figure keeps that. RTT's
+    # purple is the fifth: dark enough to read against the white panel and
+    # distinguishable from the blue it will sit closest to.
+    colors = {
+        "TOA": "blue",
+        "RTT": "purple",
+        "TDOA": "red",
+        "AOA": "green",
+        "RSS": "orange",
+    }
     # One dash pattern per method, used by every line panel below. TOA and
-    # TDOA now trace each other almost exactly -- they carry the same
-    # information once TOA estimates its clock, which is the point of this
-    # figure -- so with four solid lines only the last one drawn is visible
-    # and the panel reads as "TOA is missing". The success-rate panel already
-    # had this problem for the same reason and solved it this way.
-    dashes = ["-", "--", "-.", ":"]
+    # TDOA trace each other almost exactly -- they carry the same information
+    # once TOA estimates its clock, which is the point of this figure -- so
+    # with solid lines only the last one drawn is visible and the panel reads
+    # as "TOA is missing". The success-rate panel already had this problem for
+    # the same reason and solved it this way. RTT joins the pile-up at level 1,
+    # where every range method is exact, and at 100 % in panel 4, so it needs
+    # a pattern of its own rather than a shade of one.
+    dashes = {
+        "TOA": "-",
+        "RTT": (0, (3, 1, 1, 1)),
+        "TDOA": "--",
+        "AOA": "-.",
+        "RSS": ":",
+    }
+    series_colors = [colors[name] for name in methods]
+    series_dashes = [dashes[name] for name in methods]
 
     n_levels = len(results[methods[0]])
     level_index = np.arange(1, n_levels + 1)
@@ -1128,15 +1327,30 @@ def _level_caption(idx):
     # tolerance, so this panel was drawing the accuracy of a quarter of its
     # sample, selected by fit. The failures are in panel 4, where they can be
     # read as failures.
+    #
+    # **The axis stays linear, and a log axis was rendered before deciding
+    # that.** RSS at 8 m does crush TOA/RTT/TDOA into a band near zero, which
+    # is the shape `.cursor/rules/030` warns about, and the sibling
+    # `plot_geometry_comparison` fixes exactly that with `set_yscale("log")`.
+    # It does not work here: level 1 is noiseless, so the range and bearing
+    # medians there are roundoff -- 3.5e-10 to 1.6e-08 m -- and a log axis
+    # spends ten of its twelve decades on that, compressing levels 2-5 into
+    # the top fifth of the panel. The TOA/RTT gap is no more readable after
+    # the change than before it, and the panel acquires a meaningless cliff.
+    #
+    # So do not look for the cost of the clock state in this panel: it is
+    # 2.6% between two series that also share a noise schedule, and no scaling
+    # of a five-method overview will show it. The console table prints the
+    # TOA/RTT ratio per level, which is where that comparison lives.
     ax1 = axes[0, 0]
     for (method, schedule, unit), color, dash in zip(
-        level_schedules, colors, dashes, strict=True
+        level_schedules, series_colors, series_dashes, strict=True
     ):
         medians = [outcome.median_m for outcome in results[method]]
         ax1.plot(
             level_index,
             medians,
-            dash,
+            linestyle=dash,
             marker="o",
             label=_schedule_label(method, schedule, unit),
             color=color,
@@ -1153,14 +1367,21 @@ def _level_caption(idx):
     ax2 = axes[0, 1]
     noise_idx = 2
     for (method, _, _), color, dash in zip(
-        level_schedules, colors, dashes, strict=True
+        level_schedules, series_colors, series_dashes, strict=True
     ):
         errors = results[method][noise_idx].errors
         errors = errors[np.isfinite(errors)]
         if len(errors) > 0:
             sorted_errors = np.sort(errors)
             cdf = np.arange(1, len(sorted_errors) + 1) / len(sorted_errors)
-            ax2.plot(sorted_errors, cdf, dash, label=method, color=color, linewidth=2)
+            ax2.plot(
+                sorted_errors,
+                cdf,
+                linestyle=dash,
+                label=method,
+                color=color,
+                linewidth=2,
+            )
     ax2.set_xlabel("Position Error (m)")
     ax2.set_ylabel("CDF")
     ax2.set_title(f"Error CDF at level {noise_idx + 1}\n({_level_caption(noise_idx)})")
@@ -1178,7 +1399,12 @@ def _level_caption(idx):
             data.append(errors)
             labels.append(method)
     bp = ax3.boxplot(data, tick_labels=labels, patch_artist=True, showfliers=False)
-    for patch, color in zip(bp["boxes"], colors[: len(data)], strict=True):
+    # Keyed off the surviving labels, not sliced off the front of the colour
+    # list: a method dropped for having no finite errors would otherwise shift
+    # every box after it onto the wrong colour. `strict=True` is the guard that
+    # the two lists still line up.
+    box_colors = [colors[name] for name in labels]
+    for patch, color in zip(bp["boxes"], box_colors, strict=True):
         patch.set_facecolor(color)
         patch.set_alpha(0.6)
     ax3.set_ylabel("Position Error (m)")
@@ -1189,7 +1415,7 @@ def _level_caption(idx):
 
     # 4. Success Rate
     #
-    # Three of the four methods now sit on 100 % for most of the sweep, so a
+    # Four of the five methods now sit on 100 % for most of the sweep, so a
     # solid line per method would hide all but the last one drawn -- the same
     # "reads as absent" failure this file warns about elsewhere, arrived at
     # from the opposite direction. Distinct dash patterns keep every method
@@ -1197,7 +1423,7 @@ def _level_caption(idx):
     # position.
     ax4 = axes[1, 1]
     for (method, schedule, unit), color, dash in zip(
-        level_schedules, colors, dashes, strict=True
+        level_schedules, series_colors, series_dashes, strict=True
     ):
         rates = [
             100.0 * outcome.solved.sum() / outcome.n for outcome in results[method]
@@ -1205,7 +1431,7 @@ def _level_caption(idx):
         ax4.plot(
             level_index,
             rates,
-            dash,
+            linestyle=dash,
             marker="o",
             label=_schedule_label(method, schedule, unit),
             color=color,
diff --git a/ch4_rf_point_positioning/figs/ch4_rf_comparison.pdf b/ch4_rf_point_positioning/figs/ch4_rf_comparison.pdf
index e622d0c17156cc41b206cffe43dbf15fdbb30d6c..f6b9215b801193fe4ccb60ce6cfec019c85ec96a 100644
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diff --git a/ch4_rf_point_positioning/figs/ch4_rf_comparison.png b/ch4_rf_point_positioning/figs/ch4_rf_comparison.png
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diff --git a/tests/ch4_rf_point_positioning/test_rtt_matched_comparison.py b/tests/ch4_rf_point_positioning/test_rtt_matched_comparison.py
new file mode 100644
index 0000000..9529a49
--- /dev/null
+++ b/tests/ch4_rf_point_positioning/test_rtt_matched_comparison.py
@@ -0,0 +1,396 @@
+"""What estimating the clock costs, measured against a beacon that answers back.
+
+Chapter 4's inline comparison carries a one-way TOA series whose state is
+`(x, y, c*dt)` -- Eqs. (4.24)-(4.26), because a one-way pseudorange contains an
+unknown receiver clock offset -- beside a two-way TOA (RTT) series whose state
+is `(x, y)`, because Eq. (4.7)'s `d = c(t_rtt - t_proc)/2` is timed on one
+clock and cancels the offset by construction.
+
+**The two rows are matched so that the gap between them means something.** Same
+anchors, same 50 test points, same seed, same sigma ladder, and the same
+realisation drawn from it -- `run_inline_comparison` calls `draw_range_noise`
+once per level and hands the array to both. Only the state vector differs. The
+repo has been bitten by the unpaired version of exactly this: `aoa_bearings`
+exists because the weighted and unweighted AOA columns used to draw their own
+noise, so the ratio between them carried a full realisation on each side.
+
+The RTT row is deliberately *not* noised through processing time and clock
+drift, though `simulate_rtt_measurement` offers that and
+`example_toa_positioning`'s Example 5 uses it. That would vary the error budget
+as well as the state vector, and this row exists to vary one thing.
+
+**The size of the effect is small, and that is the finding these tests are
+shaped around.** Adding the clock column to H inflates the position covariance
+by a factor set by the geometry alone -- it does not grow with sigma. For this
+square array and these 50 points the analytic inflation is a median 1.026 over
+per-point ratios (1.033 as a ratio of medians), and a 50-point median has a
+sampling spread of roughly +/-7%. So "the RTT median is below the TOA median"
+is **not** a law that a single noise level can demonstrate: over 40 repeats at
+each sigma the ratio of medians ranged 0.87 to 1.27. It happens to hold at
+every noisy level of the shipped seed, and that is pinned below as a property
+of the shipped realisation rather than as physics. The physics is pinned twice
+over instead -- once analytically, where it is exact, and once over a sample
+large enough to resolve 2.6%.
+
+Author: Li-Ta Hsu
+References: Chapter 4, Eqs. (4.6)-(4.9), (4.14)-(4.23), (4.24)-(4.26)
+"""
+
+import contextlib
+import functools
+import io
+import unittest
+
+import matplotlib
+
+matplotlib.use("Agg")  # headless: no display during tests
+
+import numpy as np
+
+from ch4_rf_point_positioning.example_comparison import (
+    INLINE_SEED,
+    draw_range_noise,
+    generate_scenario,
+    rtt_positioning_test,
+    run_inline_comparison,
+    toa_positioning_test,
+)
+from core.rf import STALL_M, TOAPositioner, toa_solve_with_clock_bias
+
+#: The clock bias `run_inline_comparison` injects into the one-way ranges.
+CLOCK_BIAS_M = 1.5
+
+#: Points per repeat in the statistical arm. The effect is 2.6% and a
+#: 50-point median carries ~7% of sampling noise, so the shipped sweep's own
+#: sample cannot resolve it -- measuring it needs its own, larger one.
+STAT_POINTS = 1200
+STAT_SEEDS = (0, 1, 2, 3)
+STAT_SIGMA_M = 0.2
+
+
+@functools.lru_cache(maxsize=1)
+def _sweep():
+    """The inline sweep, run once and shared.
+
+    Six seconds a call, and four tests want it. `.cursor/rules/030` asks for
+    exactly this (`tests/ch7_slam/slam_example_runner.py` is the pattern).
+    No assertion lives in here: `lru_cache` does not cache exceptions, so a
+    memoised helper that asserts re-runs its whole body for every caller once
+    anything goes wrong.
+    """
+    with contextlib.redirect_stdout(io.StringIO()):
+        return run_inline_comparison()
+
+
+def _paired_medians(sigma_m, n_points, seed, rtt_gets_the_clock_state=False):
+    """Median one-way and two-way error over `n_points`, on one noise draw.
+
+    Both estimators see the identical ranges up to the injected bias, which is
+    the whole design. `rtt_gets_the_clock_state` is the mutation this file's
+    tolerance was justified against -- see
+    `test_the_measured_cost_is_the_state_vector_and_not_the_noise`.
+    """
+    anchors, _ = generate_scenario(seed=42)
+    rng = np.random.default_rng(seed)
+    points = rng.uniform(1, 9, size=(n_points, 2))
+    true_ranges = np.linalg.norm(
+        points[:, None, :] - anchors[None, :, :], axis=2
+    )
+    noise = rng.standard_normal(true_ranges.shape) * sigma_m
+
+    seed_3 = np.concatenate([INLINE_SEED, [0.0]])
+    one_way, two_way = [], []
+    for i, point in enumerate(points):
+        est, _bias, _info = toa_solve_with_clock_bias(
+            anchors, true_ranges[i] + CLOCK_BIAS_M + noise[i], seed_3
+        )
+        one_way.append(float(np.linalg.norm(est - point)))
+
+        if rtt_gets_the_clock_state:
+            est, _bias, _info = toa_solve_with_clock_bias(
+                anchors, true_ranges[i] + noise[i], seed_3
+            )
+        else:
+            est, _info = TOAPositioner(anchors, method="iterative_ls").solve(
+                true_ranges[i] + noise[i], INLINE_SEED
+            )
+        two_way.append(float(np.linalg.norm(est - point)))
+
+    return float(np.median(one_way)), float(np.median(two_way))
+
+
+def _analytic_inflation():
+    """Per-point ratio of 3-state to 2-state position DOP, over the 50 points.
+
+    Sigma factors out of both sides, which is the point: the cost of the clock
+    column is a property of the geometry and not of the noise.
+    """
+    anchors, truth = generate_scenario(seed=42)
+    two_state, three_state = [], []
+    for point in truth:
+        offsets = point - anchors
+        unit = offsets / np.linalg.norm(offsets, axis=1, keepdims=True)
+        h_2 = unit
+        # A one-way pseudorange is d + b, so the clock column is +1.
+        h_3 = np.hstack([unit, np.ones((len(anchors), 1))])
+        two_state.append(np.sqrt(np.trace(np.linalg.inv(h_2.T @ h_2))))
+        three_state.append(
+            np.sqrt(np.trace(np.linalg.inv(h_3.T @ h_3)[:2, :2]))
+        )
+    return np.array(three_state) / np.array(two_state)
+
+
+class RttSeriesIsPresentAndMatched(unittest.TestCase):
+    """The series exists, and the things held equal are actually equal."""
+
+    def test_the_sweep_returns_a_two_way_series(self):
+        schedules, results = _sweep()
+        self.assertIn("RTT", results)
+
+        names = [name for name, _, _ in schedules]
+        self.assertIn("RTT", names)
+        self.assertEqual(len(results["RTT"]), len(results["TOA"]))
+
+    def test_rtt_shares_toa_s_noise_ladder(self):
+        """A matched pair whose sigma schedules differ is not matched.
+
+        Pinned rather than trusted because the two schedules are separate
+        lists in the source, and a plausible future edit widens one of them.
+        """
+        schedules, _ = _sweep()
+        ladders = {name: schedule for name, schedule, _ in schedules}
+        units = {name: unit for name, _, unit in schedules}
+
+        self.assertEqual(ladders["RTT"], ladders["TOA"])
+        self.assertEqual(units["RTT"], "m")
+
+    def test_the_two_range_series_solve_the_same_noise_draw(self):
+        """Unpaired draws would make the gap a comparison of realisations.
+
+        Checked by handing both functions a noise array of our own: if either
+        one drew its own instead, the errors below would not reproduce.
+        """
+        anchors, truth = generate_scenario(seed=42)
+        np.random.seed(7)
+        noise = draw_range_noise(len(truth), len(anchors), 0.2)
+
+        toa_a = toa_positioning_test(
+            anchors, truth, 0.2, clock_bias_m=CLOCK_BIAS_M, range_noise=noise
+        )
+        rtt_a = rtt_positioning_test(anchors, truth, 0.2, range_noise=noise)
+
+        # Same arrays again, from a global RNG left somewhere else entirely.
+        np.random.seed(999)
+        toa_b = toa_positioning_test(
+            anchors, truth, 0.2, clock_bias_m=CLOCK_BIAS_M, range_noise=noise
+        )
+        rtt_b = rtt_positioning_test(anchors, truth, 0.2, range_noise=noise)
+
+        np.testing.assert_array_equal(toa_a.errors, toa_b.errors)
+        np.testing.assert_array_equal(rtt_a.errors, rtt_b.errors)
+
+    def test_a_zero_sigma_level_draws_nothing_at_all(self):
+        """The noiseless level must not consume the global stream.
+
+        This is what keeps the four pre-existing series byte-stable: the
+        in-line `if range_noise_std_m > 0` this replaced did not draw either,
+        so every draw later in the sweep still lands where it did.
+        """
+        self.assertIsNone(draw_range_noise(50, 4, 0.0))
+
+        np.random.seed(3)
+        before = np.random.randn(5).copy()
+        np.random.seed(3)
+        self.assertIsNone(draw_range_noise(50, 4, 0.0))
+        np.testing.assert_array_equal(before, np.random.randn(5))
+
+
+class TheClockStateIsWhatCostsAccuracy(unittest.TestCase):
+    """The physics, pinned where it is exact and where it is measurable."""
+
+    def test_the_clock_column_inflates_position_dop_at_every_test_point(self):
+        """Exact, tolerance-free, and the reason the RTT row is worth having.
+
+        Estimating a fourth quantity from the same four ranges leaves one
+        degree of freedom instead of two. The position block of
+        `(H3' H3)^-1` is the Schur complement of the clock column, so it
+        cannot be smaller than `(H2' H2)^-1` -- at any geometry, at any sigma.
+        """
+        inflation = _analytic_inflation()
+
+        self.assertTrue(np.all(inflation > 1.0))
+        # Recorded so a geometry change that flattens the effect is visible
+        # rather than merely still-passing: this array is 4 anchors on a 10 m
+        # square with the agent inside, and the cost is a few percent.
+        self.assertAlmostEqual(float(np.median(inflation)), 1.026, places=2)
+
+    def test_the_measured_cost_is_the_state_vector_and_not_the_noise(self):
+        """Measured over a sample big enough to resolve 2.6%, then mutated.
+
+        **Both halves of the bound were measured.** Against the noise: over ten
+        seeds at these settings the ratio ran 1.0115 to 1.0376, mean 1.0257,
+        std 0.0070, so a floor of 1.010 on the mean of four seeds sits about
+        four standard errors below it. Against the defect: giving the two-way
+        arm the clock state as well -- the one mutation that makes this test
+        meaningless -- returns **exactly** 1.0000, because a common bias is
+        absorbed whole into `c*dt` and the position estimates become
+        bit-identical. The ceiling catches the opposite error, a two-way arm
+        accidentally handed less noise than the one-way arm.
+        """
+        ratios = [
+            (lambda pair: pair[0] / pair[1])(
+                _paired_medians(STAT_SIGMA_M, STAT_POINTS, seed)
+            )
+            for seed in STAT_SEEDS
+        ]
+        measured = float(np.mean(ratios))
+
+        self.assertGreater(measured, 1.010)
+        self.assertLess(measured, 1.045)
+
+        one_way, two_way = _paired_medians(
+            STAT_SIGMA_M, 300, STAT_SEEDS[0], rtt_gets_the_clock_state=True
+        )
+        self.assertAlmostEqual(one_way / two_way, 1.0, places=9)
+
+    def test_the_bias_is_unobservable_to_position_in_the_three_state_solve(self):
+        """Why the one-way row can carry a 1.5 m bias and still be comparable.
+
+        The bias goes entirely into `c*dt`, so the one-way *position* is the
+        same whether or not it was injected, and the clock estimate moves by
+        exactly the injected amount. That is the claim the comparison rests
+        on: the two rows differ in their state vector and in nothing else, and
+        the 1.5 m is not a handicap loaded onto TOA.
+
+        **What is exact here is the difference, not the estimate.** A first
+        draft asserted the recovered bias was 1.5 m on noisy data and read
+        1.4642 -- the solve absorbs the injected bias *and* whatever common
+        part the range noise happens to have, which at sigma = 0.2 m over four
+        anchors is a few centimetres. The example's own docstring already said
+        "+1.500 m on **noiseless** data" and it means it; both cases are
+        separated below rather than sharing one tolerance.
+        """
+        anchors, truth = generate_scenario(seed=42)
+        rng = np.random.default_rng(5)
+        clean = np.linalg.norm(truth[:, None, :] - anchors[None, :, :], axis=2)
+        noisy = clean + rng.standard_normal(clean.shape) * 0.2
+        seed_3 = np.concatenate([INLINE_SEED, [0.0]])
+
+        # Noiseless: the bias comes back as itself.
+        for row, point in zip(clean, truth, strict=True):
+            est, bias, _ = toa_solve_with_clock_bias(
+                anchors, row + CLOCK_BIAS_M, seed_3
+            )
+            self.assertAlmostEqual(bias, CLOCK_BIAS_M, places=6)
+            self.assertLess(float(np.linalg.norm(est - point)), 1e-6)
+
+        # Noisy: the position does not move and the clock absorbs the whole
+        # of the shift, both exactly.
+        worst_position, worst_bias = 0.0, 0.0
+        for row in noisy:
+            with_bias, bias_hi, _ = toa_solve_with_clock_bias(
+                anchors, row + CLOCK_BIAS_M, seed_3
+            )
+            without, bias_lo, _ = toa_solve_with_clock_bias(anchors, row, seed_3)
+            worst_position = max(
+                worst_position, float(np.linalg.norm(with_bias - without))
+            )
+            worst_bias = max(
+                worst_bias, abs((bias_hi - bias_lo) - CLOCK_BIAS_M)
+            )
+
+        self.assertLess(worst_position, 1e-9)
+        self.assertLess(worst_bias, 1e-9)
+
+
+class WhatTheShippedSweepActuallyReports(unittest.TestCase):
+    """Pins on the committed table and figure, labelled as realisation pins."""
+
+    def test_rtt_is_at_or_below_toa_at_every_noisy_level_of_this_seed(self):
+        """True at seed 42; **not** a law, and the docstring is the point.
+
+        The effect is 2.6% and a 50-point median carries roughly 7% of
+        sampling noise, so this ordering is not something one noise level can
+        establish -- over 40 repeats per sigma the ratio of medians ranged
+        0.87 to 1.27. It holds at every noisy level here, which is what the
+        README table and the figure show, so it is pinned as a regression
+        guard on the shipped numbers.
+        `test_the_measured_cost_is_the_state_vector_and_not_the_noise` is
+        where the underlying claim is actually established.
+        """
+        schedules, results = _sweep()
+        ladder = {name: schedule for name, schedule, _ in schedules}["RTT"]
+
+        for level, sigma in enumerate(ladder):
+            if sigma == 0:
+                continue
+            with self.subTest(level=level + 1, sigma=sigma):
+                self.assertLessEqual(
+                    results["RTT"][level].median_m,
+                    results["TOA"][level].median_m,
+                )
+
+    def test_both_range_series_are_exact_on_noiseless_measurements(self):
+        """The one row with a known right answer, and neither wins it.
+
+        At sigma = 0 both medians are roundoff -- 3.5e-10 m and 9.9e-10 m --
+        so the two-way series is in fact the *larger* of the two, by 0.6
+        nanometres. Nothing is measured there, which is why the example prints
+        no ratio for level 1 and why the ordering test above skips it. An
+        earlier draft of that print guarded on `median > 0` and duly reported
+        "0.35x" as the cost of the clock state.
+        """
+        _, results = _sweep()
+
+        self.assertLess(results["TOA"][0].median_m, STALL_M)
+        self.assertLess(results["RTT"][0].median_m, STALL_M)
+
+    def test_failures_are_accounted_by_solve_batch(self):
+        """Median over every fix, failure count beside it -- not an RMSE.
+
+        `solve_batch`'s four conditions: raised, reported `converged=False`,
+        never left the seed, or landed over 100 m away. The two-way series has
+        to go through it like the other four, so that a level where it stops
+        solving reports as a failure rather than as an accuracy over the
+        survivors.
+        """
+        _, results = _sweep()
+
+        for level, outcome in enumerate(results["RTT"]):
+            with self.subTest(level=level + 1):
+                self.assertEqual(outcome.n, 50)
+                self.assertEqual(len(outcome.errors), 50)
+                self.assertEqual(outcome.solved.shape, (50,))
+                self.assertEqual(
+                    outcome.n_failed, int((~outcome.solved).sum())
+                )
+                self.assertGreaterEqual(outcome.n - outcome.n_failed, 45)
+
+        # The one failure in the sweep is a step-tolerance stop, not a
+        # divergence: its error is an ordinary sub-metre fix. `converged` here
+        # means "finished inside max_iters", which is why the failure count
+        # and the median are reported separately rather than one filtering the
+        # other.
+        for outcome in results["RTT"]:
+            unsolved = outcome.errors[~outcome.solved]
+            self.assertTrue(np.all(np.isfinite(unsolved)))
+            self.assertTrue(np.all(unsolved < 100.0))
+
+    def test_running_the_sweep_twice_gives_the_same_numbers(self):
+        """An unseeded series cannot regenerate its own committed figure."""
+        schedules_a, results_a = _sweep()
+        with contextlib.redirect_stdout(io.StringIO()):
+            schedules_b, results_b = run_inline_comparison()
+
+        self.assertEqual(schedules_a, schedules_b)
+        for method in results_a:
+            for level, (a, b) in enumerate(
+                zip(results_a[method], results_b[method], strict=True)
+            ):
+                with self.subTest(method=method, level=level + 1):
+                    np.testing.assert_array_equal(a.errors, b.errors)
+                    np.testing.assert_array_equal(a.solved, b.solved)
+
+
+if __name__ == "__main__":
+    unittest.main()

From ef782cb91d9e2cd545fe466e3c6a999d04955381 Mon Sep 17 00:00:00 2001
From: qmohsu <qmohsu@gmail.com>
Date: Wed, 2 Sep 2026 13:27:58 +0800
Subject: [PATCH 2/2] style: black-format the new RTT comparison test

`tests/test_black_is_actually_run.py` caught it on CI and nothing caught it
here: black is in the `dev` extra and was simply not installed on this
machine, so the guard could not run at all. CLAUDE.md's own line for this --
"a configured checker that is not run is indistinguishable from one that
passes" -- with the twist that the checker was absent rather than unconfigured.
black 26.5.1 installed to match the pin in `pyproject.toml`, which exists
because versions disagree: black 26 reformats files black 25 accepts.

Whitespace only: four wrapped expressions rejoined onto single lines. No
assertion, tolerance or number moved, and `tests/ch4_rf_point_positioning`
still reads 78 passed. `example_comparison.py` needed no change.
---
 .../test_rtt_matched_comparison.py               | 16 ++++------------
 1 file changed, 4 insertions(+), 12 deletions(-)

diff --git a/tests/ch4_rf_point_positioning/test_rtt_matched_comparison.py b/tests/ch4_rf_point_positioning/test_rtt_matched_comparison.py
index 9529a49..8421236 100644
--- a/tests/ch4_rf_point_positioning/test_rtt_matched_comparison.py
+++ b/tests/ch4_rf_point_positioning/test_rtt_matched_comparison.py
@@ -93,9 +93,7 @@ def _paired_medians(sigma_m, n_points, seed, rtt_gets_the_clock_state=False):
     anchors, _ = generate_scenario(seed=42)
     rng = np.random.default_rng(seed)
     points = rng.uniform(1, 9, size=(n_points, 2))
-    true_ranges = np.linalg.norm(
-        points[:, None, :] - anchors[None, :, :], axis=2
-    )
+    true_ranges = np.linalg.norm(points[:, None, :] - anchors[None, :, :], axis=2)
     noise = rng.standard_normal(true_ranges.shape) * sigma_m
 
     seed_3 = np.concatenate([INLINE_SEED, [0.0]])
@@ -134,9 +132,7 @@ def _analytic_inflation():
         # A one-way pseudorange is d + b, so the clock column is +1.
         h_3 = np.hstack([unit, np.ones((len(anchors), 1))])
         two_state.append(np.sqrt(np.trace(np.linalg.inv(h_2.T @ h_2))))
-        three_state.append(
-            np.sqrt(np.trace(np.linalg.inv(h_3.T @ h_3)[:2, :2]))
-        )
+        three_state.append(np.sqrt(np.trace(np.linalg.inv(h_3.T @ h_3)[:2, :2])))
     return np.array(three_state) / np.array(two_state)
 
 
@@ -295,9 +291,7 @@ def test_the_bias_is_unobservable_to_position_in_the_three_state_solve(self):
             worst_position = max(
                 worst_position, float(np.linalg.norm(with_bias - without))
             )
-            worst_bias = max(
-                worst_bias, abs((bias_hi - bias_lo) - CLOCK_BIAS_M)
-            )
+            worst_bias = max(worst_bias, abs((bias_hi - bias_lo) - CLOCK_BIAS_M))
 
         self.assertLess(worst_position, 1e-9)
         self.assertLess(worst_bias, 1e-9)
@@ -361,9 +355,7 @@ def test_failures_are_accounted_by_solve_batch(self):
                 self.assertEqual(outcome.n, 50)
                 self.assertEqual(len(outcome.errors), 50)
                 self.assertEqual(outcome.solved.shape, (50,))
-                self.assertEqual(
-                    outcome.n_failed, int((~outcome.solved).sum())
-                )
+                self.assertEqual(outcome.n_failed, int((~outcome.solved).sum()))
                 self.assertGreaterEqual(outcome.n - outcome.n_failed, 45)
 
         # The one failure in the sweep is a step-tolerance stop, not a