diff --git a/ROADMAP.md b/ROADMAP.md
index 77f0fd4e..9998226b 100644
--- a/ROADMAP.md
+++ b/ROADMAP.md
@@ -42,7 +42,7 @@ increment that most needs a picture to check against
 | 20 | Quality start: before DEM refinement, Steiner points at the DEM node nearest each bad triangle's circumcentre until the start mesh has a 25° minimum angle (`--start-min-angle`, 0 is off); geometry only, input segments not split, serial and deterministic. Removes increment 16's boundary fans | landed with 20b as interim (C1-C3 open, to 20c; C4 -> 20b) | `docs/increments/20-start-quality.md` |
 | 20b | Minimum insertion distance from constraints: when refinement's worst node lies within ε = clamp(tol / slope, cell/100, cell/2) of a constraint segment, insert the foot on the segment (off-node, bilinear z) instead; a footed node still above tolerance is inserted after all, so the tolerance guarantee is unchanged. Removes the 0.0117° needle at 1 m (increment 20's C4) | shipped with 20 (#97) | `docs/increments/20b-min-insertion-distance.md` |
 | — | `tools/bench.py`: the 1 m benchmark and the thread-scaling sweep from one checked-in command, with power state, quality and commit recorded per run; the one-off scripts in `docs/benchmarks/2026-09-26/` are its specification. Rule 2's acceptance run needs it | shipped with branch `tools-bench`'s PR | `docs/benchmarks/bench-py.md` |
-| — | The serial phase: profile refine's serial insert-and-flip phase, then parallelise what the profile blames. Scaling tops out at about 2.0-2.2×. Profiled 2026-09-27: serial part about a third of 1-thread refine, mostly Lawson legalisation; the scan stops near 5× from load imbalance | profiled; designed as increment 21 (Ola's rulings 2026-09-27: L1 determinism, one path, at most 2 % more triangles). **21a shipped with branch `increment21a-quick-wins`'s PR**: dynamic scan blocks, active merge, reused flip stack; mesh bit-identical; on AC refine -10 to -15 % at 8 threads, ceiling 2.1x -> 2.3x (`docs/benchmarks/2026-09-27/21a-acceptance.md`). 21b next, then 21c measurements, then 21d | `docs/increments/21-parallel-refine.md`, `docs/benchmarks/2026-09-27/serial-profile/README.md` |
+| — | The serial phase: profile refine's serial insert-and-flip phase, then parallelise what the profile blames. Scaling tops out at about 2.0-2.2×. Profiled 2026-09-27: serial part about a third of 1-thread refine, mostly Lawson legalisation; the scan stops near 5× from load imbalance | profiled; designed as increment 21 (Ola's rulings 2026-09-27: L1 determinism, one path, at most 2 % more triangles). **21a shipped with branch `increment21a-quick-wins`'s PR**: dynamic scan blocks, active merge, reused flip stack; mesh bit-identical; on AC refine -10 to -15 % at 8 threads, ceiling 2.1x -> 2.3x (`docs/benchmarks/2026-09-27/21a-acceptance.md`). **21b shipped with branch `increment21b-lattice-incircle`'s PR**: an int64 lattice incircle answers 99.97-100 % of refine's incircle tests; mesh bit-identical; on battery refine -15 % at 8 threads, ceiling 2.3x -> 2.5x (`docs/benchmarks/2026-09-27/21b-acceptance.md`). 21c measurements next, then 21d | `docs/increments/21-parallel-refine.md`, `docs/benchmarks/2026-09-27/serial-profile/README.md` |
 | 16b | Interior polygons and polylines as constraints ("terrain polygons": lakes, land cover, roads, rivers): `--features PATH`, a GeoJSON `FeatureCollection`, each feature naming a vocabulary property; closed or open `Breakline`s, crossings noded, off-node vertices with bilinear z. **Its working example is real data** (Ola, 2026-09-27): CORINE Land Cover 2018 over the benchmark tile `7908_3_10m_z33.tif`. The source is Ola's local copy, `rasputin_data/corine_sql/.../U2018_CLC2018_V2020_20u1.gpkg` (8.2 GB, EPSG:3035, a sibling of this repository, which also holds the 254-tile DTM10 archive for gap 6). It reads without GDAL: sqlite3 over its R-tree, the GeoPackage blob header stripped, `shapely.wkb`, then `pyproj` to 25833, so CRS stops in Python as before. Probed 2026-09-27 in 0.2 s: 60 polygons in 8 classes (heath, bare rock, sparse vegetation, bogs, intertidal flats, water, sea, urban), 11 068 vertices clipped to the tile, median segment 54 m against 10 m cells. The EEA's public ArcGIS service (`image.discomap.eea.europa.eu`, `Corine/CLC2018_WM`) returns the same 11 068 clipped vertices and is the route for anyone without the file. What it forces on 16b's design: neighbouring polygons share their boundaries, so each shared edge arrives twice; the polygons run past the domain and must be clipped; the extract is committed as a fixture with the Copernicus attribution. Placed before 20c because 20c may split constraint segments and should be designed and measured on inputs that have interior ones | designed in 16's R6, no increment file yet; after the serial phase, before 20c | `docs/increments/16-domain-polygon.md` (R6) |
 | 20c | Soft quality criterion: a penalty that each Steiner node or constraint split must pay for in angle gained, instead of 20's hard 25°; applied at the start and during DEM refinement; may split constraint segments when that improves the mesh. Ola's rulings on 20's C1-C3 | to design after 16b (`@architect` measures cost against 20 first) | `docs/increments/20-start-quality.md` (Ola's rulings) |
 | — | Auto-catchment: the watershed upstream of a coordinate, computed from the DEM and handed to `--domain`, so a catchment no longer has to be supplied as a file (Ola, 2026-09-27: "not far into the future"). The textbook route is depression handling (Priority-Flood, Barnes, Lehman and Mulla 2014), D8 flow directions (O'Callaghan and Mark 1984) and accumulation, the pour point snapped to the strongest flow nearby, the upstream cells traced and their outline turned into a polygon; the literature check is `@architect`'s. Open for its design: whether it runs in the C++ core (a 10 m tile is 25 M cells); how a stair-stepped cell outline becomes a domain polygon, which meets input coarsening; and that a real catchment crosses tile edges, so it needs gap 6 (a DEM in several tiles) first. Legacy has nothing on it (`grep -rliE "watershed|flow.?acc|flow.?dir|pour.?point|catchment" legacy` returns no files) | to design; after gap 6, which it needs; placed after 20c, can move ahead of it on Ola's word | none yet |
diff --git a/docs/benchmarks/2026-09-27/21b-acceptance.md b/docs/benchmarks/2026-09-27/21b-acceptance.md
new file mode 100644
index 00000000..5d99fa23
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-acceptance.md
@@ -0,0 +1,228 @@
+# Increment 21b acceptance run (@perf, 2026-09-27)
+
+**Verdict: ACCEPTED.** On battery, measured back to back against the previous
+merge (`071e4f3`, 21a): refine at 8 threads is 14.5-14.8 % faster on both
+domains, in both pairs. The mesh sha256 is identical on both domains. No
+measure regressed.
+
+One thing to act on: the head `a03ae65` has a split phase 4-6 % slower than
+21b's green commit `f707322`. The only production change between them is the
+non-finite refusal in `lattice_incircle` (see "Green against head"). Fixed in
+`4be156c`; see the Addendum.
+
+## Method
+
+- **Why a re-measured base.** The Mac was on battery. The stored `bench.py`
+  runs of 21a (`21a*/`) are AC. The only battery `bench.py` run,
+  `serial-profile-bench/`, predates 21a. So there was no comparable baseline.
+  Master `071e4f3` (the 21a merge, #102) was checked out in a temporary
+  `git worktree` and measured with `--tree`, back to back with 21b's head
+  `a03ae65`. The worktree was removed afterwards.
+- **`--baseline` is explicit.** `071e4f3` is not an ancestor of `a03ae65`: the
+  branch forked from 21a's head, not from the merge
+  (`git merge-base --is-ancestor 071e4f3 a03ae65` exits 1). `find_baseline`
+  would therefore skip it.
+- **Two pairs**, run in the order base, 21b, base, 21b between 18:56 and
+  19:05 CEST. The verdicts can be reproduced from the stored evidence:
+
+  ```
+  .venv/bin/python tools/bench.py compare docs/benchmarks/2026-09-27/21b \
+      --baseline docs/benchmarks/2026-09-27/21b-base-071e4f3          # ACCEPTED
+  .venv/bin/python tools/bench.py compare docs/benchmarks/2026-09-27/21b-r2 \
+      --baseline docs/benchmarks/2026-09-27/21b-base-071e4f3-r2       # ACCEPTED
+  .venv/bin/python tools/bench.py compare docs/benchmarks/2026-09-27/21b-base-071e4f3-r2 \
+      --baseline docs/benchmarks/2026-09-27/21b-base-071e4f3          # REGRESSION (same build; see Spread)
+  .venv/bin/python tools/bench.py compare docs/benchmarks/2026-09-27/21b-r2 \
+      --baseline docs/benchmarks/2026-09-27/21b                       # ACCEPTED
+  ```
+
+  The two cross pairs (`21b-r2` against `21b-base-071e4f3`, and `21b` against
+  `21b-base-071e4f3-r2`) are also ACCEPTED.
+- **Command.** All four runs used the main tree's `tools/bench.py`, blob
+  `77765b1`, the same as 21a's. Each run did its own Release build into
+  `<tree>/build-bench`.
+
+  ```
+  .venv/bin/python tools/bench.py run --label <label> [--tree <worktree of 071e4f3>] \
+      [--baseline docs/benchmarks/2026-09-27/<base label>] --mesh-dir <scratch> \
+      --domain docs/benchmarks/2026-09-26/quarter.geojson --domain tile \
+      --dem tests/fixtures/dem_archive/7908_3_10m_z33.tif --tolerance 1 \
+      --repeats 5 --threads 1,2,...,20
+  ```
+- **Machine and power.** Apple M1 Max (8 P + 2 E), 32 GiB, macOS 27.0. All four
+  runs were on **battery**, discharging: 90 % to 86 % over the sequence. The
+  `pmset -g batt` text from before and after each run is in its `run.json`.
+  `caffeinate -ims` was held for all runs and for the scratch drivers.
+- **Builds.** The base `_core` hashes to `654a2442…` in both of its runs. That
+  is the same `.so` as 21a's acceptance run of `cade752`. 21b's hashes to
+  `dfdeb6ca…` in both runs.
+- **The `(dirty)` on `21b` and `21b-r2` is not a source change.** As in 21a,
+  the only untracked files were the evidence directories of the runs just
+  made. `git status --porcelain --untracked-files=no` was empty.
+- **Phase times** (scan, split) are not in `bench.py`'s child record. They were
+  measured afterwards, on battery (86 %), by a scratch driver that is not
+  checked in. It uses the child's technique: import the tree's
+  `build-bench/pkg`, force `threads` into `cli.refine`, and read
+  `RefineOutcome.scan_seconds` and `split_seconds`. That was 5 repeats at 1, 8
+  and 16 threads, with the two trees interleaved.
+- **Meshes** stayed in the session scratchpad and are not kept. To regenerate
+  one, rerun `bench.py` as above.
+
+## Results (refine seconds, median of 5; ms)
+
+Pair 1 (`21b-base-071e4f3` then `21b`); pair 2 in brackets. The default thread
+count is the CLI's (`threads == 0`, 10 on this machine).
+
+| domain | threads | base 071e4f3 | 21b | change |
+|---|---:|---:|---:|---:|
+| tile | 1 | 507.1 [519.6] | 488.5 [471.4] | -3.7 % [-9.3 %] |
+| tile | 8 | 226.4 [227.4] | 193.0 [194.1] | -14.8 % [-14.6 %] |
+| tile | 16 | 223.6 [225.8] | 197.0 [193.9] | -11.9 % [-14.1 %] |
+| tile | default (10) | 226.3 [229.5] | 194.1 [193.8] | -14.2 % [-15.5 %] |
+| tile | best | 222.8 @12 [225.6 @13] | 191.8 @9 [192.4 @13] | -13.9 % [-14.7 %] |
+| quarter | 1 | 447.7 [471.7] | 432.8 [424.8] | -3.3 % [-9.9 %] |
+| quarter | 8 | 200.7 [201.2] | 171.1 [172.0] | -14.8 % [-14.5 %] |
+| quarter | 16 | 202.9 [204.0] | 173.8 [173.5] | -14.4 % [-15.0 %] |
+| quarter | default (10) | 203.1 [203.5] | 171.8 [172.7] | -15.4 % [-15.1 %] |
+| quarter | best | 200.4 @20 [201.2 @8] | 170.7 @14 [171.9 @12] | -14.8 % [-14.6 %] |
+
+Every swept thread count from 1 to 20 is faster in both pairs. In pair 1 the
+gain is 3.7-15.5 % on the tile and 3.3-16.1 % on the quarter circle. In pair 2
+it is 9.3-15.8 % and 9.9-16.5 %. The 1-thread gain is the least stable figure:
+it is -3 % to -4 % in pair 1 and -9 % to -10 % in pair 2, because the 1-thread
+cell is the noisiest one on battery (see "Spread"). The phase driver's
+interleaved samples put it at -6 % (tile) and -8 % (quarter).
+
+**Phase times** (scratch driver, median of 5, min-max in brackets, ms):
+
+| domain, threads | refine | scan | split |
+|---|---:|---:|---:|
+| quarter, 1 | 453.3 -> 419.1 (-8 %) | 310.7 -> 305.6 (-2 %) | 123.0 (122.2-123.8) -> 95.7 (94.7-98.1) (-22 %) |
+| quarter, 8 | 198.5 -> 170.8 (-14 %) | 47.1 -> 47.4 (+1 %) | 131.9 (131.4-134.6) -> 103.8 (102.4-105.0) (-21 %) |
+| quarter, 16 | 203.1 -> 174.3 (-14 %) | 45.2 -> 45.3 (0 %) | 138.2 (131.8-138.9) -> 109.4 (102.5-110.0) (-21 %) |
+| tile, 1 | 501.2 -> 471.1 (-6 %) | 345.4 -> 345.5 (0 %) | 124.5 (123.4-125.4) -> 96.3 (95.0-96.5) (-23 %) |
+| tile, 8 | 225.1 -> 191.7 (-15 %) | 57.7 -> 58.2 (+1 %) | 134.5 (133.3-134.7) -> 102.6 (101.6-103.4) (-24 %) |
+| tile, 16 | 227.3 -> 193.9 (-15 %) | 56.1 -> 55.9 (0 %) | 137.7 (134.5-139.0) -> 106.9 (104.8-108.0) (-22 %) |
+
+The whole gain is in the split phase: 27-32 ms at every thread count. The scan
+is unchanged. The start mesh's `legalise_all` on the tile also fell from 2.6
+to 0.6 ms. It goes through the same `must_flip`, so it takes the integer path
+too. Rounds, insertions and flips are the same in both trees: quarter 41 /
+213,464 / 445,657, tile 53 / 219,837 / 445,675.
+
+**Ceiling** (speed-up over 1 thread; best, and at 20 threads):
+
+| domain | base 071e4f3 | 21b | 21a acceptance (AC) | 2026-09-26 (battery) |
+|---|---|---|---|---|
+| tile | 2.28x @12 (2.26x @20) [2.30x, 2.28x] | 2.55x @9 (2.53x @20) [2.45x, 2.40x] | 2.28x | 2.12x @8 (2.10x @20) |
+| quarter | 2.23x @20 (2.23x @20) [2.34x, 2.32x] | 2.54x @14 (2.48x @20) [2.47x, 2.46x] | 2.29x | 2.09x @15 (2.07x @20) |
+
+The ceiling rises because the split phase, which does not scale, got shorter.
+The 1-thread refine fell by 30 ms and the 8-thread refine by 30-33 ms, so the
+ratio rises. The ceiling depends on the noisy 1-thread cell, so it moves by
+about 0.1x between the two pairs. From 8 threads on, the curve is flat: every
+median from 8 to 20 threads is within 1.4-3.2 % of that run's best. The
+2026-09-26 battery figures (medians of `scaling/raw_battery.tsv`, master
+`8f47e7e`) came from a different method (`scale.py`, the refine call alone)
+and are context here, not a baseline.
+
+**Quality: identical.** Both trees, both pairs:
+
+| domain | worst angle | max degree | within tolerance | Delaunay violations | mesh sha256 (ASCII POINTS on) |
+|---|---:|---:|---|---:|---|
+| quarter | 0.3955 deg | 18 | yes | 0 of 641,791 | `1e531976f33b38dd883f9cf1945f6303dbac94c92f30b52b8bd7b46db7eab9ee` |
+| tile | 0.6296 deg | 74 | yes | 0 of 692,056 | `11741a81adfa17b34a1ea56875ec9e791414d17625eb1c58305e7c0cf10e005c` |
+
+On both domains, the mesh sha256 of 21b equals the base's, and also the 21a
+acceptance run's figures. That is what a bit-identical increment requires. The
+scratch driver's binary meshes also hash the same for base, green and head:
+quarter `ff705683…`, tile `645919aa…`. These are the hashes @developer
+reported at green.
+
+## Spread of the median of 5 (battery)
+
+For each tree, the two runs' medians were compared over all 42 cells (2
+domains × thread counts 0-20), as |median(run 1) / median(run 2) - 1|:
+
+| tree | median abs. difference | 90th percentile | max |
+|---|---:|---:|---:|
+| base 071e4f3 | 0.89 % | 2.74 % | 5.45 % (quarter, 3 threads) |
+| 21b | 0.69 % | 1.77 % | 3.62 % (tile, 1 thread) |
+
+Within a single cell, (max - min) / median of the 5 samples had a median of
+3.5-4.6 % per run, against 1.1-1.9 % on AC in 21a. There were also single
+samples far from the rest. One base tile sample at the default thread count
+took 546 ms, against 222-230 ms for the others. One base tile 1-thread sample
+took 719 ms, against 499-522 ms. The median absorbed them. Their cause has not
+been measured.
+
+Measured the other way round, base run 2 against base run 1, `bench.py`
+reports REGRESSION on two quarter cells of the **same build**: +5.4 % at 1
+thread and +5.8 % at 3 threads (the `compare` above). This is the second time
+the median of 5 of an unchanged build has moved by about 5 % (21a on AC:
+4.6 %). On battery it went over the 5 % threshold. A single-cell verdict at
+about 5 % is therefore noise until that cell is rerun. 21b's gain at 8 threads
+and above is about 15 % in every cell of both pairs, several times this
+spread.
+
+## Green against head
+
+Head's split phase is 4-6 % slower than green's. The @developer quick sweep
+below reported faster 21b figures than this run measured, so the green commit
+`f707322` was built in a temporary worktree, since removed (the same `build()`, `.so` `2e30f2d8…`) and interleaved
+with head `a03ae65` by the scratch driver. Battery, 86 %, 5 repeats, ms:
+
+| domain, threads | refine, green -> head | split, green -> head |
+|---|---:|---:|
+| quarter, 1 | 414.0 -> 421.4 (+1.8 %) | 89.9 (89.5-90.9) -> 95.5 (94.4-118.7) (+6.3 %) |
+| quarter, 8 | 174.4 -> 170.2 (-2.4 %) | 99.7 (97.7-106.4) -> 103.6 (102.1-121.4) (+4.0 %) |
+| tile, 1 | 468.3 -> 472.5 (+0.9 %) | 91.2 (90.2-98.3) -> 96.6 (95.5-99.1) (+5.9 %) |
+| tile, 8 | 185.7 -> 193.0 (+4.0 %) | 97.3 (97.0-98.1) -> 102.8 (102.7-104.4) (+5.6 %) |
+
+The split phase went up by 4-6 ms in all four cells, and the min-max ranges
+do not overlap. The refine totals are less clean: green's quarter 8-thread
+samples were noisy, with a scan range of 47-61 ms. The only production-code
+change from `f707322` to `a03ae65` is the `lawson.hpp` hunk that refuses
+non-finite corners in `lattice_incircle` (`git diff --stat f707322 a03ae65 --
+include src_cpp src_python`). So the 5 ms is attributed to that hunk by
+elimination. How it costs 5 ms has not been profiled. The loss is about a
+sixth of 21b's gain, and 21b still passes with a wide margin.
+
+## Against the estimates
+
+| source | measure | estimate or report | measured (pair 1 [pair 2]) |
+|---|---|---|---|
+| design, QW2 "Expected gain" | refine, every thread count | ~-18 ms: -4 % at 1 thread, -8 % at 8 | 8 threads: tile -33.4 [-33.3] ms (-14.8 %), quarter -29.6 [-29.2] ms (-14.8 %); 1 thread: -6 to -8 % (phase driver) |
+| design, "What the quick wins add up to" | 8-thread refine after 21a + 21b | 0.229 -> ~0.18 s (battery profile) | 0.193 / 0.171 s (tile / quarter); 21a's AC base `227522c` was 0.249 / 0.230 s |
+| @developer quick sweep (battery, vs 21a head `d3ee2ce`, at green) | quarter, 8 threads | 0.198 -> 0.164 s (-17 %) | 0.201 -> 0.171 s (-14.8 %) [0.201 -> 0.172] |
+| same | tile, 8 threads | 0.225 -> 0.187 s (-17 %) | 0.226 -> 0.193 s (-14.8 %) [0.227 -> 0.194] |
+| same | split phase, 8 threads | quarter 132 -> 98 ms (-26 %), tile 134 -> 97 ms (-28 %) | quarter 131.9 -> 103.8 ms (-21 %), tile 134.5 -> 102.6 ms (-24 %) |
+
+The base figures agree with @developer's within 1.5 %. The 21b figures are
+3-4 % slower than his, and his split figures match green's here (98 and 97 ms
+against 99.7 and 97.3 ms). The difference is the green-to-head change above.
+The design's estimate was too low by a factor of about 2 at 8 threads. As the
+increment file records, the integer path is also cheaper than the filtered
+path it replaces for all-node quads, not only than the exact path.
+
+## Addendum: the head slowdown, fixed in 4be156c (@developer, 2026-09-27)
+
+"Green against head" above measured `a03ae65`, whose non-finite refusal (four
+`std::isfinite` pairs per `lattice_incircle` call) cost 4-6 % in the split
+phase. `4be156c` folds that refusal into the spread test
+(`!(std::fabs(x) <= bound)` refuses inf and NaN alike) and drops the explicit
+checks; `[nonfinite]` still passes and UBSan stays quiet on it. Timed by
+@developer with a scratch driver (not checked in), green `f707322` and head
+`4be156c` alternating per repeat, 5 repeats, battery 86 %, discharging, ms,
+median (min-max):
+
+| domain, threads | refine, green -> head | split, green -> head |
+|---|---:|---:|
+| quarter, 1 | 410.3 (409.5-412.9) -> 409.5 (408.8-411.5) | 90.4 (89.1-90.7) -> 89.2 (89.0-90.5) |
+| quarter, 8 | 163.9 (163.0-164.0) -> 163.0 (162.1-165.2) | 97.8 (97.4-98.2) -> 97.5 (96.3-99.1) |
+| tile, 1 | 462.8 (460.2-466.4) -> 462.9 (460.9-465.5) | 90.6 (89.7-91.2) -> 90.4 (89.8-91.8) |
+| tile, 8 | 184.4 (183.8-184.8) -> 185.8 (183.2-186.2) | 96.6 (96.1-96.9) -> 97.5 (95.9-97.6) |
+
+Every range overlaps and every median is within 1.4 %: the gap is closed. The
+bench.py verdict above was measured on `a03ae65`, so it understates 21b by
+about that gap; it was not re-run on `4be156c`. Meshes equal on both domains.
diff --git a/docs/benchmarks/2026-09-27/21b-base-071e4f3-r2/README.md b/docs/benchmarks/2026-09-27/21b-base-071e4f3-r2/README.md
new file mode 100644
index 00000000..b5567fae
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-base-071e4f3-r2/README.md
@@ -0,0 +1,80 @@
+# Benchmark run `21b-base-071e4f3-r2`
+
+Generated by `tools/bench.py`.
+
+## Verdict
+
+- REGRESSION: quarter refine_s[t=1] 0.4477 -> 0.4717 (+5.4 %)
+- REGRESSION: quarter refine_s[t=3] 0.2536 -> 0.2682 (+5.8 %)
+
+Time threshold: 5.0 %.
+
+## Method
+
+- Started 2026-09-27T19:00:58.501425+02:00; tree `071e4f3910db061e263e3b46aa6d3201e4b0ee5b`; bench.py blob `77765b181714c31714da476c2335c799c6a84021`. Build: Release, AppleClang 21.0.0.21000101, `-O3 -DNDEBUG`, _core sha256 `654a2442f55e341f290ecca53b2761e57a4ea3a09bdc273fcf6c27d42874daaa`.
+- Apple M1 Max, 8 P + 2 E cores, 32 GiB, macOS 27.0, Python 3.14.7, numpy 2.5.3.
+- Power **battery** (87%), `pmset -g batt` before and after (in run.json).
+- DEM `/Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif` (sha256 `aabd0cbc28471ce8593e4c278381811bec3fbf4e1194058b3b4888d00c3af575`), tolerance 1.0, extra mesh args ``; domains: `quarter` (/Users/skavhaug/projects/rasputin/docs/benchmarks/2026-09-26/quarter.geojson), `tile` (the whole tile).
+- One child per sample, `/Users/skavhaug/projects/rasputin/.venv/bin/python /Users/skavhaug/projects/rasputin/tools/bench.py _child --pkg /private/tmp/claude-501/-Users-skavhaug-projects-rasputin/38487caf-f56e-46a3-b05b-867af1fb1619/scratchpad/base-071e4f3/build-bench/pkg --threads <N> -- mesh --dem /Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif --tolerance 1 --out <path> --binary`, repeats interleaved over thread counts; t=0 is the CLI's default, other counts are forced into `refine`. Quality: one `--ascii` run per domain at t=0, kept out of the repository; rerun the child with `--ascii --out PATH` to regenerate it.
+
+## `quarter`
+
+| threads | n | median s | min s | max s |
+|---:|---:|---:|---:|---:|
+| 0 | 5 | 0.2035 | 0.2005 | 0.2047 |
+| 1 | 5 | 0.4717 | 0.4590 | 0.4856 |
+| 2 | 5 | 0.3131 | 0.3052 | 0.3215 |
+| 3 | 5 | 0.2682 | 0.2543 | 0.2781 |
+| 4 | 5 | 0.2457 | 0.2371 | 0.4392 |
+| 5 | 5 | 0.2248 | 0.2202 | 0.2592 |
+| 6 | 5 | 0.2140 | 0.2096 | 0.2453 |
+| 7 | 5 | 0.2071 | 0.2026 | 0.2717 |
+| 8 | 5 | 0.2012 | 0.1985 | 0.2129 |
+| 9 | 5 | 0.2027 | 0.2007 | 0.2101 |
+| 10 | 5 | 0.2026 | 0.2005 | 0.2071 |
+| 11 | 5 | 0.2057 | 0.2022 | 0.2157 |
+| 12 | 5 | 0.2021 | 0.2014 | 0.2260 |
+| 13 | 5 | 0.2041 | 0.2011 | 0.2084 |
+| 14 | 5 | 0.2063 | 0.2020 | 0.2113 |
+| 15 | 5 | 0.2027 | 0.1977 | 0.2136 |
+| 16 | 5 | 0.2040 | 0.2020 | 0.2099 |
+| 17 | 5 | 0.2017 | 0.1998 | 0.2095 |
+| 18 | 5 | 0.2041 | 0.2003 | 0.2060 |
+| 19 | 5 | 0.2041 | 0.2032 | 0.4034 |
+| 20 | 5 | 0.2035 | 0.2020 | 0.2083 |
+
+Ceiling: 2.32x at 20 threads over 1, best 2.34x at 8; 2026-09-26: about 2.2x, flat from about 7.
+
+Quality: worst angle 0.3955 deg, median 45.00, share under 1 deg 0.00003, max degree 18, within tolerance True, Delaunay 0 violations of 641791 edges (80136 decided exactly), mesh sha256 `1e531976f33b38dd883f9cf1945f6303dbac94c92f30b52b8bd7b46db7eab9ee`.
+
+## `tile`
+
+| threads | n | median s | min s | max s |
+|---:|---:|---:|---:|---:|
+| 0 | 5 | 0.2295 | 0.2258 | 0.2310 |
+| 1 | 5 | 0.5196 | 0.5115 | 0.5774 |
+| 2 | 5 | 0.3464 | 0.3452 | 0.3483 |
+| 3 | 5 | 0.2986 | 0.2949 | 0.2999 |
+| 4 | 5 | 0.2697 | 0.2684 | 0.2708 |
+| 5 | 5 | 0.2527 | 0.2507 | 0.2560 |
+| 6 | 5 | 0.2412 | 0.2403 | 0.2460 |
+| 7 | 5 | 0.2309 | 0.2303 | 0.2565 |
+| 8 | 5 | 0.2274 | 0.2254 | 0.2347 |
+| 9 | 5 | 0.2277 | 0.2245 | 0.2388 |
+| 10 | 5 | 0.2263 | 0.2235 | 0.2283 |
+| 11 | 5 | 0.2259 | 0.2249 | 0.2272 |
+| 12 | 5 | 0.2274 | 0.2258 | 0.2302 |
+| 13 | 5 | 0.2256 | 0.2250 | 0.2294 |
+| 14 | 5 | 0.2287 | 0.2248 | 0.2330 |
+| 15 | 5 | 0.2275 | 0.2272 | 0.2285 |
+| 16 | 5 | 0.2258 | 0.2240 | 0.2285 |
+| 17 | 5 | 0.2279 | 0.2220 | 0.2288 |
+| 18 | 5 | 0.2281 | 0.2250 | 0.2300 |
+| 19 | 5 | 0.2267 | 0.2262 | 0.2503 |
+| 20 | 5 | 0.2280 | 0.2263 | 0.2304 |
+
+Ceiling: 2.28x at 20 threads over 1, best 2.30x at 13; 2026-09-26: about 2.2x, flat from about 7.
+
+Quality: worst angle 0.6296 deg, median 45.00, share under 1 deg 0.00000, max degree 74, within tolerance True, Delaunay 0 violations of 692056 edges (94469 decided exactly), mesh sha256 `11741a81adfa17b34a1ea56875ec9e791414d17625eb1c58305e7c0cf10e005c`.
+
+<!-- bench.py: generated above this line; hand-written prose below is kept -->
diff --git a/docs/benchmarks/2026-09-27/21b-base-071e4f3-r2/raw.tsv b/docs/benchmarks/2026-09-27/21b-base-071e4f3-r2/raw.tsv
new file mode 100644
index 00000000..c52520a8
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-base-071e4f3-r2/raw.tsv
@@ -0,0 +1,210 @@
+quarter	0	0	0.200459	battery
+quarter	1	0	0.485582	battery
+quarter	2	0	0.311852	battery
+quarter	3	0	0.268191	battery
+quarter	4	0	0.255414	battery
+quarter	5	0	0.224823	battery
+quarter	6	0	0.218775	battery
+quarter	7	0	0.206270	battery
+quarter	8	0	0.200308	battery
+quarter	9	0	0.200724	battery
+quarter	10	0	0.202624	battery
+quarter	11	0	0.202155	battery
+quarter	12	0	0.201572	battery
+quarter	13	0	0.202155	battery
+quarter	14	0	0.206264	battery
+quarter	15	0	0.201372	battery
+quarter	16	0	0.201963	battery
+quarter	17	0	0.202723	battery
+quarter	18	0	0.200339	battery
+quarter	19	0	0.403431	battery
+quarter	20	0	0.206785	battery
+quarter	0	1	0.204110	battery
+quarter	1	1	0.459004	battery
+quarter	2	1	0.305214	battery
+quarter	3	1	0.254312	battery
+quarter	4	1	0.237127	battery
+quarter	5	1	0.220227	battery
+quarter	6	1	0.209550	battery
+quarter	7	1	0.202557	battery
+quarter	8	1	0.198464	battery
+quarter	9	1	0.202391	battery
+quarter	10	1	0.202658	battery
+quarter	11	1	0.207216	battery
+quarter	12	1	0.202120	battery
+quarter	13	1	0.204660	battery
+quarter	14	1	0.202190	battery
+quarter	15	1	0.202678	battery
+quarter	16	1	0.204170	battery
+quarter	17	1	0.209489	battery
+quarter	18	1	0.204085	battery
+quarter	19	1	0.204079	battery
+quarter	20	1	0.203463	battery
+quarter	0	2	0.202652	battery
+quarter	1	2	0.471674	battery
+quarter	2	2	0.321496	battery
+quarter	3	2	0.278132	battery
+quarter	4	2	0.439194	battery
+quarter	5	2	0.259238	battery
+quarter	6	2	0.245332	battery
+quarter	7	2	0.271666	battery
+quarter	8	2	0.212885	battery
+quarter	9	2	0.210065	battery
+quarter	10	2	0.207113	battery
+quarter	11	2	0.215739	battery
+quarter	12	2	0.226002	battery
+quarter	13	2	0.208420	battery
+quarter	14	2	0.211313	battery
+quarter	15	2	0.213639	battery
+quarter	16	2	0.202291	battery
+quarter	17	2	0.200527	battery
+quarter	18	2	0.205990	battery
+quarter	19	2	0.203163	battery
+quarter	20	2	0.202049	battery
+quarter	0	3	0.204737	battery
+quarter	1	3	0.464410	battery
+quarter	2	3	0.313105	battery
+quarter	3	3	0.262664	battery
+quarter	4	3	0.245690	battery
+quarter	5	3	0.224979	battery
+quarter	6	3	0.214014	battery
+quarter	7	3	0.207119	battery
+quarter	8	3	0.201220	battery
+quarter	9	3	0.202717	battery
+quarter	10	3	0.200512	battery
+quarter	11	3	0.205701	battery
+quarter	12	3	0.202536	battery
+quarter	13	3	0.201102	battery
+quarter	14	3	0.201958	battery
+quarter	15	3	0.205876	battery
+quarter	16	3	0.204023	battery
+quarter	17	3	0.201718	battery
+quarter	18	3	0.205416	battery
+quarter	19	3	0.204126	battery
+quarter	20	3	0.208344	battery
+quarter	0	4	0.203521	battery
+quarter	1	4	0.480052	battery
+quarter	2	4	0.313865	battery
+quarter	3	4	0.269578	battery
+quarter	4	4	0.239599	battery
+quarter	5	4	0.224790	battery
+quarter	6	4	0.213109	battery
+quarter	7	4	0.208681	battery
+quarter	8	4	0.210579	battery
+quarter	9	4	0.202930	battery
+quarter	10	4	0.202131	battery
+quarter	11	4	0.204161	battery
+quarter	12	4	0.201438	battery
+quarter	13	4	0.204145	battery
+quarter	14	4	0.211258	battery
+quarter	15	4	0.197724	battery
+quarter	16	4	0.209924	battery
+quarter	17	4	0.199821	battery
+quarter	18	4	0.201075	battery
+quarter	19	4	0.205292	battery
+quarter	20	4	0.203324	battery
+tile	0	0	0.231036	battery
+tile	1	0	0.520778	battery
+tile	2	0	0.345944	battery
+tile	3	0	0.298592	battery
+tile	4	0	0.270133	battery
+tile	5	0	0.255796	battery
+tile	6	0	0.243458	battery
+tile	7	0	0.230270	battery
+tile	8	0	0.228004	battery
+tile	9	0	0.224461	battery
+tile	10	0	0.226486	battery
+tile	11	0	0.225944	battery
+tile	12	0	0.226456	battery
+tile	13	0	0.227303	battery
+tile	14	0	0.224840	battery
+tile	15	0	0.227932	battery
+tile	16	0	0.228540	battery
+tile	17	0	0.228092	battery
+tile	18	0	0.229984	battery
+tile	19	0	0.226247	battery
+tile	20	0	0.229510	battery
+tile	0	1	0.229506	battery
+tile	1	1	0.516637	battery
+tile	2	1	0.348291	battery
+tile	3	1	0.299730	battery
+tile	4	1	0.269688	battery
+tile	5	1	0.252408	battery
+tile	6	1	0.240632	battery
+tile	7	1	0.230873	battery
+tile	8	1	0.227333	battery
+tile	9	1	0.229541	battery
+tile	10	1	0.226338	battery
+tile	11	1	0.227173	battery
+tile	12	1	0.225821	battery
+tile	13	1	0.225161	battery
+tile	14	1	0.232958	battery
+tile	15	1	0.227153	battery
+tile	16	1	0.225887	battery
+tile	17	1	0.228808	battery
+tile	18	1	0.228060	battery
+tile	19	1	0.250340	battery
+tile	20	1	0.230400	battery
+tile	0	2	0.229494	battery
+tile	1	2	0.519648	battery
+tile	2	2	0.345191	battery
+tile	3	2	0.294877	battery
+tile	4	2	0.269346	battery
+tile	5	2	0.250668	battery
+tile	6	2	0.241225	battery
+tile	7	2	0.231232	battery
+tile	8	2	0.227433	battery
+tile	9	2	0.225887	battery
+tile	10	2	0.228292	battery
+tile	11	2	0.226492	battery
+tile	12	2	0.227363	battery
+tile	13	2	0.225552	battery
+tile	14	2	0.229303	battery
+tile	15	2	0.227531	battery
+tile	16	2	0.225251	battery
+tile	17	2	0.227905	battery
+tile	18	2	0.229013	battery
+tile	19	2	0.226724	battery
+tile	20	2	0.226564	battery
+tile	0	3	0.226004	battery
+tile	1	3	0.511511	battery
+tile	2	3	0.348199	battery
+tile	3	3	0.297863	battery
+tile	4	3	0.270763	battery
+tile	5	3	0.252715	battery
+tile	6	3	0.240333	battery
+tile	7	3	0.230918	battery
+tile	8	3	0.225412	battery
+tile	9	3	0.227677	battery
+tile	10	3	0.224431	battery
+tile	11	3	0.225692	battery
+tile	12	3	0.228856	battery
+tile	13	3	0.229391	battery
+tile	14	3	0.227421	battery
+tile	15	3	0.227519	battery
+tile	16	3	0.224000	battery
+tile	17	3	0.222048	battery
+tile	18	3	0.226161	battery
+tile	19	3	0.229516	battery
+tile	20	3	0.228050	battery
+tile	0	4	0.225756	battery
+tile	1	4	0.577365	battery
+tile	2	4	0.346433	battery
+tile	3	4	0.299856	battery
+tile	4	4	0.268390	battery
+tile	5	4	0.256050	battery
+tile	6	4	0.245968	battery
+tile	7	4	0.256545	battery
+tile	8	4	0.234697	battery
+tile	9	4	0.238841	battery
+tile	10	4	0.223546	battery
+tile	11	4	0.224906	battery
+tile	12	4	0.230229	battery
+tile	13	4	0.224951	battery
+tile	14	4	0.228675	battery
+tile	15	4	0.228530	battery
+tile	16	4	0.225757	battery
+tile	17	4	0.224592	battery
+tile	18	4	0.225015	battery
+tile	19	4	0.226618	battery
+tile	20	4	0.226290	battery
diff --git a/docs/benchmarks/2026-09-27/21b-base-071e4f3-r2/run.json b/docs/benchmarks/2026-09-27/21b-base-071e4f3-r2/run.json
new file mode 100644
index 00000000..7dfe3a9f
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-base-071e4f3-r2/run.json
@@ -0,0 +1,2956 @@
+{
+ "label": "21b-base-071e4f3-r2",
+ "started": "2026-09-27T19:00:58.501425+02:00",
+ "tree": {
+  "commit": "071e4f3910db061e263e3b46aa6d3201e4b0ee5b",
+  "dirty": false
+ },
+ "bench_blob": "77765b181714c31714da476c2335c799c6a84021",
+ "child_argv": [
+  "/Users/skavhaug/projects/rasputin/.venv/bin/python",
+  "/Users/skavhaug/projects/rasputin/tools/bench.py",
+  "_child",
+  "--pkg",
+  "/private/tmp/claude-501/-Users-skavhaug-projects-rasputin/38487caf-f56e-46a3-b05b-867af1fb1619/scratchpad/base-071e4f3/build-bench/pkg",
+  "--threads",
+  "<N>",
+  "--",
+  "mesh",
+  "--dem",
+  "/Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif",
+  "--tolerance",
+  "1",
+  "--out",
+  "<path>",
+  "--binary"
+ ],
+ "build": {
+  "no_build": false,
+  "type": "Release",
+  "cxx_flags_release": "-O3 -DNDEBUG",
+  "compiler": "AppleClang 21.0.0.21000101",
+  "so_sha256": "654a2442f55e341f290ecca53b2761e57a4ea3a09bdc273fcf6c27d42874daaa"
+ },
+ "machine": {
+  "cpu_brand": "Apple M1 Max",
+  "p_cores": 8,
+  "e_cores": 2,
+  "memory_bytes": 34359738368,
+  "macos": "27.0",
+  "python": "3.14.7",
+  "numpy": "2.5.3"
+ },
+ "power": {
+  "state": "battery",
+  "percent": 87,
+  "raw": "Now drawing from 'Battery Power'\n -InternalBattery-0 (id=7929955)\t88%; discharging; 4:15 remaining present: true\n--- after ---\nNow drawing from 'Battery Power'\n -InternalBattery-0 (id=7929955)\t87%; discharging; 3:45 remaining present: true\n"
+ },
+ "inputs": {
+  "dem": "/Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif",
+  "dem_sha256": "aabd0cbc28471ce8593e4c278381811bec3fbf4e1194058b3b4888d00c3af575",
+  "domains": [
+   {
+    "name": "quarter",
+    "path": "/Users/skavhaug/projects/rasputin/docs/benchmarks/2026-09-26/quarter.geojson",
+    "sha256": "b7f7bbe96848383c77863ce29ee28c45bce087601a1738ee82ddb4ffb8279d1e"
+   },
+   {
+    "name": "tile",
+    "path": null,
+    "sha256": null
+   }
+  ],
+  "tolerance": 1.0,
+  "extra_args": []
+ },
+ "samples": [
+  {
+   "domain": "quarter",
+   "threads": 0,
+   "repeat": 0,
+   "refine_s": 0.20045908296015114,
+   "app_s": 0.29912745801266283,
+   "proc_s": 0.7424882500199601,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 1,
+   "repeat": 0,
+   "refine_s": 0.48558204201981425,
+   "app_s": 0.587119875010103,
+   "proc_s": 0.8479538340470754,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 2,
+   "repeat": 0,
+   "refine_s": 0.3118523749872111,
+   "app_s": 0.4132430420140736,
+   "proc_s": 0.6703048329800367,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 3,
+   "repeat": 0,
+   "refine_s": 0.26819075003731996,
+   "app_s": 0.3686289170291275,
+   "proc_s": 0.6177374999970198,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 4,
+   "repeat": 0,
+   "refine_s": 0.2554139170097187,
+   "app_s": 0.35949529201025143,
+   "proc_s": 0.6240787910064682,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 5,
+   "repeat": 0,
+   "refine_s": 0.22482312499778345,
+   "app_s": 0.32935429201461375,
+   "proc_s": 0.5997380409971811,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 6,
+   "repeat": 0,
+   "refine_s": 0.2187749170116149,
+   "app_s": 0.3204351250315085,
+   "proc_s": 0.5704745419789106,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 7,
+   "repeat": 0,
+   "refine_s": 0.20626950002042577,
+   "app_s": 0.30797908402746543,
+   "proc_s": 0.5609417500090785,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 8,
+   "repeat": 0,
+   "refine_s": 0.20030795800266787,
+   "app_s": 0.2990692919702269,
+   "proc_s": 0.5462660419871099,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 9,
+   "repeat": 0,
+   "refine_s": 0.20072391699068248,
+   "app_s": 0.2987687080167234,
+   "proc_s": 0.5472577909822576,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 10,
+   "repeat": 0,
+   "refine_s": 0.20262391702271998,
+   "app_s": 0.3021687500295229,
+   "proc_s": 0.5465909159975126,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 11,
+   "repeat": 0,
+   "refine_s": 0.2021552079822868,
+   "app_s": 0.30302804097300395,
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+   "mesh_sha256": "1e531976f33b38dd883f9cf1945f6303dbac94c92f30b52b8bd7b46db7eab9ee"
+  },
+  "tile": {
+   "worst_angle": 0.6295986084102668,
+   "angle_median": 45.0,
+   "share_under_1": 2.1538262723728703e-6,
+   "max_degree": 74,
+   "within_tolerance": true,
+   "delaunay_checked": 692056,
+   "delaunay_ambiguous": 94469,
+   "delaunay_violations": 0,
+   "mesh_sha256": "11741a81adfa17b34a1ea56875ec9e791414d17625eb1c58305e7c0cf10e005c"
+  }
+ },
+ "accept_quality": false,
+ "threshold_pct": 5.0,
+ "verdict": [
+  "REGRESSION: quarter refine_s[t=1] 0.4477 -> 0.4717 (+5.4 %)",
+  "REGRESSION: quarter refine_s[t=3] 0.2536 -> 0.2682 (+5.8 %)"
+ ]
+}
diff --git a/docs/benchmarks/2026-09-27/21b-base-071e4f3/README.md b/docs/benchmarks/2026-09-27/21b-base-071e4f3/README.md
new file mode 100644
index 00000000..c4ef0b6b
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-base-071e4f3/README.md
@@ -0,0 +1,79 @@
+# Benchmark run `21b-base-071e4f3`
+
+Generated by `tools/bench.py`.
+
+## Verdict
+
+- NO BASELINE: no comparable stored run
+
+Time threshold: 5.0 %.
+
+## Method
+
+- Started 2026-09-27T18:56:04.773520+02:00; tree `071e4f3910db061e263e3b46aa6d3201e4b0ee5b`; bench.py blob `77765b181714c31714da476c2335c799c6a84021`. Build: Release, AppleClang 21.0.0.21000101, `-O3 -DNDEBUG`, _core sha256 `654a2442f55e341f290ecca53b2761e57a4ea3a09bdc273fcf6c27d42874daaa`.
+- Apple M1 Max, 8 P + 2 E cores, 32 GiB, macOS 27.0, Python 3.14.7, numpy 2.5.3.
+- Power **battery** (89%), `pmset -g batt` before and after (in run.json).
+- DEM `/Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif` (sha256 `aabd0cbc28471ce8593e4c278381811bec3fbf4e1194058b3b4888d00c3af575`), tolerance 1.0, extra mesh args ``; domains: `quarter` (/Users/skavhaug/projects/rasputin/docs/benchmarks/2026-09-26/quarter.geojson), `tile` (the whole tile).
+- One child per sample, `/Users/skavhaug/projects/rasputin/.venv/bin/python /Users/skavhaug/projects/rasputin/tools/bench.py _child --pkg /private/tmp/claude-501/-Users-skavhaug-projects-rasputin/38487caf-f56e-46a3-b05b-867af1fb1619/scratchpad/base-071e4f3/build-bench/pkg --threads <N> -- mesh --dem /Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif --tolerance 1 --out <path> --binary`, repeats interleaved over thread counts; t=0 is the CLI's default, other counts are forced into `refine`. Quality: one `--ascii` run per domain at t=0, kept out of the repository; rerun the child with `--ascii --out PATH` to regenerate it.
+
+## `quarter`
+
+| threads | n | median s | min s | max s |
+|---:|---:|---:|---:|---:|
+| 0 | 5 | 0.2031 | 0.2013 | 0.2063 |
+| 1 | 5 | 0.4477 | 0.4463 | 0.4566 |
+| 2 | 5 | 0.3003 | 0.2984 | 0.3041 |
+| 3 | 5 | 0.2536 | 0.2526 | 0.2614 |
+| 4 | 5 | 0.2349 | 0.2334 | 0.2448 |
+| 5 | 5 | 0.2203 | 0.2187 | 0.2253 |
+| 6 | 5 | 0.2097 | 0.2094 | 0.2125 |
+| 7 | 5 | 0.2041 | 0.2010 | 0.2200 |
+| 8 | 5 | 0.2007 | 0.2003 | 0.2073 |
+| 9 | 5 | 0.2026 | 0.1974 | 0.2107 |
+| 10 | 5 | 0.2038 | 0.2018 | 0.2046 |
+| 11 | 5 | 0.2014 | 0.1988 | 0.2034 |
+| 12 | 5 | 0.2015 | 0.1995 | 0.2043 |
+| 13 | 5 | 0.2024 | 0.1990 | 0.2037 |
+| 14 | 5 | 0.2006 | 0.1954 | 0.2025 |
+| 15 | 5 | 0.2051 | 0.2019 | 0.2104 |
+| 16 | 5 | 0.2029 | 0.1958 | 0.2049 |
+| 17 | 5 | 0.2050 | 0.2023 | 0.2056 |
+| 18 | 5 | 0.2032 | 0.1956 | 0.2048 |
+| 19 | 5 | 0.2045 | 0.2006 | 0.2081 |
+| 20 | 5 | 0.2004 | 0.1972 | 0.2045 |
+
+Ceiling: 2.23x at 20 threads over 1, best 2.23x at 20; 2026-09-26: about 2.2x, flat from about 7.
+
+Quality: worst angle 0.3955 deg, median 45.00, share under 1 deg 0.00003, max degree 18, within tolerance True, Delaunay 0 violations of 641791 edges (80136 decided exactly), mesh sha256 `1e531976f33b38dd883f9cf1945f6303dbac94c92f30b52b8bd7b46db7eab9ee`.
+
+## `tile`
+
+| threads | n | median s | min s | max s |
+|---:|---:|---:|---:|---:|
+| 0 | 5 | 0.2263 | 0.2222 | 0.5463 |
+| 1 | 5 | 0.5071 | 0.4992 | 0.7191 |
+| 2 | 5 | 0.3428 | 0.3371 | 0.3948 |
+| 3 | 5 | 0.2895 | 0.2857 | 0.3160 |
+| 4 | 5 | 0.2680 | 0.2642 | 0.2846 |
+| 5 | 5 | 0.2507 | 0.2475 | 0.2618 |
+| 6 | 5 | 0.2380 | 0.2371 | 0.2446 |
+| 7 | 5 | 0.2305 | 0.2264 | 0.2330 |
+| 8 | 5 | 0.2264 | 0.2235 | 0.2296 |
+| 9 | 5 | 0.2263 | 0.2230 | 0.2299 |
+| 10 | 5 | 0.2252 | 0.2194 | 0.2295 |
+| 11 | 5 | 0.2252 | 0.2198 | 0.2321 |
+| 12 | 5 | 0.2228 | 0.2204 | 0.2329 |
+| 13 | 5 | 0.2270 | 0.2246 | 0.2282 |
+| 14 | 5 | 0.2266 | 0.2205 | 0.2335 |
+| 15 | 5 | 0.2272 | 0.2186 | 0.2329 |
+| 16 | 5 | 0.2236 | 0.2216 | 0.2293 |
+| 17 | 5 | 0.2279 | 0.2188 | 0.2392 |
+| 18 | 5 | 0.2265 | 0.2238 | 0.2271 |
+| 19 | 5 | 0.2252 | 0.2232 | 0.2339 |
+| 20 | 5 | 0.2247 | 0.2202 | 0.2331 |
+
+Ceiling: 2.26x at 20 threads over 1, best 2.28x at 12; 2026-09-26: about 2.2x, flat from about 7.
+
+Quality: worst angle 0.6296 deg, median 45.00, share under 1 deg 0.00000, max degree 74, within tolerance True, Delaunay 0 violations of 692056 edges (94469 decided exactly), mesh sha256 `11741a81adfa17b34a1ea56875ec9e791414d17625eb1c58305e7c0cf10e005c`.
+
+<!-- bench.py: generated above this line; hand-written prose below is kept -->
diff --git a/docs/benchmarks/2026-09-27/21b-base-071e4f3/raw.tsv b/docs/benchmarks/2026-09-27/21b-base-071e4f3/raw.tsv
new file mode 100644
index 00000000..7371854b
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-base-071e4f3/raw.tsv
@@ -0,0 +1,210 @@
+quarter	0	0	0.206342	battery
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diff --git a/docs/benchmarks/2026-09-27/21b-base-071e4f3/run.json b/docs/benchmarks/2026-09-27/21b-base-071e4f3/run.json
new file mode 100644
index 00000000..916dc080
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-base-071e4f3/run.json
@@ -0,0 +1,2955 @@
+{
+ "label": "21b-base-071e4f3",
+ "started": "2026-09-27T18:56:04.773520+02:00",
+ "tree": {
+  "commit": "071e4f3910db061e263e3b46aa6d3201e4b0ee5b",
+  "dirty": false
+ },
+ "bench_blob": "77765b181714c31714da476c2335c799c6a84021",
+ "child_argv": [
+  "/Users/skavhaug/projects/rasputin/.venv/bin/python",
+  "/Users/skavhaug/projects/rasputin/tools/bench.py",
+  "_child",
+  "--pkg",
+  "/private/tmp/claude-501/-Users-skavhaug-projects-rasputin/38487caf-f56e-46a3-b05b-867af1fb1619/scratchpad/base-071e4f3/build-bench/pkg",
+  "--threads",
+  "<N>",
+  "--",
+  "mesh",
+  "--dem",
+  "/Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif",
+  "--tolerance",
+  "1",
+  "--out",
+  "<path>",
+  "--binary"
+ ],
+ "build": {
+  "no_build": false,
+  "type": "Release",
+  "cxx_flags_release": "-O3 -DNDEBUG",
+  "compiler": "AppleClang 21.0.0.21000101",
+  "so_sha256": "654a2442f55e341f290ecca53b2761e57a4ea3a09bdc273fcf6c27d42874daaa"
+ },
+ "machine": {
+  "cpu_brand": "Apple M1 Max",
+  "p_cores": 8,
+  "e_cores": 2,
+  "memory_bytes": 34359738368,
+  "macos": "27.0",
+  "python": "3.14.7",
+  "numpy": "2.5.3"
+ },
+ "power": {
+  "state": "battery",
+  "percent": 89,
+  "raw": "Now drawing from 'Battery Power'\n -InternalBattery-0 (id=7929955)\t90%; discharging; 4:35 remaining present: true\n--- after ---\nNow drawing from 'Battery Power'\n -InternalBattery-0 (id=7929955)\t89%; discharging; 4:45 remaining present: true\n"
+ },
+ "inputs": {
+  "dem": "/Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif",
+  "dem_sha256": "aabd0cbc28471ce8593e4c278381811bec3fbf4e1194058b3b4888d00c3af575",
+  "domains": [
+   {
+    "name": "quarter",
+    "path": "/Users/skavhaug/projects/rasputin/docs/benchmarks/2026-09-26/quarter.geojson",
+    "sha256": "b7f7bbe96848383c77863ce29ee28c45bce087601a1738ee82ddb4ffb8279d1e"
+   },
+   {
+    "name": "tile",
+    "path": null,
+    "sha256": null
+   }
+  ],
+  "tolerance": 1.0,
+  "extra_args": []
+ },
+ "samples": [
+  {
+   "domain": "quarter",
+   "threads": 0,
+   "repeat": 0,
+   "refine_s": 0.20634187496034428,
+   "app_s": 0.3182133749942295,
+   "proc_s": 0.8246584999724291,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 1,
+   "repeat": 0,
+   "refine_s": 0.44894795899745077,
+   "app_s": 0.5505178750026971,
+   "proc_s": 0.794157666969113,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 2,
+   "repeat": 0,
+   "refine_s": 0.2992964590084739,
+   "app_s": 0.399226792040281,
+   "proc_s": 0.6434074160060845,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 3,
+   "repeat": 0,
+   "refine_s": 0.2529656669939868,
+   "app_s": 0.3528619999997318,
+   "proc_s": 0.5971264170366339,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 4,
+   "repeat": 0,
+   "refine_s": 0.23450291698100045,
+   "app_s": 0.33351133298128843,
+   "proc_s": 0.5762154590338469,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 5,
+   "repeat": 0,
+   "refine_s": 0.2252621249645017,
+   "app_s": 0.3275257089990191,
+   "proc_s": 0.5756914590019733,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 6,
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+   "max": 0.2335126670077443
+  },
+  {
+   "domain": "tile",
+   "threads": 15,
+   "n": 5,
+   "median": 0.22718854201957583,
+   "min": 0.21860779100097716,
+   "max": 0.23291791701922193
+  },
+  {
+   "domain": "tile",
+   "threads": 16,
+   "n": 5,
+   "median": 0.22355283296201378,
+   "min": 0.22160725004505366,
+   "max": 0.2293331249966286
+  },
+  {
+   "domain": "tile",
+   "threads": 17,
+   "n": 5,
+   "median": 0.2279202500358224,
+   "min": 0.2188132909941487,
+   "max": 0.23917637503473088
+  },
+  {
+   "domain": "tile",
+   "threads": 18,
+   "n": 5,
+   "median": 0.22654483397491276,
+   "min": 0.2238081669784151,
+   "max": 0.22709508403204381
+  },
+  {
+   "domain": "tile",
+   "threads": 19,
+   "n": 5,
+   "median": 0.22518200002377853,
+   "min": 0.2232382500078529,
+   "max": 0.23388974997214973
+  },
+  {
+   "domain": "tile",
+   "threads": 20,
+   "n": 5,
+   "median": 0.2246745410375297,
+   "min": 0.22023845801595598,
+   "max": 0.2331381250405684
+  }
+ ],
+ "quality": {
+  "quarter": {
+   "worst_angle": 0.3955267116350698,
+   "angle_median": 45.0,
+   "share_under_1": 0.000028023175165862168,
+   "max_degree": 18,
+   "within_tolerance": true,
+   "delaunay_checked": 641791,
+   "delaunay_ambiguous": 80136,
+   "delaunay_violations": 0,
+   "mesh_sha256": "1e531976f33b38dd883f9cf1945f6303dbac94c92f30b52b8bd7b46db7eab9ee"
+  },
+  "tile": {
+   "worst_angle": 0.6295986084102668,
+   "angle_median": 45.0,
+   "share_under_1": 2.1538262723728703e-6,
+   "max_degree": 74,
+   "within_tolerance": true,
+   "delaunay_checked": 692056,
+   "delaunay_ambiguous": 94469,
+   "delaunay_violations": 0,
+   "mesh_sha256": "11741a81adfa17b34a1ea56875ec9e791414d17625eb1c58305e7c0cf10e005c"
+  }
+ },
+ "accept_quality": false,
+ "threshold_pct": 5.0,
+ "verdict": [
+  "NO BASELINE: no comparable stored run"
+ ]
+}
diff --git a/docs/benchmarks/2026-09-27/21b-r2/README.md b/docs/benchmarks/2026-09-27/21b-r2/README.md
new file mode 100644
index 00000000..8f87acde
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-r2/README.md
@@ -0,0 +1,79 @@
+# Benchmark run `21b-r2`
+
+Generated by `tools/bench.py`.
+
+## Verdict
+
+- ACCEPTED
+
+Time threshold: 5.0 %.
+
+## Method
+
+- Started 2026-09-27T19:03:19.106071+02:00; tree `a03ae65f06773d7191f7bcb66088a8ce087892d7` (dirty); bench.py blob `77765b181714c31714da476c2335c799c6a84021`. Build: Release, AppleClang 21.0.0.21000101, `-O3 -DNDEBUG`, _core sha256 `dfdeb6ca078bae61f3881d6bffda8a19e2cf5dac4cca8a27a39327f938fef080`.
+- Apple M1 Max, 8 P + 2 E cores, 32 GiB, macOS 27.0, Python 3.14.7, numpy 2.5.3.
+- Power **battery** (86%), `pmset -g batt` before and after (in run.json).
+- DEM `/Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif` (sha256 `aabd0cbc28471ce8593e4c278381811bec3fbf4e1194058b3b4888d00c3af575`), tolerance 1.0, extra mesh args ``; domains: `quarter` (/Users/skavhaug/projects/rasputin/docs/benchmarks/2026-09-26/quarter.geojson), `tile` (the whole tile).
+- One child per sample, `/Users/skavhaug/projects/rasputin/.venv/bin/python /Users/skavhaug/projects/rasputin/tools/bench.py _child --pkg /Users/skavhaug/projects/rasputin/build-bench/pkg --threads <N> -- mesh --dem /Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif --tolerance 1 --out <path> --binary`, repeats interleaved over thread counts; t=0 is the CLI's default, other counts are forced into `refine`. Quality: one `--ascii` run per domain at t=0, kept out of the repository; rerun the child with `--ascii --out PATH` to regenerate it.
+
+## `quarter`
+
+| threads | n | median s | min s | max s |
+|---:|---:|---:|---:|---:|
+| 0 | 5 | 0.1727 | 0.1722 | 0.1741 |
+| 1 | 5 | 0.4248 | 0.4158 | 0.4598 |
+| 2 | 5 | 0.2819 | 0.2715 | 0.2873 |
+| 3 | 5 | 0.2264 | 0.2245 | 0.2281 |
+| 4 | 5 | 0.2071 | 0.2064 | 0.2090 |
+| 5 | 5 | 0.1935 | 0.1907 | 0.1990 |
+| 6 | 5 | 0.1818 | 0.1805 | 0.1854 |
+| 7 | 5 | 0.1741 | 0.1730 | 0.1769 |
+| 8 | 5 | 0.1720 | 0.1680 | 0.1757 |
+| 9 | 5 | 0.1741 | 0.1715 | 0.1780 |
+| 10 | 5 | 0.1719 | 0.1704 | 0.1761 |
+| 11 | 5 | 0.1738 | 0.1727 | 0.1774 |
+| 12 | 5 | 0.1719 | 0.1696 | 0.1724 |
+| 13 | 5 | 0.1733 | 0.1668 | 0.1780 |
+| 14 | 5 | 0.1722 | 0.1680 | 0.1762 |
+| 15 | 5 | 0.1723 | 0.1661 | 0.1758 |
+| 16 | 5 | 0.1735 | 0.1733 | 0.1857 |
+| 17 | 5 | 0.1749 | 0.1718 | 0.1770 |
+| 18 | 5 | 0.1732 | 0.1724 | 0.1759 |
+| 19 | 5 | 0.1742 | 0.1716 | 0.1775 |
+| 20 | 5 | 0.1728 | 0.1726 | 0.1738 |
+
+Ceiling: 2.46x at 20 threads over 1, best 2.47x at 12; 2026-09-26: about 2.2x, flat from about 7.
+
+Quality: worst angle 0.3955 deg, median 45.00, share under 1 deg 0.00003, max degree 18, within tolerance True, Delaunay 0 violations of 641791 edges (80136 decided exactly), mesh sha256 `1e531976f33b38dd883f9cf1945f6303dbac94c92f30b52b8bd7b46db7eab9ee`.
+
+## `tile`
+
+| threads | n | median s | min s | max s |
+|---:|---:|---:|---:|---:|
+| 0 | 5 | 0.1938 | 0.1926 | 0.2006 |
+| 1 | 5 | 0.4714 | 0.4692 | 0.5157 |
+| 2 | 5 | 0.3113 | 0.3100 | 0.3137 |
+| 3 | 5 | 0.2596 | 0.2562 | 0.2672 |
+| 4 | 5 | 0.2352 | 0.2319 | 0.2420 |
+| 5 | 5 | 0.2185 | 0.2166 | 0.2190 |
+| 6 | 5 | 0.2051 | 0.2044 | 0.2141 |
+| 7 | 5 | 0.1970 | 0.1963 | 0.2028 |
+| 8 | 5 | 0.1941 | 0.1916 | 0.2035 |
+| 9 | 5 | 0.1937 | 0.1925 | 0.1944 |
+| 10 | 5 | 0.1938 | 0.1927 | 0.1944 |
+| 11 | 5 | 0.1926 | 0.1900 | 0.1937 |
+| 12 | 5 | 0.1934 | 0.1867 | 0.1949 |
+| 13 | 5 | 0.1924 | 0.1878 | 0.1993 |
+| 14 | 5 | 0.1926 | 0.1871 | 0.1950 |
+| 15 | 5 | 0.1939 | 0.1922 | 0.1980 |
+| 16 | 5 | 0.1939 | 0.1924 | 0.1977 |
+| 17 | 5 | 0.1931 | 0.1898 | 0.1955 |
+| 18 | 5 | 0.1930 | 0.1892 | 0.2021 |
+| 19 | 5 | 0.1946 | 0.1882 | 0.1985 |
+| 20 | 5 | 0.1963 | 0.1934 | 0.2011 |
+
+Ceiling: 2.40x at 20 threads over 1, best 2.45x at 13; 2026-09-26: about 2.2x, flat from about 7.
+
+Quality: worst angle 0.6296 deg, median 45.00, share under 1 deg 0.00000, max degree 74, within tolerance True, Delaunay 0 violations of 692056 edges (94469 decided exactly), mesh sha256 `11741a81adfa17b34a1ea56875ec9e791414d17625eb1c58305e7c0cf10e005c`.
+
+<!-- bench.py: generated above this line; hand-written prose below is kept -->
diff --git a/docs/benchmarks/2026-09-27/21b-r2/raw.tsv b/docs/benchmarks/2026-09-27/21b-r2/raw.tsv
new file mode 100644
index 00000000..605c1ffc
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-r2/raw.tsv
@@ -0,0 +1,210 @@
+quarter	0	0	0.174138	battery
+quarter	1	0	0.459843	battery
+quarter	2	0	0.287283	battery
+quarter	3	0	0.226407	battery
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+quarter	20	0	0.173760	battery
+quarter	0	1	0.172720	battery
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+quarter	19	1	0.174246	battery
+quarter	20	1	0.172795	battery
+quarter	0	2	0.174087	battery
+quarter	1	2	0.416042	battery
+quarter	2	2	0.271839	battery
+quarter	3	2	0.224458	battery
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+quarter	10	2	0.171940	battery
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+quarter	12	2	0.171897	battery
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+quarter	14	2	0.171826	battery
+quarter	15	2	0.175838	battery
+quarter	16	2	0.177047	battery
+quarter	17	2	0.176984	battery
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+quarter	19	2	0.171550	battery
+quarter	20	2	0.172734	battery
+quarter	0	3	0.172202	battery
+quarter	1	3	0.424784	battery
+quarter	2	3	0.271466	battery
+quarter	3	3	0.226839	battery
+quarter	4	3	0.207144	battery
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+quarter	8	3	0.173457	battery
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+quarter	10	3	0.170365	battery
+quarter	11	3	0.173394	battery
+quarter	12	3	0.169565	battery
+quarter	13	3	0.166795	battery
+quarter	14	3	0.167957	battery
+quarter	15	3	0.166127	battery
+quarter	16	3	0.185653	battery
+quarter	17	3	0.171810	battery
+quarter	18	3	0.172711	battery
+quarter	19	3	0.172879	battery
+quarter	20	3	0.173437	battery
+quarter	0	4	0.172404	battery
+quarter	1	4	0.415798	battery
+quarter	2	4	0.281926	battery
+quarter	3	4	0.224698	battery
+quarter	4	4	0.206495	battery
+quarter	5	4	0.190667	battery
+quarter	6	4	0.180865	battery
+quarter	7	4	0.172994	battery
+quarter	8	4	0.168908	battery
+quarter	9	4	0.175085	battery
+quarter	10	4	0.174012	battery
+quarter	11	4	0.173762	battery
+quarter	12	4	0.171906	battery
+quarter	13	4	0.177991	battery
+quarter	14	4	0.173746	battery
+quarter	15	4	0.172342	battery
+quarter	16	4	0.173480	battery
+quarter	17	4	0.175773	battery
+quarter	18	4	0.174010	battery
+quarter	19	4	0.174828	battery
+quarter	20	4	0.172566	battery
+tile	0	0	0.198318	battery
+tile	1	0	0.469162	battery
+tile	2	0	0.309961	battery
+tile	3	0	0.259805	battery
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diff --git a/docs/benchmarks/2026-09-27/21b-r2/run.json b/docs/benchmarks/2026-09-27/21b-r2/run.json
new file mode 100644
index 00000000..9663a1bb
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-r2/run.json
@@ -0,0 +1,2955 @@
+{
+ "label": "21b-r2",
+ "started": "2026-09-27T19:03:19.106071+02:00",
+ "tree": {
+  "commit": "a03ae65f06773d7191f7bcb66088a8ce087892d7",
+  "dirty": true
+ },
+ "bench_blob": "77765b181714c31714da476c2335c799c6a84021",
+ "child_argv": [
+  "/Users/skavhaug/projects/rasputin/.venv/bin/python",
+  "/Users/skavhaug/projects/rasputin/tools/bench.py",
+  "_child",
+  "--pkg",
+  "/Users/skavhaug/projects/rasputin/build-bench/pkg",
+  "--threads",
+  "<N>",
+  "--",
+  "mesh",
+  "--dem",
+  "/Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif",
+  "--tolerance",
+  "1",
+  "--out",
+  "<path>",
+  "--binary"
+ ],
+ "build": {
+  "no_build": false,
+  "type": "Release",
+  "cxx_flags_release": "-O3 -DNDEBUG",
+  "compiler": "AppleClang 21.0.0.21000101",
+  "so_sha256": "dfdeb6ca078bae61f3881d6bffda8a19e2cf5dac4cca8a27a39327f938fef080"
+ },
+ "machine": {
+  "cpu_brand": "Apple M1 Max",
+  "p_cores": 8,
+  "e_cores": 2,
+  "memory_bytes": 34359738368,
+  "macos": "27.0",
+  "python": "3.14.7",
+  "numpy": "2.5.3"
+ },
+ "power": {
+  "state": "battery",
+  "percent": 86,
+  "raw": "Now drawing from 'Battery Power'\n -InternalBattery-0 (id=7929955)\t87%; discharging; 3:45 remaining present: true\n--- after ---\nNow drawing from 'Battery Power'\n -InternalBattery-0 (id=7929955)\t86%; discharging; 3:45 remaining present: true\n"
+ },
+ "inputs": {
+  "dem": "/Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif",
+  "dem_sha256": "aabd0cbc28471ce8593e4c278381811bec3fbf4e1194058b3b4888d00c3af575",
+  "domains": [
+   {
+    "name": "quarter",
+    "path": "/Users/skavhaug/projects/rasputin/docs/benchmarks/2026-09-26/quarter.geojson",
+    "sha256": "b7f7bbe96848383c77863ce29ee28c45bce087601a1738ee82ddb4ffb8279d1e"
+   },
+   {
+    "name": "tile",
+    "path": null,
+    "sha256": null
+   }
+  ],
+  "tolerance": 1.0,
+  "extra_args": []
+ },
+ "samples": [
+  {
+   "domain": "quarter",
+   "threads": 0,
+   "repeat": 0,
+   "refine_s": 0.17413837503409013,
+   "app_s": 0.279501834011171,
+   "proc_s": 0.8468572910060175,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 1,
+   "repeat": 0,
+   "refine_s": 0.4598433749633841,
+   "app_s": 0.5692267090198584,
+   "proc_s": 0.8248299169936217,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
+   "domain": "quarter",
+   "threads": 2,
+   "repeat": 0,
+   "refine_s": 0.28728316602064297,
+   "app_s": 0.3943518750020303,
+   "proc_s": 0.6471626249840483,
+   "max_error": 0.9999796549479072,
+   "rounds": 41,
+   "inserted": 213464,
+   "flips": 445657
+  },
+  {
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+   "max": 0.20347508299164474
+  },
+  {
+   "domain": "tile",
+   "threads": 9,
+   "n": 5,
+   "median": 0.193747750017792,
+   "min": 0.19253195804776624,
+   "max": 0.194360958004836
+  },
+  {
+   "domain": "tile",
+   "threads": 10,
+   "n": 5,
+   "median": 0.1937961659859866,
+   "min": 0.1926926250453107,
+   "max": 0.1943662500125356
+  },
+  {
+   "domain": "tile",
+   "threads": 11,
+   "n": 5,
+   "median": 0.19258350000018254,
+   "min": 0.18997612496605143,
+   "max": 0.19373358297161758
+  },
+  {
+   "domain": "tile",
+   "threads": 12,
+   "n": 5,
+   "median": 0.1934349579969421,
+   "min": 0.1867351250257343,
+   "max": 0.1948923339950852
+  },
+  {
+   "domain": "tile",
+   "threads": 13,
+   "n": 5,
+   "median": 0.19238208403112367,
+   "min": 0.18776508298469707,
+   "max": 0.19929324998520315
+  },
+  {
+   "domain": "tile",
+   "threads": 14,
+   "n": 5,
+   "median": 0.19257062498945743,
+   "min": 0.18705441697966307,
+   "max": 0.1950259170262143
+  },
+  {
+   "domain": "tile",
+   "threads": 15,
+   "n": 5,
+   "median": 0.19385916600003839,
+   "min": 0.1921607909607701,
+   "max": 0.19800166704226285
+  },
+  {
+   "domain": "tile",
+   "threads": 16,
+   "n": 5,
+   "median": 0.19394916703458875,
+   "min": 0.19238533300813287,
+   "max": 0.1977054169983603
+  },
+  {
+   "domain": "tile",
+   "threads": 17,
+   "n": 5,
+   "median": 0.19311808299971744,
+   "min": 0.1898172500077635,
+   "max": 0.19553862500470132
+  },
+  {
+   "domain": "tile",
+   "threads": 18,
+   "n": 5,
+   "median": 0.19303141703130677,
+   "min": 0.1892026670393534,
+   "max": 0.2021426249993965
+  },
+  {
+   "domain": "tile",
+   "threads": 19,
+   "n": 5,
+   "median": 0.19463408400770277,
+   "min": 0.188183874997776,
+   "max": 0.19850595900788903
+  },
+  {
+   "domain": "tile",
+   "threads": 20,
+   "n": 5,
+   "median": 0.19625312503194436,
+   "min": 0.19335650000721216,
+   "max": 0.20112016599159688
+  }
+ ],
+ "quality": {
+  "quarter": {
+   "worst_angle": 0.3955267116350698,
+   "angle_median": 45.0,
+   "share_under_1": 0.000028023175165862168,
+   "max_degree": 18,
+   "within_tolerance": true,
+   "delaunay_checked": 641791,
+   "delaunay_ambiguous": 80136,
+   "delaunay_violations": 0,
+   "mesh_sha256": "1e531976f33b38dd883f9cf1945f6303dbac94c92f30b52b8bd7b46db7eab9ee"
+  },
+  "tile": {
+   "worst_angle": 0.6295986084102668,
+   "angle_median": 45.0,
+   "share_under_1": 2.1538262723728703e-6,
+   "max_degree": 74,
+   "within_tolerance": true,
+   "delaunay_checked": 692056,
+   "delaunay_ambiguous": 94469,
+   "delaunay_violations": 0,
+   "mesh_sha256": "11741a81adfa17b34a1ea56875ec9e791414d17625eb1c58305e7c0cf10e005c"
+  }
+ },
+ "accept_quality": false,
+ "threshold_pct": 5.0,
+ "verdict": [
+  "ACCEPTED"
+ ]
+}
diff --git a/docs/benchmarks/2026-09-27/21b-ties/README.md b/docs/benchmarks/2026-09-27/21b-ties/README.md
new file mode 100644
index 00000000..7b27c1da
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/README.md
@@ -0,0 +1,203 @@
+# 21b tie classification: which incircle tests could QW2's integer path answer?
+
+This is the measurement that `docs/increments/21-parallel-refine.md` §3 QW2 asks
+for before 21b is designed in detail. It counts cases and does not time
+anything.
+
+**Finding.** In the refine loop, every exact-path incircle test is one of these
+ties: a quad with four DEM-node corners, a lattice determinant of exactly 0, and
+all of QW2's conditions met. That is 146,962 of 146,962 on the quarter circle
+and 154,502 of 154,502 on the tile. The 146,962 is the same count the
+2026-09-27 profile gave. The ties are not mostly grid rectangles: on the quarter
+circle 38 % are axis-aligned rectangles or squares, and 62 % are other
+lattice-cocircular quads. Of **all** incircle tests in the refine loop, QW2's
+conditions hold on 99.973 % (quarter circle) and 100 % (tile).
+
+## Method
+
+- **Commit** `d3ee2ce` (branch `increment21b-lattice-incircle`, 21a's head).
+  It was measured in a scratch worktree (`git worktree add --detach <wt> d3ee2ce`)
+  with `scripts/instrument.patch` applied. That is a local patch and was never
+  committed; it is kept here unapplied. `git apply --check
+  scripts/instrument.patch` passes on `d3ee2ce`.
+- **What the patch records.** `FilteredKernel::incircle_ccw` sets a flag
+  saying whether it took the exact (`E::incircle_ccw`) path.
+  `detail::must_flip` passes the four `MeshVertex` corners of each incircle
+  call to `mesh::prof_ties::record` (new header
+  `include/terrain/mesh/prof_ties.hpp`), in the order they went to the kernel
+  (`a, b, c` counter-clockwise in the frame, then `d`), together with the
+  kernel's answer and that flag. `refine` tags the phase: `p0` is
+  `legalise_all`, `p1` the quality pass, `p2` the refine loop. At the end of
+  `refine` it appends one JSON line to `$RASPUTIN_TIES_OUT`. For each call it
+  records:
+  1. the number of corners that are nodes (`MeshVertex::is_node`);
+  2. for four nodes, the lattice determinant on `(col, -row)` translated to
+     `d`, computed in `__int128` so it is exact at any spread, with its sign
+     compared against the kernel's answer;
+  3. QW2's conditions, per call: `dx == dy`; every coordinate difference from
+     `d` at most 2^14 nodes; the frame exact, meaning `col * dx` and
+     `row * dy` are exact for all four corners (checked with `fma(a, b, -a*b)
+     == 0`). Once per refine call it also records the sufficient condition
+     QW2 states: the significant bits of `dx` plus
+     `bit_width(max(rows, cols) - 1)` are at most 53;
+  4. for four nodes and a zero determinant, the shape. A quad is an
+     axis-aligned square or rectangle when it has two distinct rows and two
+     distinct columns, and the size is recorded. It is a rotated square or
+     rectangle when its diagonals have equal midpoints and equal lengths. It
+     is an isosceles trapezoid when two disjoint chords are parallel: along
+     rows or columns, or in another direction. Anything else is "other
+     cyclic". A quad with three collinear corners is counted separately.
+- **Probes that can fail.** `scripts/classifier_selftest.cpp` checks the shape
+  classifier on hand-made quads, one per class, including a circle of radius
+  5. `scripts/counter_selftest.cpp` feeds `record` a case for each refusal or
+  mismatch counter: an inexact frame at `dx = 0.1`, `dx != dy`, a spread above
+  2^14, a nonzero lattice determinant on the exact path, a lattice sign that
+  disagrees with the kernel, and three node corners. Every one of those
+  counters fired. To build either one:
+  `c++ -std=c++20 -Wall -Wextra -Wpedantic -Werror -I<wt>/include <file>`.
+- **The instrumented build produces the same output.** The quarter circle's
+  binary VTK hashes to `ff705683…`, the value the serial profile got. Hashed
+  from `POINTS` onward, the way `bench.py` does it, the ASCII meshes match the
+  21a acceptance run (`docs/benchmarks/2026-09-27/21a/run.json`): quarter
+  circle `1e531976…`, tile `11741a81…`. Rounds, insertions and flips match as
+  well.
+- **Build**: Release, configured the way `bench.py`'s `build()` does it
+  (`-DCMAKE_BUILD_TYPE=Release -DRASPUTIN_BUILD_PYTHON=ON
+  -DRASPUTIN_BUILD_TESTS=OFF`), in `<wt>/build-instr`. The `_core` target was
+  built (exit 0) and copied into a fresh `build-instr/pkg/tin_engine/`
+  (symlinks to `src_python/tin_engine/*` plus the `.so`). The `.so` sha256 is
+  `731a9609…`.
+- **Runs.** The inputs are the 1 m benchmark: DEM
+  `tests/fixtures/dem_archive/7908_3_10m_z33.tif` (5051 × 5051 nodes,
+  `dx = dy = 10`), tolerance 1, all other CLI defaults. The domains are
+  `docs/benchmarks/2026-09-26/quarter.geojson` and the whole tile (no
+  `--domain`). The command runs through `scripts/prof_driver.py`, copied
+  verbatim from `serial-profile/scripts/`:
+
+  ```bash
+  RASPUTIN_TIES_OUT=<out>/ties_<domain>_t<N>.jsonl .venv/bin/python scripts/prof_driver.py \
+      --pkg <wt>/build-instr/pkg --threads <N> --repeat 1 -- \
+      mesh --dem tests/fixtures/dem_archive/7908_3_10m_z33.tif --tolerance 1 \
+      [--domain docs/benchmarks/2026-09-26/quarter.geojson] --out <out>/<domain>_t<N>.vtk --binary
+  python scripts/tables.py data/ties_quarter_t1.jsonl data/ties_tile_t1.jsonl > data/tables.md
+  ```
+
+  Each domain was run at 1 and 8 threads. The count files are byte-identical
+  between the two thread counts (`cmp`), as they should be, since the split
+  phase is serial.
+- **Machine**: Apple M1 Max, macOS 27.0. **Power: AC**, 100 %
+  (`data/pmset-start.txt`). The counts do not depend on power; it is recorded
+  because the persona requires it.
+- **Timings** printed by the driver are inflated by the instrumentation, which
+  does a `std::map` update per call. None is reported.
+
+## Results
+
+`data/tables.md` is produced by `scripts/tables.py` from `data/ties_*_t1.jsonl`.
+The rows for the refine loop:
+
+| measure | quarter | tile |
+|---|---:|---:|
+| incircle calls | 1,615,895 | 1,639,699 |
+| exact path | 146,962 (9.095 %) | 154,502 (9.423 %) |
+| exact path, answer Cocircular | 146,962 | 154,502 |
+| exact path, four node corners | 146,962 | 154,502 |
+| exact path, lattice det = 0 | 146,962 | 154,502 |
+| exact path, lattice det != 0 | 0 | 0 |
+| exact path, all QW2 conditions hold | 146,962 | 154,502 |
+| filtered path, four node corners | 1,468,493 | 1,485,197 |
+| filtered path, all QW2 conditions hold | 1,468,493 | 1,485,197 |
+| **all calls QW2 would answer** | **1,615,455 (99.973 %)** | **1,639,699 (100 %)** |
+| calls with fewer than four node corners | 440 | 0 |
+| lattice sign differs from the kernel's answer (four nodes) | 0 | 0 |
+| max coordinate difference from `d`, four-node calls | 951 nodes | 88 nodes |
+| max coordinate difference from `d`, exact-path calls | 146 nodes | 48 nodes |
+| frame-exact sufficient condition | 3 + 13 = 16 ≤ 53 | 3 + 13 = 16 ≤ 53 |
+
+The per-call conditions never refused a four-node quad. Every four-node call
+had `dx == dy`, a spread at most 2^14 and an exact frame, on both domains and
+in every phase.
+
+The shapes of the exact-path quads in the refine loop:
+
+| shape | quarter | tile |
+|---|---:|---:|
+| other cyclic quad (no parallel sides) | 47,547 (32.4 %) | 47,269 (30.6 %) |
+| axis-aligned square | 39,977 (27.2 %) | 39,911 (25.8 %) |
+| rotated square | 17,100 (11.6 %) | 17,203 (11.1 %) |
+| axis-aligned rectangle, not square | 16,261 (11.1 %) | 16,567 (10.7 %) |
+| isosceles trapezoid, parallel sides on rows or columns | 12,697 (8.6 %) | 20,146 (13.0 %) |
+| isosceles trapezoid, other direction | 11,169 (7.6 %) | 11,080 (7.2 %) |
+| rotated rectangle, not square | 2,211 (1.5 %) | 2,326 (1.5 %) |
+| three corners collinear | 0 | 0 |
+
+The most common axis-aligned sizes are 1×1 (37,887 on the quarter circle), 1×2
+(13,119), 2×2 (1,845), 1×3 and 2×3. `data/tables.md` has the per-size lists and
+the other phases, which also go through `must_flip`:
+
+- On the quarter circle, `legalise_all` makes 533 calls, of which QW2 would
+  answer 122. The quality pass makes 7,677, of which QW2 would answer 4,622.
+  The remaining 411 and 3,055 calls have an off-node corner (the arc
+  vertices).
+- On the tile, QW2 would answer all of `legalise_all`'s 48,133 calls, 16,129
+  of them exact ties: 15,876 are 40×40 squares, 252 are 10×40 rectangles and
+  one is a 10×10 square,
+  from the starting mesh. It would also answer all 4,017 of the quality pass's
+  calls.
+
+## What is measured and what is inferred
+
+Measured:
+
+- QW2's premise holds. All 146,962 (quarter circle) and 154,502 (tile)
+  exact-path ties in the refine loop have four node corners and a zero int64
+  lattice determinant on `(col, -row)`, and each of QW2's per-call conditions
+  holds for each of them.
+- On every four-node call, in every phase and on both domains (3.3 million
+  calls), the lattice sign agreed with `FilteredKernel<DetriaExact>`'s answer
+  on the frame doubles. This is QW2's bit-identity argument checked on this
+  input. It does not prove the argument.
+- The share of all refine-loop incircle calls that QW2's conditions would
+  admit: 99.973 % and 100 %.
+- The ties are lattice-cocircular quads of many shapes. Only 38 % (quarter
+  circle) and 37 % (tile) are axis-aligned rectangles.
+
+Not measured, and so inferred or open:
+
+- **The time QW2 saves.** Answering 99.97 % of calls says nothing about the
+  cost of an int64 determinant compared with the filtered double determinant
+  plus its permanent. Whether QW2 reaches past the 5.0 % exact-path cost into
+  the rest of the 13.2 % flip-test cost has to be measured on an
+  implementation.
+- **Why the ties occur.** One conjecture is that small cocircular quads are
+  common because refinement inserts nodes near one another on a grid. That
+  has not been checked.
+- **Other DEMs.** This covers one DEM, with `dx = dy = 10`. A DEM with
+  `dx != dy`, or with a `dx` like 0.1 whose frame is not exact, would get
+  nothing from QW2 by design. How common such DEMs are was not surveyed.
+
+## What bears on QW2's design
+
+1. **The general determinant is needed.** An axis-aligned-rectangle shortcut
+   would handle only 38 % of the ties, so QW2's general int64 determinant is
+   the right shape. The "points on a circle of radius 5" case in the planned
+   suite matches what the data shows.
+2. **The spread bound never binds here.** The largest spread was 951 nodes
+   against a limit of 2^14. By arithmetic, not measurement: for a spread of at
+   most 2^12, every intermediate of the plain double determinant on the
+   integer `(col, -row)` differences is an integer below 2^53, so it would be
+   exact in doubles too. The choice between int64 and doubles can be decided
+   by timing, not by correctness.
+3. **The conditions are per refine, not per call, on this data.** The per-call
+   checks for `dx == dy` and an exact frame never refused anything. The
+   once-per-refine decision QW2 already plans is enough, leaving the node check
+   and the spread check as the only per-call tests.
+4. **`must_flip` is also used outside the refine loop.** `legalise_all`
+   (16,129 ties on the tile) and the quality pass also go through it, so
+   QW2's call at the top of `must_flip` covers them without extra work.
+
+## Not in the repository
+
+The meshes (15-17 MB binary, 22-23 MB ASCII) and the scratch worktree with its build
+stayed in the session scratchpad and are gone with it. To regenerate them,
+apply `scripts/instrument.patch` to `d3ee2ce` and follow the Method above.
diff --git a/docs/benchmarks/2026-09-27/21b-ties/data/pmset-start.txt b/docs/benchmarks/2026-09-27/21b-ties/data/pmset-start.txt
new file mode 100644
index 00000000..62504dbf
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/data/pmset-start.txt
@@ -0,0 +1,2 @@
+Now drawing from 'AC Power'
+ -InternalBattery-0 (id=7929955)	100%; finishing charge; 0:13 remaining present: true
diff --git a/docs/benchmarks/2026-09-27/21b-ties/data/tables.md b/docs/benchmarks/2026-09-27/21b-ties/data/tables.md
new file mode 100644
index 00000000..8415df32
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/data/tables.md
@@ -0,0 +1,60 @@
+| measure | quarter | tile |
+|---|---:|---:|
+| refine loop: incircle calls | 1,615,895 | 1,639,699 |
+| refine loop: exact path | 146,962 (9.095 %) | 154,502 (9.423 %) |
+| refine loop: exact path, Cocircular | 146,962 | 154,502 |
+| refine loop: exact path, four node corners | 146,962 | 154,502 |
+| refine loop: exact path, lattice det = 0 | 146,962 | 154,502 |
+| refine loop: exact path, lattice det != 0 | 0 | 0 |
+| refine loop: exact path, QW2 conditions hold | 146,962 | 154,502 |
+| refine loop: filtered path, four node corners | 1,468,493 | 1,485,197 |
+| refine loop: filtered path, QW2 conditions hold | 1,468,493 | 1,485,197 |
+| refine loop: all calls QW2 would answer | 1,615,455 (99.973 %) | 1,639,699 (100.000 %) |
+| refine loop: calls with fewer than four node corners | 440 | 0 |
+| refine loop: lattice sign differs from kernel (four nodes) | 0 | 0 |
+| legalise_all: incircle calls | 533 | 48,133 |
+| legalise_all: exact path | 61 (11.445 %) | 16,129 (33.509 %) |
+| legalise_all: exact path, Cocircular | 61 | 16,129 |
+| legalise_all: exact path, four node corners | 61 | 16,129 |
+| legalise_all: exact path, lattice det = 0 | 61 | 16,129 |
+| legalise_all: exact path, lattice det != 0 | 0 | 0 |
+| legalise_all: exact path, QW2 conditions hold | 61 | 16,129 |
+| legalise_all: filtered path, four node corners | 61 | 32,004 |
+| legalise_all: filtered path, QW2 conditions hold | 61 | 32,004 |
+| legalise_all: all calls QW2 would answer | 122 (22.889 %) | 48,133 (100.000 %) |
+| legalise_all: calls with fewer than four node corners | 411 | 0 |
+| legalise_all: lattice sign differs from kernel (four nodes) | 0 | 0 |
+| quality pass: incircle calls | 7,677 | 4,017 |
+| quality pass: exact path | 22 (0.287 %) | 0 (0.000 %) |
+| quality pass: exact path, Cocircular | 22 | 0 |
+| quality pass: exact path, four node corners | 22 | 0 |
+| quality pass: exact path, lattice det = 0 | 22 | 0 |
+| quality pass: exact path, lattice det != 0 | 0 | 0 |
+| quality pass: exact path, QW2 conditions hold | 22 | 0 |
+| quality pass: filtered path, four node corners | 4,600 | 4,017 |
+| quality pass: filtered path, QW2 conditions hold | 4,600 | 4,017 |
+| quality pass: all calls QW2 would answer | 4,622 (60.206 %) | 4,017 (100.000 %) |
+| quality pass: calls with fewer than four node corners | 3,055 | 0 |
+| quality pass: lattice sign differs from kernel (four nodes) | 0 | 0 |
+| max spread from d, four-node calls, refine loop (nodes) | 951 | 88 |
+| max spread from d, exact-path calls, refine loop (nodes) | 146 | 48 |
+| dx, dy; rows x cols | 10, 10; 5051 x 5051 | 10, 10; 5051 x 5051 |
+| frame-exact sufficient condition (bits of dx + bit_width) <= 53 | 3 + 13 = 16: True | 3 + 13 = 16: True |
+| rounds, inserted, flips | 41, 213,464, 445,657 | 53, 219,837, 445,675 |
+
+Shapes of the exact-path (tie) quads, refine loop:
+
+| shape | quarter | tile |
+|---|---:|---:|
+| other cyclic quad (no parallel sides) | 47,547 (32.353 %) | 47,269 (30.594 %) |
+| axis-aligned square | 39,977 (27.202 %) | 39,911 (25.832 %) |
+| rotated square | 17,100 (11.636 %) | 17,203 (11.134 %) |
+| axis-aligned rectangle, not square | 16,261 (11.065 %) | 16,567 (10.723 %) |
+| isosceles trapezoid, parallel sides on rows or columns | 12,697 (8.640 %) | 20,146 (13.039 %) |
+| isosceles trapezoid, other direction | 11,169 (7.600 %) | 11,080 (7.171 %) |
+| rotated rectangle, not square | 2,211 (1.504 %) | 2,326 (1.505 %) |
+
+Most frequent axis-aligned sizes (short x long side, nodes), refine loop exact path:
+
+- quarter: square_1x1 37,887, rectangle_1x2 13,119, square_2x2 1,845, rectangle_1x3 1,182, rectangle_2x3 1,107, rectangle_1x4 240
+- tile: square_1x1 37,866, rectangle_1x2 13,103, square_2x2 1,794, rectangle_2x3 1,133, rectangle_1x3 1,131, rectangle_1x4 207
diff --git a/docs/benchmarks/2026-09-27/21b-ties/data/ties_quarter_t1.jsonl b/docs/benchmarks/2026-09-27/21b-ties/data/ties_quarter_t1.jsonl
new file mode 100644
index 00000000..939baf10
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/data/ties_quarter_t1.jsonl
@@ -0,0 +1 @@
+{"dx": 10, "dy": 10, "rows": 5051, "cols": 5051, "dx_significant_bits": 3, "index_bit_width": 13, "global_frame_exact_condition": true, "rounds": 41, "inserted": 213464, "flips": 445657, "max_spread_all4_refine": 951, "max_spread_exact4_refine": 146, "counts": {"p0.calls": 533, "p0.exact.calls": 61, "p0.exact.n4.dx_eq_dy": 61, "p0.exact.n4.frame_exact": 61, "p0.exact.n4.lattice_det_zero": 61, "p0.exact.n4.lattice_sign_agrees": 61, "p0.exact.n4.qw2_qualifies": 61, "p0.exact.n4.shape.isosceles_trapezoid_other": 61, "p0.exact.n4.spread_le_2^14": 61, "p0.exact.nodes4": 61, "p0.exact.nodes4.result.cocircular": 61, "p0.exact.result.cocircular": 61, "p0.filtered.calls": 472, "p0.filtered.n4.dx_eq_dy": 61, "p0.filtered.n4.frame_exact": 61, "p0.filtered.n4.lattice_det_neg": 61, "p0.filtered.n4.lattice_sign_agrees": 61, "p0.filtered.n4.qw2_qualifies": 61, "p0.filtered.n4.spread_le_2^14": 61, "p0.filtered.nodes1": 58, "p0.filtered.nodes1.result.outside": 58, "p0.filtered.nodes2": 352, 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diff --git a/docs/benchmarks/2026-09-27/21b-ties/data/ties_quarter_t8.jsonl b/docs/benchmarks/2026-09-27/21b-ties/data/ties_quarter_t8.jsonl
new file mode 100644
index 00000000..939baf10
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/data/ties_quarter_t8.jsonl
@@ -0,0 +1 @@
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diff --git a/docs/benchmarks/2026-09-27/21b-ties/data/ties_tile_t1.jsonl b/docs/benchmarks/2026-09-27/21b-ties/data/ties_tile_t1.jsonl
new file mode 100644
index 00000000..5e227269
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/data/ties_tile_t1.jsonl
@@ -0,0 +1 @@
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"p2.filtered.n4.lattice_sign_agrees": 1485197, "p2.filtered.n4.qw2_qualifies": 1485197, "p2.filtered.n4.spread_le_2^14": 1485197, "p2.filtered.nodes4": 1485197, "p2.filtered.nodes4.result.inside": 445675, "p2.filtered.nodes4.result.outside": 1039522, "p2.filtered.result.inside": 445675, "p2.filtered.result.outside": 1039522}}
diff --git a/docs/benchmarks/2026-09-27/21b-ties/data/ties_tile_t8.jsonl b/docs/benchmarks/2026-09-27/21b-ties/data/ties_tile_t8.jsonl
new file mode 100644
index 00000000..5e227269
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/data/ties_tile_t8.jsonl
@@ -0,0 +1 @@
+{"dx": 10, "dy": 10, "rows": 5051, "cols": 5051, "dx_significant_bits": 3, "index_bit_width": 13, "global_frame_exact_condition": true, "rounds": 53, "inserted": 219837, "flips": 445675, "max_spread_all4_refine": 88, "max_spread_exact4_refine": 48, "counts": {"p0.calls": 48133, "p0.exact.calls": 16129, "p0.exact.n4.dx_eq_dy": 16129, "p0.exact.n4.frame_exact": 16129, "p0.exact.n4.lattice_det_zero": 16129, "p0.exact.n4.lattice_sign_agrees": 16129, "p0.exact.n4.qw2_qualifies": 16129, "p0.exact.n4.shape.axis_rectangle_10x40": 252, "p0.exact.n4.shape.axis_square_10x10": 1, "p0.exact.n4.shape.axis_square_40x40": 15876, "p0.exact.n4.spread_le_2^14": 16129, "p0.exact.nodes4": 16129, "p0.exact.nodes4.result.cocircular": 16129, "p0.exact.result.cocircular": 16129, "p0.filtered.calls": 32004, "p0.filtered.n4.dx_eq_dy": 32004, "p0.filtered.n4.frame_exact": 32004, "p0.filtered.n4.lattice_det_neg": 32004, "p0.filtered.n4.lattice_sign_agrees": 32004, "p0.filtered.n4.qw2_qualifies": 32004, "p0.filtered.n4.spread_le_2^14": 32004, "p0.filtered.nodes4": 32004, "p0.filtered.nodes4.result.outside": 32004, "p0.filtered.result.outside": 32004, "p1.calls": 4017, "p1.filtered.calls": 4017, "p1.filtered.n4.dx_eq_dy": 4017, "p1.filtered.n4.frame_exact": 4017, "p1.filtered.n4.lattice_det_neg": 2760, "p1.filtered.n4.lattice_det_pos": 1257, "p1.filtered.n4.lattice_sign_agrees": 4017, "p1.filtered.n4.qw2_qualifies": 4017, "p1.filtered.n4.spread_le_2^14": 4017, "p1.filtered.nodes4": 4017, "p1.filtered.nodes4.result.inside": 1257, "p1.filtered.nodes4.result.outside": 2760, "p1.filtered.result.inside": 1257, "p1.filtered.result.outside": 2760, "p2.calls": 1639699, "p2.exact.calls": 154502, "p2.exact.n4.dx_eq_dy": 154502, "p2.exact.n4.frame_exact": 154502, "p2.exact.n4.lattice_det_zero": 154502, "p2.exact.n4.lattice_sign_agrees": 154502, "p2.exact.n4.qw2_qualifies": 154502, "p2.exact.n4.shape.axis_rectangle_1x11": 2, "p2.exact.n4.shape.axis_rectangle_1x12": 1, "p2.exact.n4.shape.axis_rectangle_1x13": 1, "p2.exact.n4.shape.axis_rectangle_1x2": 13103, "p2.exact.n4.shape.axis_rectangle_1x3": 1131, "p2.exact.n4.shape.axis_rectangle_1x33": 1, "p2.exact.n4.shape.axis_rectangle_1x4": 207, "p2.exact.n4.shape.axis_rectangle_1x5": 55, "p2.exact.n4.shape.axis_rectangle_1x6": 28, "p2.exact.n4.shape.axis_rectangle_1x7": 5, "p2.exact.n4.shape.axis_rectangle_1x8": 3, "p2.exact.n4.shape.axis_rectangle_1x9": 1, "p2.exact.n4.shape.axis_rectangle_2x11": 1, "p2.exact.n4.shape.axis_rectangle_2x14": 1, "p2.exact.n4.shape.axis_rectangle_2x3": 1133, "p2.exact.n4.shape.axis_rectangle_2x4": 173, "p2.exact.n4.shape.axis_rectangle_2x5": 70, "p2.exact.n4.shape.axis_rectangle_2x6": 18, "p2.exact.n4.shape.axis_rectangle_2x7": 8, "p2.exact.n4.shape.axis_rectangle_2x8": 7, "p2.exact.n4.shape.axis_rectangle_2x9": 1, "p2.exact.n4.shape.axis_rectangle_34x35": 1, "p2.exact.n4.shape.axis_rectangle_34x40": 70, "p2.exact.n4.shape.axis_rectangle_35x40": 124, "p2.exact.n4.shape.axis_rectangle_3x10": 1, "p2.exact.n4.shape.axis_rectangle_3x11": 1, "p2.exact.n4.shape.axis_rectangle_3x4": 115, "p2.exact.n4.shape.axis_rectangle_3x5": 37, "p2.exact.n4.shape.axis_rectangle_3x6": 16, "p2.exact.n4.shape.axis_rectangle_3x7": 3, "p2.exact.n4.shape.axis_rectangle_3x8": 4, "p2.exact.n4.shape.axis_rectangle_3x9": 1, "p2.exact.n4.shape.axis_rectangle_4x11": 1, "p2.exact.n4.shape.axis_rectangle_4x12": 1, "p2.exact.n4.shape.axis_rectangle_4x20": 1, "p2.exact.n4.shape.axis_rectangle_4x5": 15, "p2.exact.n4.shape.axis_rectangle_4x6": 11, "p2.exact.n4.shape.axis_rectangle_4x7": 3, "p2.exact.n4.shape.axis_rectangle_4x8": 2, "p2.exact.n4.shape.axis_rectangle_4x9": 3, "p2.exact.n4.shape.axis_rectangle_5x10": 1, "p2.exact.n4.shape.axis_rectangle_5x40": 125, "p2.exact.n4.shape.axis_rectangle_5x6": 3, "p2.exact.n4.shape.axis_rectangle_5x7": 1, "p2.exact.n4.shape.axis_rectangle_6x40": 70, "p2.exact.n4.shape.axis_rectangle_6x7": 2, "p2.exact.n4.shape.axis_rectangle_6x8": 2, "p2.exact.n4.shape.axis_rectangle_8x9": 1, "p2.exact.n4.shape.axis_rectangle_9x11": 1, "p2.exact.n4.shape.axis_rectangle_9x13": 1, "p2.exact.n4.shape.axis_square_1x1": 37866, "p2.exact.n4.shape.axis_square_2x2": 1794, "p2.exact.n4.shape.axis_square_3x3": 201, "p2.exact.n4.shape.axis_square_4x4": 37, "p2.exact.n4.shape.axis_square_5x5": 12, "p2.exact.n4.shape.axis_square_7x7": 1, "p2.exact.n4.shape.isosceles_trapezoid_axis": 20146, "p2.exact.n4.shape.isosceles_trapezoid_other": 11080, "p2.exact.n4.shape.other_cyclic": 47269, "p2.exact.n4.shape.rotated_rectangle": 2326, "p2.exact.n4.shape.rotated_square": 17203, "p2.exact.n4.spread_le_2^14": 154502, "p2.exact.nodes4": 154502, "p2.exact.nodes4.result.cocircular": 154502, "p2.exact.result.cocircular": 154502, "p2.filtered.calls": 1485197, "p2.filtered.n4.dx_eq_dy": 1485197, "p2.filtered.n4.frame_exact": 1485197, "p2.filtered.n4.lattice_det_neg": 1039522, "p2.filtered.n4.lattice_det_pos": 445675, "p2.filtered.n4.lattice_sign_agrees": 1485197, "p2.filtered.n4.qw2_qualifies": 1485197, "p2.filtered.n4.spread_le_2^14": 1485197, "p2.filtered.nodes4": 1485197, "p2.filtered.nodes4.result.inside": 445675, "p2.filtered.nodes4.result.outside": 1039522, "p2.filtered.result.inside": 445675, "p2.filtered.result.outside": 1039522}}
diff --git a/docs/benchmarks/2026-09-27/21b-ties/scripts/classifier_selftest.cpp b/docs/benchmarks/2026-09-27/21b-ties/scripts/classifier_selftest.cpp
new file mode 100644
index 00000000..f471dea8
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/scripts/classifier_selftest.cpp
@@ -0,0 +1,21 @@
+#include <terrain/mesh/prof_ties.hpp>
+#include <cstdio>
+using namespace terrain::mesh;
+using namespace terrain::mesh::prof_ties;
+static MeshVertex V(double c, double r) { return MeshVertex{c, r}; }
+int main() {
+    // shape on (col, row); the classifier reads them as a set
+    std::printf("%s\n", shape({V(0,0), V(1,0), V(1,1), V(0,1)}).c_str());        // axis_square_1x1
+    std::printf("%s\n", shape({V(0,0), V(2,0), V(2,1), V(0,1)}).c_str());        // axis_rectangle_1x2
+    std::printf("%s\n", shape({V(0,0), V(1,1), V(0,2), V(-1,1)}).c_str());       // rotated_square
+    std::printf("%s\n", shape({V(0,0), V(2,2), V(1,3), V(-1,1)}).c_str());       // rotated_rectangle
+    std::printf("%s\n", shape({V(0,0), V(3,0), V(2,1), V(1,1)}).c_str());        // trapezoid axis? (not cyclic in general, shape only)
+    std::printf("%s\n", shape({V(5,0), V(3,4), V(-4,3), V(0,-5)}).c_str());      // other_cyclic (r = 5)
+    std::printf("%s\n", shape({V(0,0), V(1,0), V(2,0), V(0,1)}).c_str());        // collinear_triple
+    // det: unit square cocircular -> 0; d at centre-ish inside -> positive
+    std::printf("det sq %d\n", (int)lattice_det({V(0,0), V(1,0), V(1,1), V(0,1)}));
+    // a,b,c CCW in (col,-row): (0,0),(2,0) at row 0; (1,-1) is row 1 -> (1,1) in world up? use row -> -y
+    std::printf("det in %d\n", (int)(lattice_det({V(0,0), V(2,0), V(1,-2), V(1,-1)}) > 0));  // world (0,0),(2,0),(1,2), query (1,1): inside
+    std::printf("fma 0.1*3 exact %d, 10*4999 exact %d\n", product_exact(3, 0.1), product_exact(4999, 10));
+    std::printf("bits 10=%d 0.1=%d 1=%d\n", significant_bits(10), significant_bits(0.1), significant_bits(1));
+}
diff --git a/docs/benchmarks/2026-09-27/21b-ties/scripts/counter_selftest.cpp b/docs/benchmarks/2026-09-27/21b-ties/scripts/counter_selftest.cpp
new file mode 100644
index 00000000..03a7dd89
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/scripts/counter_selftest.cpp
@@ -0,0 +1,16 @@
+#include <terrain/mesh/prof_ties.hpp>
+#include <cstdio>
+using namespace terrain::mesh;
+using namespace terrain::mesh::prof_ties;
+static MeshVertex V(double c, double r) { return MeshVertex{c, r}; }
+int main() {
+    std::printf("%s\n", shape({V(5,0), V(3,4), V(-5,0), V(0,-5)}).c_str());  // other_cyclic
+    phase = 2;
+    record(V(0,0), V(1,0), V(1,1), V(0,1), 0.1, 0.1, 0, true);         // frame inexact? 0.1*1 exact; row 1 -> exact
+    record(V(0,3), V(7,0), V(7,3), V(0,0), 0.1, 0.1, 0, true);         // 7*0.1, 3*0.1 inexact
+    record(V(0,0), V(1,0), V(1,1), V(0,1), 10, 20, 0, true);           // dx != dy
+    record(V(0,0), V(20000,0), V(1,1), V(0,1), 10, 10, 0, true);       // spread > 2^14, det != 0
+    record(V(0,0), V(1,0), V(1,1), V(0,1), 10, 10, 1, false);          // sign differs
+    record(V(0.5,0), V(1,0), V(1,1), V(0,1), 10, 10, 0, true);         // 3 nodes
+    for (auto& [k, n] : counts) std::printf("%s %llu\n", k.c_str(), n);
+}
diff --git a/docs/benchmarks/2026-09-27/21b-ties/scripts/instrument.patch b/docs/benchmarks/2026-09-27/21b-ties/scripts/instrument.patch
new file mode 100644
index 00000000..6b3652fc
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/scripts/instrument.patch
@@ -0,0 +1,315 @@
+diff --git a/include/terrain/mesh/lawson.hpp b/include/terrain/mesh/lawson.hpp
+index 2796055..c4dae58 100644
+--- a/include/terrain/mesh/lawson.hpp
++++ b/include/terrain/mesh/lawson.hpp
+@@ -24,6 +24,8 @@
+ #include <terrain/mesh/lattice_mesh.hpp>
+ #include <terrain/predicates/kernel.hpp>
+ 
++#include <terrain/mesh/prof_ties.hpp>
++
+ #include <cstddef>
+ #include <cstdint>
+ #include <span>
+@@ -67,10 +69,18 @@ template <pred::GeometryKernel K>
+     // the frame. Both branches flip exactly when the same exact determinant,
+     // on the frame points of the quad in mesh CCW order, is positive; a flip
+     // lowers the sum of signed lifted volumes, so the process terminates.
+-    if (K::orient2d(a, b, c) == pred::Orientation::CounterClockwise)
+-        return K::incircle(a, b, c, d) == pred::Incircle::Inside;
+-    return K::orient2d(b, a, d) == pred::Orientation::CounterClockwise
+-        && K::incircle(b, a, d, c) == pred::Incircle::Inside;
++    const MeshVertex va = v[tri[e]], vb = v[tri[(e + 1) % 3]], vc = v[tri[(e + 2) % 3]],
++                     vd = v[m.triangles()[u][(j + 2) % 3]];
++    if (K::orient2d(a, b, c) == pred::Orientation::CounterClockwise) {
++        const auto r = K::incircle(a, b, c, d);
++        prof_ties::record(va, vb, vc, vd, f.dx, f.dy, static_cast<int>(r), pred::prof_ties::last_exact);
++        return r == pred::Incircle::Inside;
++    }
++    if (K::orient2d(b, a, d) != pred::Orientation::CounterClockwise)
++        return false;
++    const auto r = K::incircle(b, a, d, c);
++    prof_ties::record(vb, va, vd, vc, f.dx, f.dy, static_cast<int>(r), pred::prof_ties::last_exact);
++    return r == pred::Incircle::Inside;
+ }
+ 
+ }  // namespace detail
+diff --git a/include/terrain/mesh/prof_ties.hpp b/include/terrain/mesh/prof_ties.hpp
+new file mode 100644
+index 0000000..5d516f4
+--- /dev/null
++++ b/include/terrain/mesh/prof_ties.hpp
+@@ -0,0 +1,212 @@
++#pragma once
++
++// 21b tie classification (docs/increments/21-parallel-refine.md, section 3 QW2).
++// Local measurement instrumentation, never committed: kept as
++// docs/benchmarks/2026-09-27/21b-ties/scripts/instrument.patch.
++//
++// At every incircle call made by must_flip, record, on the four MeshVertex
++// corners as passed to the kernel (a, b, c counter-clockwise in the frame,
++// d the query point):
++//   - how many are DEM nodes (MeshVertex::is_node);
++//   - for four nodes: the exact lattice determinant on (col, -row) translated
++//     to d (in __int128, so it is exact whatever the spread), QW2's
++//     conditions (dx == dy; every |difference from d| <= 2^14; the frame
++//     products col * dx and row * dy exact, checked per call with fma), the
++//     agreement of the lattice sign with the kernel's answer, and the shape;
++//   - whether the kernel took the exact path, and its answer.
++// Counters are keyed by phase (0 legalise_all, 1 quality, 2 refine loop). The
++// callers are serial (the split phase), so plain globals suffice.
++
++#include <terrain/mesh/lattice_mesh.hpp>
++
++#include <algorithm>
++#include <array>
++#include <bit>
++#include <cmath>
++#include <cstdint>
++#include <cstdio>
++#include <cstdlib>
++#include <map>
++#include <set>
++#include <string>
++#include <utility>
++
++namespace terrain::mesh::prof_ties {
++
++inline int phase = -1;
++inline std::map<std::string, unsigned long long> counts;
++inline double g_dx = 0, g_dy = 0;
++inline std::size_t g_rows = 0, g_cols = 0;
++inline long long max_spread_all4 = 0, max_spread_exact4 = 0;
++
++inline void reset(double dx, double dy, std::size_t rows, std::size_t cols) {
++    counts.clear();
++    g_dx = dx;
++    g_dy = dy;
++    g_rows = rows;
++    g_cols = cols;
++    max_spread_all4 = max_spread_exact4 = 0;
++}
++
++// Significant bits of a positive finite double: 53 less the trailing zeros of
++// its 53-bit integer significand.
++inline int significant_bits(double x) {
++    int e = 0;
++    const double m = std::frexp(x, &e);
++    const auto s = static_cast<std::uint64_t>(std::ldexp(m, 53));
++    return 53 - std::countr_zero(s);
++}
++
++inline bool product_exact(double a, double b) {
++    const double p = a * b;
++    return std::fma(a, b, -p) == 0.0;
++}
++
++__extension__ typedef __int128 I;
++
++inline I lattice_det(const std::array<MeshVertex, 4>& q) {
++    const auto X = [](const MeshVertex& v) { return static_cast<I>(static_cast<long long>(v.col)); };
++    const auto Y = [](const MeshVertex& v) { return -static_cast<I>(static_cast<long long>(v.row)); };
++    const auto& d = q[3];
++    const I adx = X(q[0]) - X(d), ady = Y(q[0]) - Y(d);
++    const I bdx = X(q[1]) - X(d), bdy = Y(q[1]) - Y(d);
++    const I cdx = X(q[2]) - X(d), cdy = Y(q[2]) - Y(d);
++    const I al = adx * adx + ady * ady, bl = bdx * bdx + bdy * bdy, cl = cdx * cdx + cdy * cdy;
++    return al * (bdx * cdy - cdx * bdy) + bl * (cdx * ady - adx * cdy) + cl * (adx * bdy - bdx * ady);
++}
++
++inline long long orient_ll(const MeshVertex& a, const MeshVertex& b, const MeshVertex& c) {
++    const auto ax = static_cast<long long>(a.col), ay = -static_cast<long long>(a.row);
++    const auto bx = static_cast<long long>(b.col), by = -static_cast<long long>(b.row);
++    const auto cx = static_cast<long long>(c.col), cy = -static_cast<long long>(c.row);
++    return (bx - ax) * (cy - ay) - (by - ay) * (cx - ax);
++}
++
++// Shape of a four-node quad with a zero lattice determinant.
++inline std::string shape(const std::array<MeshVertex, 4>& q) {
++    for (int i = 0; i < 4; ++i)
++        for (int j = i + 1; j < 4; ++j)
++            for (int k = j + 1; k < 4; ++k)
++                if (orient_ll(q[i], q[j], q[k]) == 0)
++                    return "collinear_triple";
++    std::set<long long> cols, rows;
++    for (const auto& v : q) {
++        cols.insert(static_cast<long long>(v.col));
++        rows.insert(static_cast<long long>(v.row));
++    }
++    const auto c = [&](int i) { return std::pair{static_cast<long long>(q[i].col), static_cast<long long>(q[i].row)}; };
++    constexpr std::array<std::array<int, 4>, 3> pairings{{{0, 1, 2, 3}, {0, 2, 1, 3}, {0, 3, 1, 2}}};
++    for (const auto& p : pairings) {
++        const auto [x0, y0] = c(p[0]);
++        const auto [x1, y1] = c(p[1]);
++        const auto [x2, y2] = c(p[2]);
++        const auto [x3, y3] = c(p[3]);
++        const bool same_mid = x0 + x1 == x2 + x3 && y0 + y1 == y2 + y3;
++        const bool same_len = (x1 - x0) * (x1 - x0) + (y1 - y0) * (y1 - y0)
++                           == (x3 - x2) * (x3 - x2) + (y3 - y2) * (y3 - y2);
++        if (!same_mid || !same_len)
++            continue;
++        // Rectangle with diagonals (p0,p1), (p2,p3); sides p0-p2 and p0-p3.
++        const auto s1 = (x2 - x0) * (x2 - x0) + (y2 - y0) * (y2 - y0);
++        const auto s2 = (x3 - x0) * (x3 - x0) + (y3 - y0) * (y3 - y0);
++        const bool square = s1 == s2;
++        if (cols.size() == 2 && rows.size() == 2) {
++            const long long w = *cols.rbegin() - *cols.begin(), h = *rows.rbegin() - *rows.begin();
++            return std::string(square ? "axis_square" : "axis_rectangle") + "_" + std::to_string(std::min(w, h))
++                 + "x" + std::to_string(std::max(w, h));
++        }
++        return square ? "rotated_square" : "rotated_rectangle";
++    }
++    if (cols.size() == 2 || rows.size() == 2)
++        return "isosceles_trapezoid_axis";  // two parallel sides on rows or columns
++    // A cyclic quad with a pair of parallel (disjoint) chords is an isosceles
++    // trapezoid; the diagonals cross, so testing all three pairings is safe.
++    for (const auto& p : pairings) {
++        const auto [x0, y0] = c(p[0]);
++        const auto [x1, y1] = c(p[1]);
++        const auto [x2, y2] = c(p[2]);
++        const auto [x3, y3] = c(p[3]);
++        if ((x1 - x0) * (y3 - y2) - (y1 - y0) * (x3 - x2) == 0)
++            return "isosceles_trapezoid_other";
++    }
++    return "other_cyclic";
++}
++
++inline void add(const std::string& key, unsigned long long n = 1) { counts[key] += n; }
++
++inline void record(MeshVertex a, MeshVertex b, MeshVertex c, MeshVertex d, double dx, double dy, int result,
++                   bool exact) {
++    const std::string P = "p" + std::to_string(phase) + ".";
++    const std::string R = result > 0 ? "inside" : result < 0 ? "outside" : "cocircular";
++    const std::array<MeshVertex, 4> q{a, b, c, d};
++    int nodes = 0;
++    for (const auto& v : q)
++        nodes += v.is_node() ? 1 : 0;
++    const std::string path = exact ? "exact" : "filtered";
++    add(P + "calls");
++    add(P + path + ".calls");
++    add(P + path + ".result." + R);
++    add(P + path + ".nodes" + std::to_string(nodes));
++    add(P + path + ".nodes" + std::to_string(nodes) + ".result." + R);
++    if (nodes != 4)
++        return;
++    long long spread = 0;
++    for (int i = 0; i < 3; ++i) {
++        spread = std::max(spread, std::llabs(static_cast<long long>(q[i].col) - static_cast<long long>(d.col)));
++        spread = std::max(spread, std::llabs(static_cast<long long>(q[i].row) - static_cast<long long>(d.row)));
++    }
++    if (phase == 2) {
++        max_spread_all4 = std::max(max_spread_all4, spread);
++        if (exact)
++            max_spread_exact4 = std::max(max_spread_exact4, spread);
++    }
++    bool frame_exact = true;
++    for (const auto& v : q)
++        frame_exact = frame_exact && product_exact(v.col, dx) && product_exact(v.row, dy);
++    const bool eq = dx == dy;
++    const bool small = spread <= (1LL << 14);
++    const bool qualifies = eq && small && frame_exact;
++    const I det = lattice_det(q);
++    const int s = det > 0 ? 1 : det < 0 ? -1 : 0;
++    const std::string N = P + path + ".n4.";
++    add(N + (eq ? "dx_eq_dy" : "dx_ne_dy"));
++    add(N + (small ? "spread_le_2^14" : "spread_gt_2^14"));
++    add(N + (frame_exact ? "frame_exact" : "frame_inexact"));
++    add(N + (qualifies ? "qw2_qualifies" : "qw2_refuses"));
++    add(N + "lattice_det_" + (s > 0 ? "pos" : s < 0 ? "neg" : "zero"));
++    add(N + (s == result ? "lattice_sign_agrees" : "lattice_sign_differs"));
++    if (qualifies && s != result)
++        add(N + "qw2_qualifies_and_differs");
++    if (s == 0)
++        add(N + "shape." + shape(q));
++    if (exact && s != 0)
++        add(N + "exact_path_lattice_nonzero");
++}
++
++inline void dump(std::size_t rounds, std::size_t inserted, std::size_t flips) {
++    const char* path = std::getenv("RASPUTIN_TIES_OUT");
++    if (path == nullptr)
++        return;
++    std::FILE* f = std::fopen(path, "a");
++    if (f == nullptr)
++        return;
++    const int bits = significant_bits(g_dx);
++    const int width = std::bit_width(std::max(g_rows, g_cols) - 1);
++    std::fprintf(f,
++                 "{\"dx\": %.17g, \"dy\": %.17g, \"rows\": %zu, \"cols\": %zu, \"dx_significant_bits\": %d, "
++                 "\"index_bit_width\": %d, \"global_frame_exact_condition\": %s, \"rounds\": %zu, "
++                 "\"inserted\": %zu, \"flips\": %zu, \"max_spread_all4_refine\": %lld, "
++                 "\"max_spread_exact4_refine\": %lld, \"counts\": {",
++                 g_dx, g_dy, g_rows, g_cols, bits, width,
++                 (g_dx == g_dy && bits + width <= 53) ? "true" : "false", rounds, inserted, flips,
++                 max_spread_all4, max_spread_exact4);
++    bool first = true;
++    for (const auto& [k, n] : counts) {
++        std::fprintf(f, "%s\"%s\": %llu", first ? "" : ", ", k.c_str(), n);
++        first = false;
++    }
++    std::fprintf(f, "}}\n");
++    std::fclose(f);
++}
++
++}  // namespace terrain::mesh::prof_ties
+diff --git a/include/terrain/predicates/kernel.hpp b/include/terrain/predicates/kernel.hpp
+index b794353..a4f7eff 100644
+--- a/include/terrain/predicates/kernel.hpp
++++ b/include/terrain/predicates/kernel.hpp
+@@ -33,6 +33,12 @@ concept GeometryKernel = requires(Point2 a, Point2 b, Point2 c, Point2 d) {
+     { K::incircle(a, b, c, d) } -> std::same_as<Incircle>;
+ };
+ 
++// 21b tie classification (local instrumentation, never committed): whether the
++// last FilteredKernel incircle call took the exact path.
++namespace prof_ties {
++inline bool last_exact = false;
++}  // namespace prof_ties
++
+ namespace detail {
+ 
+ // The naive 3x3 lifted determinant, translated so d is the origin. Positive iff
+@@ -172,8 +178,10 @@ private:
+         const double permanent = detail::incircle_permanent(a, b, c, d);
+ 
+         if (std::fabs(det) > incircle_bound_a * permanent) {
++            prof_ties::last_exact = false;
+             return incircle_of_sign(det);
+         }
++        prof_ties::last_exact = true;
+         return E::incircle_ccw(a, b, c, d);
+     }
+ };
+diff --git a/include/terrain/refinement/refine.hpp b/include/terrain/refinement/refine.hpp
+index c0c6a25..9003583 100644
+--- a/include/terrain/refinement/refine.hpp
++++ b/include/terrain/refinement/refine.hpp
+@@ -282,9 +282,12 @@ template <raster::RasterSource R>
+     const auto since = [](clock::time_point t0) {
+         return std::chrono::duration<double>(clock::now() - t0).count();
+     };
++    mesh::prof_ties::reset(g.delta_x(), g.delta_y(), g.rows(), g.cols());
++    mesh::prof_ties::phase = 0;
+     auto t0 = clock::now();
+     out.flips = mesh::legalise_all<pred::DefaultKernel>(m, frame, [](std::uint32_t) {});
+     out.legalise_seconds = since(t0);
++    mesh::prof_ties::phase = 1;
+     if (options.min_angle_deg > 0.0) {
+         t0 = clock::now();
+         const auto q = mesh::improve<pred::DefaultKernel>(
+@@ -297,6 +300,7 @@ template <raster::RasterSource R>
+     std::vector<ScanResult> results;
+     std::set<std::pair<std::uint32_t, std::uint32_t>> footed;  // (row, col), R2 step 5
+     mesh::FlipStack flip_stack;  // one buffer for every legalise_around
++    mesh::prof_ties::phase = 2;
+     std::vector<std::uint32_t> active(m.triangle_count());
+     for (std::uint32_t t = 0; t < active.size(); ++t)
+         active[t] = t;
+@@ -378,6 +382,7 @@ template <raster::RasterSource R>
+         detail::rebuild_active(touched, skipped, active);
+     }
+ 
++    mesh::prof_ties::dump(out.rounds, out.inserted, out.flips);
+     // By the stopping rule a void triangle holds no valid node, so `uncovered`
+     // sums zeros unless that rule changes; it is reported so a change shows.
+     for (const ScanResult& r : results) {
diff --git a/docs/benchmarks/2026-09-27/21b-ties/scripts/prof_driver.py b/docs/benchmarks/2026-09-27/21b-ties/scripts/prof_driver.py
new file mode 100644
index 00000000..ca8eacd5
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/scripts/prof_driver.py
@@ -0,0 +1,51 @@
+"""prof_driver.py --pkg DIR --threads N --repeat K [--pause S] -- <rasputin argv>
+
+bench.py's child technique (pkg first on sys.path, wrap cli.refine), but the
+wrapped refine is called K times with the same arguments, each timed, and the
+RefineOutcome phase seconds printed per call as one JSON line (PHASES ...).
+--pause S sleeps S seconds before the first call and prints the pid, so
+`sample` can attach to a steady window of refine calls.
+"""
+import json
+import os
+import sys
+import time
+
+argv = sys.argv[1:]
+split = argv.index("--")
+own, rasputin = argv[:split], argv[split + 1 :]
+pkg = own[own.index("--pkg") + 1]
+threads = int(own[own.index("--threads") + 1])
+repeat = int(own[own.index("--repeat") + 1])
+pause = float(own[own.index("--pause") + 1]) if "--pause" in own else 0.0
+sys.meta_path[:] = [f for f in sys.meta_path if "ScikitBuild" not in type(f).__name__]
+sys.path.insert(0, pkg)
+import tin_engine._core as core  # noqa: E402
+import tin_engine.cli as cli  # noqa: E402
+
+assert core.__file__.startswith(pkg), core.__file__
+real = vars(cli)["refine"]
+
+
+def refine(*args, **kwargs):
+    kwargs["threads"] = threads
+    if pause:
+        print(f"PID {os.getpid()}", file=sys.stderr, flush=True)
+        time.sleep(pause)
+    out = None
+    for i in range(repeat):
+        t0 = time.perf_counter()
+        out = real(*args, **kwargs)
+        wall = time.perf_counter() - t0
+        rec = dict(call=i, threads=threads, refine_s=wall, legalise_s=out.legalise_seconds,
+                   quality_s=out.quality_seconds, scan_s=out.scan_seconds,
+                   split_s=out.split_seconds, rounds=out.rounds, inserted=out.inserted,
+                   flips=out.flips, max_error=out.max_error,
+                   triangles=len(out.triangles), vertices=len(out.vertices))
+        rec["rest_s"] = wall - rec["legalise_s"] - rec["quality_s"] - rec["scan_s"] - rec["split_s"]
+        print("PHASES " + json.dumps(rec), file=sys.stderr, flush=True)
+    return out
+
+
+vars(cli)["refine"] = refine
+cli.app(args=rasputin, prog_name="rasputin", standalone_mode=False)
diff --git a/docs/benchmarks/2026-09-27/21b-ties/scripts/tables.py b/docs/benchmarks/2026-09-27/21b-ties/scripts/tables.py
new file mode 100644
index 00000000..5eda6dba
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b-ties/scripts/tables.py
@@ -0,0 +1,102 @@
+"""tables.py data/ties_quarter_t1.jsonl data/ties_tile_t1.jsonl > data/tables.md
+
+Summarises instrument.patch's RASPUTIN_TIES_OUT records (one JSON line per
+refine call) into the README's tables. Phase p2 is the refine loop, p0
+legalise_all, p1 the quality pass.
+"""
+import json
+import re
+import sys
+from pathlib import Path
+
+
+def load(path):
+    return json.loads(Path(path).read_text().splitlines()[-1])
+
+
+def shape_group(name):
+    if name.startswith("axis_square"):
+        return "axis-aligned square"
+    if name.startswith("axis_rectangle"):
+        return "axis-aligned rectangle, not square"
+    return {
+        "rotated_square": "rotated square",
+        "rotated_rectangle": "rotated rectangle, not square",
+        "isosceles_trapezoid_axis": "isosceles trapezoid, parallel sides on rows or columns",
+        "isosceles_trapezoid_other": "isosceles trapezoid, other direction",
+        "other_cyclic": "other cyclic quad (no parallel sides)",
+        "collinear_triple": "three corners collinear",
+    }[name]
+
+
+def pct(n, d):
+    return f"{100.0 * n / d:.3f} %" if d else "-"
+
+
+def main(paths):
+    recs = {Path(p).stem.split("_")[1]: load(p) for p in paths}
+    names = list(recs)
+    print("| measure | " + " | ".join(names) + " |")
+    print("|---|" + "---:|" * len(names))
+    rows = []
+    for ph, label in (("p2", "refine loop"), ("p0", "legalise_all"), ("p1", "quality pass")):
+        def g(r, k, ph=ph):
+            return r["counts"].get(f"{ph}.{k}", 0)
+
+        rows += [
+            (f"{label}: incircle calls", lambda r, g=g: f"{g(r, 'calls'):,}"),
+            (f"{label}: exact path", lambda r, g=g: f"{g(r, 'exact.calls'):,} ({pct(g(r, 'exact.calls'), g(r, 'calls'))})"),
+            (f"{label}: exact path, Cocircular", lambda r, g=g: f"{g(r, 'exact.result.cocircular'):,}"),
+            (f"{label}: exact path, four node corners", lambda r, g=g: f"{g(r, 'exact.nodes4'):,}"),
+            (f"{label}: exact path, lattice det = 0", lambda r, g=g: f"{g(r, 'exact.n4.lattice_det_zero'):,}"),
+            (f"{label}: exact path, lattice det != 0", lambda r, g=g: f"{g(r, 'exact.n4.exact_path_lattice_nonzero'):,}"),
+            (f"{label}: exact path, QW2 conditions hold", lambda r, g=g: f"{g(r, 'exact.n4.qw2_qualifies'):,}"),
+            (f"{label}: filtered path, four node corners", lambda r, g=g: f"{g(r, 'filtered.nodes4'):,}"),
+            (f"{label}: filtered path, QW2 conditions hold", lambda r, g=g: f"{g(r, 'filtered.n4.qw2_qualifies'):,}"),
+            (f"{label}: all calls QW2 would answer", lambda r, g=g: (
+                lambda q: f"{q:,} ({pct(q, g(r, 'calls'))})")(
+                g(r, 'exact.n4.qw2_qualifies') + g(r, 'filtered.n4.qw2_qualifies'))),
+            (f"{label}: calls with fewer than four node corners", lambda r, g=g: f"{sum(g(r, f'{p}.nodes{k}') for p in ('exact', 'filtered') for k in range(4)):,}"),
+            (f"{label}: lattice sign differs from kernel (four nodes)", lambda r, g=g: f"{g(r, 'exact.n4.lattice_sign_differs') + g(r, 'filtered.n4.lattice_sign_differs'):,}"),
+        ]
+    rows += [
+        ("max spread from d, four-node calls, refine loop (nodes)", lambda r: f"{r['max_spread_all4_refine']}"),
+        ("max spread from d, exact-path calls, refine loop (nodes)", lambda r: f"{r['max_spread_exact4_refine']}"),
+        ("dx, dy; rows x cols", lambda r: f"{r['dx']:g}, {r['dy']:g}; {r['rows']} x {r['cols']}"),
+        ("frame-exact sufficient condition (bits of dx + bit_width) <= 53", lambda r: f"{r['dx_significant_bits']} + {r['index_bit_width']} = {r['dx_significant_bits'] + r['index_bit_width']}: {r['global_frame_exact_condition']}"),
+        ("rounds, inserted, flips", lambda r: f"{r['rounds']}, {r['inserted']:,}, {r['flips']:,}"),
+    ]
+    for label, f in rows:
+        print(f"| {label} | " + " | ".join(f(recs[n]) for n in names) + " |")
+
+    print()
+    print("Shapes of the exact-path (tie) quads, refine loop:")
+    print()
+    print("| shape | " + " | ".join(names) + " |")
+    print("|---|" + "---:|" * len(names))
+    groups = {}
+    for n in names:
+        tot = recs[n]["counts"].get("p2.exact.calls", 0)
+        for k, v in recs[n]["counts"].items():
+            m = re.fullmatch(r"p2\.exact\.n4\.shape\.(.+)", k)
+            if m:
+                groups.setdefault(shape_group(m.group(1)), {}).setdefault(n, 0)
+                groups[shape_group(m.group(1))][n] += v
+    order = sorted(groups, key=lambda s: -groups[s].get(names[0], 0))
+    for s in order:
+        cells = []
+        for n in names:
+            v = groups[s].get(n, 0)
+            cells.append(f"{v:,} ({pct(v, recs[n]['counts'].get('p2.exact.calls', 0))})")
+        print(f"| {s} | " + " | ".join(cells) + " |")
+    print()
+    print("Most frequent axis-aligned sizes (short x long side, nodes), refine loop exact path:")
+    print()
+    for n in names:
+        sizes = sorted(((v, k.split(".")[-1]) for k, v in recs[n]["counts"].items()
+                        if k.startswith("p2.exact.n4.shape.axis_")), reverse=True)[:6]
+        print(f"- {n}: " + ", ".join(f"{s.removeprefix('axis_')} {v:,}" for v, s in sizes))
+
+
+if __name__ == "__main__":
+    main(sys.argv[1:])
diff --git a/docs/benchmarks/2026-09-27/21b/README.md b/docs/benchmarks/2026-09-27/21b/README.md
new file mode 100644
index 00000000..017bce73
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b/README.md
@@ -0,0 +1,79 @@
+# Benchmark run `21b`
+
+Generated by `tools/bench.py`.
+
+## Verdict
+
+- ACCEPTED
+
+Time threshold: 5.0 %.
+
+## Method
+
+- Started 2026-09-27T18:58:35.532606+02:00; tree `a03ae65f06773d7191f7bcb66088a8ce087892d7` (dirty); bench.py blob `77765b181714c31714da476c2335c799c6a84021`. Build: Release, AppleClang 21.0.0.21000101, `-O3 -DNDEBUG`, _core sha256 `dfdeb6ca078bae61f3881d6bffda8a19e2cf5dac4cca8a27a39327f938fef080`.
+- Apple M1 Max, 8 P + 2 E cores, 32 GiB, macOS 27.0, Python 3.14.7, numpy 2.5.3.
+- Power **battery** (88%), `pmset -g batt` before and after (in run.json).
+- DEM `/Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif` (sha256 `aabd0cbc28471ce8593e4c278381811bec3fbf4e1194058b3b4888d00c3af575`), tolerance 1.0, extra mesh args ``; domains: `quarter` (/Users/skavhaug/projects/rasputin/docs/benchmarks/2026-09-26/quarter.geojson), `tile` (the whole tile).
+- One child per sample, `/Users/skavhaug/projects/rasputin/.venv/bin/python /Users/skavhaug/projects/rasputin/tools/bench.py _child --pkg /Users/skavhaug/projects/rasputin/build-bench/pkg --threads <N> -- mesh --dem /Users/skavhaug/projects/rasputin/tests/fixtures/dem_archive/7908_3_10m_z33.tif --tolerance 1 --out <path> --binary`, repeats interleaved over thread counts; t=0 is the CLI's default, other counts are forced into `refine`. Quality: one `--ascii` run per domain at t=0, kept out of the repository; rerun the child with `--ascii --out PATH` to regenerate it.
+
+## `quarter`
+
+| threads | n | median s | min s | max s |
+|---:|---:|---:|---:|---:|
+| 0 | 5 | 0.1718 | 0.1691 | 0.1785 |
+| 1 | 5 | 0.4328 | 0.4312 | 0.4825 |
+| 2 | 5 | 0.2869 | 0.2769 | 0.3032 |
+| 3 | 5 | 0.2309 | 0.2292 | 0.2417 |
+| 4 | 5 | 0.2078 | 0.2069 | 0.2184 |
+| 5 | 5 | 0.1932 | 0.1916 | 0.2019 |
+| 6 | 5 | 0.1820 | 0.1814 | 0.1891 |
+| 7 | 5 | 0.1767 | 0.1747 | 0.1836 |
+| 8 | 5 | 0.1711 | 0.1697 | 0.1743 |
+| 9 | 5 | 0.1731 | 0.1704 | 0.2000 |
+| 10 | 5 | 0.1761 | 0.1703 | 0.1848 |
+| 11 | 5 | 0.1712 | 0.1692 | 0.1738 |
+| 12 | 5 | 0.1728 | 0.1687 | 0.1753 |
+| 13 | 5 | 0.1728 | 0.1694 | 0.1783 |
+| 14 | 5 | 0.1707 | 0.1698 | 0.1827 |
+| 15 | 5 | 0.1722 | 0.1719 | 0.1767 |
+| 16 | 5 | 0.1738 | 0.1715 | 0.1837 |
+| 17 | 5 | 0.1722 | 0.1720 | 0.1737 |
+| 18 | 5 | 0.1745 | 0.1712 | 0.1786 |
+| 19 | 5 | 0.1747 | 0.1721 | 0.1799 |
+| 20 | 5 | 0.1747 | 0.1692 | 0.1804 |
+
+Ceiling: 2.48x at 20 threads over 1, best 2.54x at 14; 2026-09-26: about 2.2x, flat from about 7.
+
+Quality: worst angle 0.3955 deg, median 45.00, share under 1 deg 0.00003, max degree 18, within tolerance True, Delaunay 0 violations of 641791 edges (80136 decided exactly), mesh sha256 `1e531976f33b38dd883f9cf1945f6303dbac94c92f30b52b8bd7b46db7eab9ee`.
+
+## `tile`
+
+| threads | n | median s | min s | max s |
+|---:|---:|---:|---:|---:|
+| 0 | 5 | 0.1941 | 0.1893 | 0.2239 |
+| 1 | 5 | 0.4885 | 0.4776 | 0.5013 |
+| 2 | 5 | 0.3221 | 0.3120 | 0.3339 |
+| 3 | 5 | 0.2633 | 0.2618 | 0.2652 |
+| 4 | 5 | 0.2361 | 0.2339 | 0.2385 |
+| 5 | 5 | 0.2197 | 0.2184 | 0.2300 |
+| 6 | 5 | 0.2074 | 0.2055 | 0.2078 |
+| 7 | 5 | 0.1986 | 0.1975 | 0.2030 |
+| 8 | 5 | 0.1930 | 0.1922 | 0.1937 |
+| 9 | 5 | 0.1918 | 0.1899 | 0.1979 |
+| 10 | 5 | 0.1920 | 0.1887 | 0.1934 |
+| 11 | 5 | 0.1926 | 0.1910 | 0.2012 |
+| 12 | 5 | 0.1925 | 0.1910 | 0.1952 |
+| 13 | 5 | 0.1936 | 0.1915 | 0.2009 |
+| 14 | 5 | 0.1947 | 0.1929 | 0.1957 |
+| 15 | 5 | 0.1923 | 0.1914 | 0.1980 |
+| 16 | 5 | 0.1970 | 0.1941 | 0.2083 |
+| 17 | 5 | 0.1927 | 0.1908 | 0.1976 |
+| 18 | 5 | 0.1937 | 0.1928 | 0.1946 |
+| 19 | 5 | 0.1944 | 0.1923 | 0.2132 |
+| 20 | 5 | 0.1931 | 0.1928 | 0.1988 |
+
+Ceiling: 2.53x at 20 threads over 1, best 2.55x at 9; 2026-09-26: about 2.2x, flat from about 7.
+
+Quality: worst angle 0.6296 deg, median 45.00, share under 1 deg 0.00000, max degree 74, within tolerance True, Delaunay 0 violations of 692056 edges (94469 decided exactly), mesh sha256 `11741a81adfa17b34a1ea56875ec9e791414d17625eb1c58305e7c0cf10e005c`.
+
+<!-- bench.py: generated above this line; hand-written prose below is kept -->
diff --git a/docs/benchmarks/2026-09-27/21b/raw.tsv b/docs/benchmarks/2026-09-27/21b/raw.tsv
new file mode 100644
index 00000000..88ee51aa
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b/raw.tsv
@@ -0,0 +1,210 @@
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diff --git a/docs/benchmarks/2026-09-27/21b/run.json b/docs/benchmarks/2026-09-27/21b/run.json
new file mode 100644
index 00000000..b8c496b2
--- /dev/null
+++ b/docs/benchmarks/2026-09-27/21b/run.json
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+ "accept_quality": false,
+ "threshold_pct": 5.0,
+ "verdict": [
+  "ACCEPTED"
+ ]
+}
diff --git a/docs/increments/21-parallel-refine.md b/docs/increments/21-parallel-refine.md
index 7597c778..34b38d34 100644
--- a/docs/increments/21-parallel-refine.md
+++ b/docs/increments/21-parallel-refine.md
@@ -487,6 +487,96 @@ is no difference beyond noise (refine 515 vs 519 ms), as QW1 predicts. The
 mesh sha256 that `bench.py` prints for the tile and the quarter circle was the same
 before and after.
 
+### Pinned by the red suite (21b)
+
+QW2 names `lattice_incircle(a, b, c, d) -> std::optional<Incircle>` but not
+where `dx == dy` and the exact-frame decision come from. `@tester` chose the
+following in the red step; `tests/cpp/unit/test_mesh_lattice_incircle.cpp`
+holds it. Both functions are in `namespace terrain::mesh` and reachable
+through `include/terrain/mesh/lawson.hpp`. Whether they live there or in a new
+header beside it that `lawson.hpp` includes is `@developer`'s choice.
+
+- `[[nodiscard]] LatticeFrame lattice_frame(double dx, double dy, std::size_t rows, std::size_t cols) noexcept`
+  returns a frame with the same `dx` and `dy` that also records, once per
+  refine call, whether the integer path may answer on it. It may answer only
+  when `dx == dy`, `dx` is finite and positive, and `col * dx` and `row * dy`
+  are exact for every node with `col < cols` and `row < rows`. It must answer
+  wherever QW2's sufficient condition holds: the significant bits of `dx` plus
+  `bit_width(max(rows, cols) - 1)` are at most 53. The suite tests exactly 53,
+  and 54 with a `dx` whose `3 * dx` rounds (it must refuse), on each axis. A
+  largest product `(max(rows, cols) - 1) * dx` that overflows is not exact:
+  `2^1023` on a 3 × 3 grid must refuse.
+  Between that condition and exactness, for example `dx = 0.1` on a
+  3 × 3 grid, which is exact but fails the condition, nothing is pinned.
+  `refine` builds its frame with this function instead of
+  `LatticeFrame{g.delta_x(), g.delta_y()}`. No test can see that, because the
+  output is bit-identical by design. The reviewer checks it by reading the
+  code, and `@perf`'s acceptance run shows it as time saved.
+- **A `LatticeFrame` built directly never enables the integer path.**
+  `LatticeFrame{dx, dy}`, the way every caller except refine builds one, keeps compiling
+  and keeps today's kernel path. The existing Lawson and quality suites
+  therefore still exercise that path, and the 21b suite uses such a frame as
+  its "without".
+- `[[nodiscard]] std::optional<pred::Incircle> lattice_incircle(MeshVertex a, MeshVertex b, MeshVertex c, MeshVertex d, const LatticeFrame& f) noexcept`.
+  Precondition: `a, b, c` strictly counter-clockwise on `(col, -row)`, which
+  `must_flip`'s triangle is (the `LatticeMesh` invariant). It returns empty
+  unless `f` came from an enabling `lattice_frame`, all four corners are nodes
+  with finite coordinates (`is_node()` is true for ±inf; the spread test
+  `!(fabs(diff) <= 2^14)` refuses every non-finite difference, ±inf or the NaN
+  of `inf - inf`, before the `int64` cast), and `|col_x - col_d|` and `|row_x - row_d|` are at most 2^14 for each `x` in
+  `a, b, c`. The bound is measured from `d`, so `a` and `b` may be 2^15 apart.
+  When it answers, the answer is the sign `DetriaExact::incircle_ccw` gives on
+  the frame points `(col * dx, -(row * dy))`.
+- **`must_flip` calls it first.** When it answers, `must_flip` returns
+  `answer == Inside` and asks the kernel nothing, not even the frame
+  `orient2d`. The suite counts kernel calls to check this. When it refuses,
+  today's path runs unchanged.
+
+At the red commit the suite was registered only once `lawson.hpp` named
+`lattice_incircle`, as in 21a; the guard was removed after green. It starts no
+threads, so it is not in the TSan job.
+
+**Mutation round** against a scratch implementation that is not committed. All
+of these were killed: the sign inverted; the determinant on `(col, row)`;
+refusal at exactly 2^14 (`>=`); a bound of 2^14 + 1; a bound of 2^15; the
+bound checked on one side only; the bound checked on columns only; the
+exact-frame check dropped; the exact-frame test at `< 53`; the exact-frame
+test on columns only, and on rows only; the node check dropped; the node check
+skipping `d`; `dx != dy` accepted; `dx <= 0` accepted; a directly built frame
+enabling the path; no call in `must_flip`. The determinant computed in doubles
+rather than `int64` was also killed, and only by the radius-8085 circle, where
+spreads come close to 2^14.
+
+Two mutants survived that round and were found by `@reviewer` after green: the
+exact-frame limit one bit loose (`bit_width(top - 1) - 1`) and the finite-extent
+clause dropped. A test amendment after green kills both, and adds the
+non-finite-corner refusal, which HEAD at the amendment did not have.
+
+### 21b: the integer path is faster (green, not acceptance)
+
+Measured by `@developer` at the green commit, back to back against 21a's head
+`d3ee2ce`, **on battery** (98 %, before and after), so it compares with no AC
+figure. Both trees were built by `bench.py`'s `build()` (Release; the base
+`.so` hashes to the 21a acceptance run's `654a2442…`), and each sample was one
+`rasputin mesh` process on the 1 m benchmark at tolerance 1, reading
+`RefineOutcome`'s phase times. Medians of 5, interleaved base/green:
+
+| domain, threads | refine, 21a -> 21b | split phase, 21a -> 21b |
+|---|---:|---:|
+| quarter, 1 | 0.444 -> 0.414 s (-6.7 %) | 0.122 -> 0.090 s (-26 %) |
+| quarter, 8 | 0.198 -> 0.164 s (-17 %) | 0.132 -> 0.098 s (-26 %) |
+| tile, 1 | 0.503 -> 0.464 s (-7.8 %) | 0.124 -> 0.091 s (-27 %) |
+| tile, 8 | 0.225 -> 0.187 s (-17 %) | 0.134 -> 0.097 s (-28 %) |
+
+The scan is unchanged, as it should be; the whole gain is in the split phase,
+and it is larger than QW2's estimate (-4 % and -8 %), so the int64 determinant
+with the frame `orient2d` it skips is cheaper than the filtered path too, not
+only than the exact path (5.0 % of refine in the profile) it replaces.
+The binary meshes hash the same as 21a's (quarter `ff705683…`, tile
+`645919aa…`), with the same flip and insertion counts. The scratch driver and
+its samples were not kept. `@perf`'s acceptance run, battery against battery,
+is `docs/benchmarks/2026-09-27/21b-acceptance.md` (ACCEPTED).
+
 ## 4. Determinism levels
 
 Today's contract (14 R5, 14b R1, tested by 14's T6 and 18's T3 golden
@@ -784,8 +874,15 @@ Each is answered or placed.
    cheaper exact decision for quads whose corners are all nodes? That is 5 %
    of refine and about 17 % of the serial phase."*
    **Answered: yes,** an int64 lattice incircle under four stated conditions,
-   bit-identical to today (QW2, 21b). The premise is still an inference; the
-   classification measurement in QW2 comes first.
+   bit-identical to today (QW2, 21b). **The premise is measured**
+   (`docs/benchmarks/2026-09-27/21b-ties/README.md`): every exact-path tie
+   has four node corners, lattice determinant 0 and meets QW2's conditions
+   (146,962 of 146,962 on the quarter circle, 154,502 of 154,502 on the tile);
+   QW2 would answer 99.97 % and 100 % of all refine-loop incircle calls. Only
+   38 % of the ties are axis-aligned rectangles, so the general determinant is
+   needed. The time saved was measured on battery at green ("21b: the
+   integer path is faster") and in `@perf`'s acceptance run
+   (`docs/benchmarks/2026-09-27/21b-acceptance.md`, ACCEPTED).
 
 4. *"The scan loses 16.5 ms of 61.8 ms to imbalance at 8 threads. Is the
    chunking free to change, given that results are written per slot? And does
@@ -862,7 +959,8 @@ suites:
 - Other tests that pin exact counts of a refined mesh (for example 14's T2
   under 14b's amendment) are for `@tester` to find in the red step.
 
-Every new suite joins the TSan job's list in `.github/workflows/main.yaml`.
+Every new suite that starts threads joins the TSan job's list in
+`.github/workflows/main.yaml`.
 
 **Acceptance for 21a, 21b and 21d** is `@perf`'s run
 (`docs/increments/README.md`, "Acceptance"): the 1 m benchmark and the thread
diff --git a/include/terrain/mesh/lawson.hpp b/include/terrain/mesh/lawson.hpp
index 27960558..6c65c2d0 100644
--- a/include/terrain/mesh/lawson.hpp
+++ b/include/terrain/mesh/lawson.hpp
@@ -24,23 +24,101 @@
 #include <terrain/mesh/lattice_mesh.hpp>
 #include <terrain/predicates/kernel.hpp>
 
+#include <algorithm>
+#include <array>
+#include <bit>
+#include <cmath>
 #include <cstddef>
 #include <cstdint>
+#include <optional>
 #include <span>
 #include <utility>
 #include <vector>
 
 namespace terrain::mesh {
 
-struct LatticeFrame {
+// A frame built directly, LatticeFrame{dx, dy}, never enables the integer
+// incircle; only lattice_frame() below can (docs/increments/21-parallel-refine.md,
+// "Pinned by the red suite (21b)").
+class LatticeFrame {
+public:
     double dx = 1.0;
     double dy = 1.0;
 
+    constexpr LatticeFrame() noexcept = default;
+    constexpr LatticeFrame(double x, double y) noexcept : dx{x}, dy{y} {}
+
     [[nodiscard]] Point2 at(MeshVertex v) const noexcept {
         return Point2{v.col * dx, -(v.row * dy)};
     }
+    // Whether lattice_incircle may answer on this frame.
+    [[nodiscard]] bool integer() const noexcept { return integer_; }
+
+private:
+    bool integer_ = false;
+    friend LatticeFrame lattice_frame(double, double, std::size_t, std::size_t) noexcept;
 };
 
+// The frame of a rows x cols grid, with the integer path enabled when the frame
+// is the lattice times one exact constant (QW2): dx == dy, dx a positive normal
+// double, and col * dx, row * dy exact for every node. Exactness is decided by
+// QW2's sufficient condition, which is also what it takes: the significant bits
+// of dx plus bit_width(max(rows, cols) - 1) are at most 53, so every product
+// fits the mantissa; the largest one must also be finite.
+[[nodiscard]] inline LatticeFrame lattice_frame(double dx, double dy, std::size_t rows,
+                                                std::size_t cols) noexcept {
+    LatticeFrame f{dx, dy};
+    if (dx != dy || !std::isnormal(dx) || dx < 0.0)
+        return f;
+    const std::uint64_t mantissa =
+        (std::bit_cast<std::uint64_t>(dx) & ((std::uint64_t{1} << 52) - 1)) | (std::uint64_t{1} << 52);
+    const auto significant = 53 - std::countr_zero(mantissa);
+    const std::size_t top = std::max(rows, cols);
+    const auto span = top == 0 ? 0 : std::bit_width(top - 1);
+    f.integer_ = significant + span <= 53
+              && std::isfinite(static_cast<double>(top == 0 ? 0 : top - 1) * dx);
+    return f;
+}
+
+// The incircle sign of the quad (a, b, c, d) from an int64 determinant on the
+// lattice (col, -row), translated to d. Answers only on an enabling frame, four
+// node corners and every difference from d at most 2^14 nodes; the determinant
+// is then below 3 * 2^58 and exact. Under those conditions it is the sign
+// DetriaExact gives on the frame points, since the frame is the lattice scaled
+// by dx > 0 without rounding. That holds only for nodes inside the rows x cols
+// grid f was built for: lattice_frame checked exactness of col * dx and
+// row * dy up to that extent and no further, so a node beyond it can have a
+// rounded frame point whose DetriaExact sign differs. Precondition: a, b, c
+// counter-clockwise on (col, -row). Otherwise empty, and the caller asks the
+// kernel. A corner with a non-finite coordinate is refused by the spread test,
+// before its cast: NaN is not a node (NaN != floor(NaN)), but is_node() is true
+// for +-inf. Every coordinate of a, b, c and d enters some difference, and a
+// difference with an infinite operand is +-inf (the other finite, or infinities
+// of opposite sign) or NaN (inf - inf). `!(fabs <= bound)` is true for both,
+// where `fabs > bound` would pass NaN on to the int64 cast (UB).
+[[nodiscard]] inline std::optional<pred::Incircle> lattice_incircle(MeshVertex a, MeshVertex b,
+                                                                    MeshVertex c, MeshVertex d,
+                                                                    const LatticeFrame& f) noexcept {
+    if (!f.integer() || !a.is_node() || !b.is_node() || !c.is_node() || !d.is_node())
+        return std::nullopt;
+    constexpr double bound = 1 << 14;
+    std::array<std::array<std::int64_t, 2>, 3> p{};  // (col, -row) minus d's
+    const std::array<MeshVertex, 3> abc{a, b, c};
+    for (std::size_t i = 0; i < 3; ++i) {
+        const double x = abc[i].col - d.col, y = d.row - abc[i].row;  // exact: integers < 2^53
+        if (!(std::fabs(x) <= bound) || !(std::fabs(y) <= bound))  // also refuses NaN
+            return std::nullopt;
+        p[i] = {static_cast<std::int64_t>(x), static_cast<std::int64_t>(y)};
+    }
+    const auto lift = [](const std::array<std::int64_t, 2>& q) { return q[0] * q[0] + q[1] * q[1]; };
+    const auto cross = [](const std::array<std::int64_t, 2>& u, const std::array<std::int64_t, 2>& v) {
+        return u[0] * v[1] - u[1] * v[0];
+    };
+    const std::int64_t det =
+        lift(p[0]) * cross(p[1], p[2]) + lift(p[1]) * cross(p[2], p[0]) + lift(p[2]) * cross(p[0], p[1]);
+    return det > 0 ? pred::Incircle::Inside : det < 0 ? pred::Incircle::Outside : pred::Incircle::Cocircular;
+}
+
 namespace detail {
 
 // Whether t's edge e must flip: interior, unconstrained, and the quad's
@@ -57,8 +135,16 @@ template <pred::GeometryKernel K>
     while (m.triangles()[u][j] != tri[(e + 1) % 3])
         ++j;
     const auto v = m.vertices();
+    const MeshVertex d_vertex = v[m.triangles()[u][(j + 2) % 3]];
+    // The integer path first; the mesh triangle is counter-clockwise on
+    // (col, -row), and it answers only where the kernel would give the same sign.
+    // Answering without consulting K assumes K's incircle sign is exact, as
+    // DefaultKernel's (FilteredKernel<DetriaExact>) is: for an inexact K the
+    // integer sign could differ from K's, and the flip sequence with it.
+    if (const auto s = lattice_incircle(v[tri[e]], v[tri[(e + 1) % 3]], v[tri[(e + 2) % 3]], d_vertex, f))
+        return *s == pred::Incircle::Inside;
     const Point2 a = f.at(v[tri[e]]), b = f.at(v[tri[(e + 1) % 3]]), c = f.at(v[tri[(e + 2) % 3]]),
-                 d = f.at(v[m.triangles()[u][(j + 2) % 3]]);
+                 d = f.at(d_vertex);
     // Orientation is exact on (col, -row), but the frame rounds col * dx and
     // row * dy, so a triangle counter-clockwise in the mesh can be collinear
     // or clockwise here. The kernel answers Cocircular for a collinear triple
diff --git a/include/terrain/refinement/refine.hpp b/include/terrain/refinement/refine.hpp
index c0c6a256..1a6a7823 100644
--- a/include/terrain/refinement/refine.hpp
+++ b/include/terrain/refinement/refine.hpp
@@ -276,7 +276,7 @@ template <raster::RasterSource R>
         return std::move(*refused);
     auto& m = std::get<mesh::LatticeMesh>(built);
 
-    const mesh::LatticeFrame frame{g.delta_x(), g.delta_y()};
+    const auto frame = mesh::lattice_frame(g.delta_x(), g.delta_y(), g.rows(), g.cols());
     RefineOutcome out;
     using clock = std::chrono::steady_clock;
     const auto since = [](clock::time_point t0) {
diff --git a/tests/cpp/CMakeLists.txt b/tests/cpp/CMakeLists.txt
index a21cc2c2..f179e0a1 100644
--- a/tests/cpp/CMakeLists.txt
+++ b/tests/cpp/CMakeLists.txt
@@ -291,3 +291,13 @@ add_terrain_backend_test(test_refinement_active unit/test_refinement_active.cpp)
 target_link_libraries(test_refinement_active PRIVATE Threads::Threads)
 add_terrain_backend_test(test_mesh_lawson_stack unit/test_mesh_lawson_stack.cpp)
 target_link_libraries(test_mesh_lawson_stack PRIVATE Threads::Threads)
+
+# The integer incircle, increment 21b (docs/increments/21-parallel-refine.md,
+# section 3 QW2 and "Pinned by the red suite (21b)"). Bit-identical: 14's T6
+# and 18's golden digests are the output oracle and stay as they are.
+# test_mesh_lattice_incircle is the invariant-critical suite: agreement with
+# DetriaExact on the frame doubles wherever lattice_incircle answers, refusal
+# wherever QW2's conditions fail, and must_flip's decisions as today's. It
+# starts no threads, so it is not in the TSan job.
+
+add_terrain_backend_test(test_mesh_lattice_incircle unit/test_mesh_lattice_incircle.cpp)
diff --git a/tests/cpp/unit/test_mesh_lattice_incircle.cpp b/tests/cpp/unit/test_mesh_lattice_incircle.cpp
new file mode 100644
index 00000000..a316bf5f
--- /dev/null
+++ b/tests/cpp/unit/test_mesh_lattice_incircle.cpp
@@ -0,0 +1,977 @@
+// Increment 21b (docs/increments/21-parallel-refine.md, section 3 QW2, section
+// 7's 21b row, and "Pinned by the red suite (21b)"): an integer incircle for
+// quads whose four corners are DEM nodes, answered at the top of must_flip.
+// This is 21b's invariant-critical suite.
+//
+// 21b is bit-identical. Where lattice_incircle answers, its answer must be the
+// sign DetriaExact computes on the frame doubles (col * dx, -(row * dy)),
+// because that is the number must_flip decides on today. Where any of QW2's
+// conditions fails it must refuse, and must_flip must then run today's path.
+//
+// Interface pinned here (reachable through include/terrain/mesh/lawson.hpp):
+//
+//   namespace terrain::mesh {
+//   // A frame for this grid that also records, once, whether QW2 may answer
+//   // on it: dx == dy, finite and > 0, and col * dx, row * dy exact for
+//   // every node with col < cols, row < rows.
+//   [[nodiscard]] LatticeFrame lattice_frame(double dx, double dy,
+//                                            std::size_t rows, std::size_t cols) noexcept;
+//   // Precondition: a, b, c strictly counter-clockwise on (col, -row).
+//   [[nodiscard]] std::optional<pred::Incircle> lattice_incircle(
+//       MeshVertex a, MeshVertex b, MeshVertex c, MeshVertex d, const LatticeFrame& f) noexcept;
+//   }
+//
+// A LatticeFrame built directly, LatticeFrame{dx, dy}, never enables the
+// integer path; refine builds its frame with lattice_frame, every other caller
+// directly.
+//
+// The oracle is DetriaExact::incircle_ccw on the frame points, computed here
+// from col * dx and -(row * dy), not through LatticeFrame::at. For today's
+// must_flip the oracle is a copy of it (lawson.hpp at 93a8066).
+
+#include <catch2/catch_test_macros.hpp>
+#include <catch2/generators/catch_generators.hpp>
+#include <catch2/generators/catch_generators_range.hpp>
+
+#include <terrain/core/point.hpp>
+#include <terrain/mesh/lattice_mesh.hpp>
+#include <terrain/mesh/lawson.hpp>
+#include <terrain/predicates/default_kernel.hpp>
+#include <terrain/predicates/detria_exact.hpp>
+
+#include <algorithm>
+#include <array>
+#include <cmath>
+#include <cstddef>
+#include <cstdint>
+#include <limits>
+#include <numbers>
+#include <optional>
+#include <random>
+#include <span>
+#include <string>
+#include <utility>
+#include <vector>
+
+using terrain::Point2;
+using terrain::TriangleIndices;
+using terrain::mesh::kNoNeighbour;
+using terrain::mesh::lattice_frame;
+using terrain::mesh::lattice_incircle;
+using terrain::mesh::LatticeFrame;
+using terrain::mesh::LatticeMesh;
+using terrain::mesh::LatticeVertex;
+using terrain::mesh::MeshVertex;
+using terrain::pred::DefaultKernel;
+using terrain::pred::DetriaExact;
+using terrain::pred::Incircle;
+using terrain::pred::Orientation;
+
+namespace {
+
+constexpr std::int64_t kBound = std::int64_t{1} << 14;  // QW2's spread bound, in nodes
+constexpr std::size_t kBench = 5051;                    // the 1 m benchmark's rows and cols
+
+// A node at (col, row). MeshVertex is (col, row); LatticeVertex is {row, col}.
+MeshVertex node(std::int64_t col, std::int64_t row) {
+    return MeshVertex{static_cast<double>(col), static_cast<double>(row)};
+}
+
+// The frame point, written out rather than taken from LatticeFrame::at.
+Point2 fp(double dx, double dy, MeshVertex v) { return Point2{v.col * dx, -(v.row * dy)}; }
+
+// Twice the signed area on (col, -row), exact for nodes of the sizes used here.
+std::int64_t iorient(MeshVertex a, MeshVertex b, MeshVertex c) {
+    const auto ax = static_cast<std::int64_t>(a.col), ay = -static_cast<std::int64_t>(a.row);
+    const auto bx = static_cast<std::int64_t>(b.col), by = -static_cast<std::int64_t>(b.row);
+    const auto cx = static_cast<std::int64_t>(c.col), cy = -static_cast<std::int64_t>(c.row);
+    return (bx - ax) * (cy - ay) - (by - ay) * (cx - ax);
+}
+
+// The oracle: DetriaExact on the frame doubles. Empty if a, b, c is not
+// counter-clockwise in the frame (DetriaExact's precondition), which under an
+// exact frame and a counter-clockwise lattice triangle cannot happen.
+std::optional<Incircle> oracle(double dx, double dy, MeshVertex a, MeshVertex b, MeshVertex c,
+                               MeshVertex d) {
+    const Point2 pa = fp(dx, dy, a), pb = fp(dx, dy, b), pc = fp(dx, dy, c), pd = fp(dx, dy, d);
+    if (DetriaExact::orient2d(pa, pb, pc) != Orientation::CounterClockwise)
+        return std::nullopt;
+    return DetriaExact::incircle_ccw(pa, pb, pc, pd);
+}
+
+std::string name(Incircle s) {
+    return s == Incircle::Inside ? "Inside" : s == Incircle::Outside ? "Outside" : "Cocircular";
+}
+std::string describe(MeshVertex v) {
+    return "(col " + std::to_string(v.col) + ", row " + std::to_string(v.row) + ")";
+}
+
+// Agreement bookkeeping: every quad the caller hands in must be answered, with
+// the oracle's sign. The first failure is kept for the report, and the three
+// signs are counted so a run that saw only one of them is visible.
+struct Agreement {
+    std::size_t quads = 0, refused = 0, disagreed = 0;
+    std::array<std::size_t, 3> by_sign{};  // Outside, Cocircular, Inside
+    std::string first;
+
+    void check(const LatticeFrame& f, double dx, MeshVertex a, MeshVertex b, MeshVertex c,
+               MeshVertex d) {
+        ++quads;
+        const auto want = oracle(dx, dx, a, b, c, d);
+        const auto got = lattice_incircle(a, b, c, d, f);
+        if (!want || !got || *got != *want) {
+            (got ? disagreed : refused) += 1;
+            if (first.empty())
+                first = describe(a) + " " + describe(b) + " " + describe(c) + " d " + describe(d)
+                      + " dx " + std::to_string(dx) + ": got "
+                      + (got ? name(*got) : std::string{"nullopt"}) + ", oracle "
+                      + (want ? name(*want) : std::string{"not ccw in frame"});
+            return;
+        }
+        ++by_sign[static_cast<std::size_t>(static_cast<int>(*want) + 1)];
+    }
+    void require_all_agree() const {
+        CAPTURE(quads, refused, disagreed, first);
+        REQUIRE(quads > 0);
+        REQUIRE(refused == 0);
+        REQUIRE(disagreed == 0);
+    }
+};
+
+// The four points in cyclic order, turned counter-clockwise on (col, -row).
+std::array<MeshVertex, 4> ccw_cycle(std::array<MeshVertex, 4> p) {
+    if (iorient(p[0], p[1], p[2]) < 0)
+        std::reverse(p.begin(), p.end());
+    return p;
+}
+
+// Every lattice point on the circle of squared radius r2 about (cx, cy), in
+// angular order.
+std::vector<MeshVertex> circle(std::int64_t cx, std::int64_t cy, std::int64_t r2) {
+    std::vector<std::pair<double, MeshVertex>> pts;
+    const auto root = [](std::int64_t v) {  // floor(sqrt(v)), exact for the sizes here
+        auto s = static_cast<std::int64_t>(std::sqrt(static_cast<double>(v)));
+        while (s * s > v)
+            --s;
+        while ((s + 1) * (s + 1) <= v)
+            ++s;
+        return s;
+    };
+    const auto r = root(r2);
+    for (std::int64_t x = -r; x <= r; ++x) {
+        const std::int64_t y = root(r2 - x * x);
+        if (x * x + y * y != r2)
+            continue;
+        for (const std::int64_t sy : {y, -y}) {
+            pts.emplace_back(std::atan2(static_cast<double>(sy), static_cast<double>(x)), node(cx + x, cy + sy));
+            if (y == 0)
+                break;
+        }
+    }
+    std::sort(pts.begin(), pts.end(), [](const auto& l, const auto& r) { return l.first < r.first; });
+    std::vector<MeshVertex> out;
+    for (const auto& [angle, v] : pts)
+        out.push_back(v);
+    return out;
+}
+
+// Every choice of four points from a cyclically ordered set, as quads in
+// cyclic order.
+std::vector<std::array<MeshVertex, 4>> quads_of(const std::vector<MeshVertex>& ring) {
+    std::vector<std::array<MeshVertex, 4>> out;
+    const std::size_t n = ring.size();
+    for (std::size_t i = 0; i < n; ++i)
+        for (std::size_t j = i + 1; j < n; ++j)
+            for (std::size_t k = j + 1; k < n; ++k)
+                for (std::size_t l = k + 1; l < n; ++l)
+                    out.push_back(ccw_cycle({ring[i], ring[j], ring[k], ring[l]}));
+    return out;
+}
+
+// A cocircular quad in each of its four rotations, (a, b, c) the triangle and
+// d the fourth corner, as must_flip hands them over.
+template <class Fn>
+void for_each_rotation(const std::array<MeshVertex, 4>& q, Fn&& fn) {
+    for (std::size_t k = 0; k < 4; ++k)
+        fn(q[k], q[(k + 1) % 4], q[(k + 2) % 4], q[(k + 3) % 4]);
+}
+
+}  // namespace
+
+// ---------------------------------------------------------------- agreement
+
+TEST_CASE("lattice_incircle agrees with DetriaExact on every quad of a 5 x 5 neighbourhood",
+          "[lattice_incircle][agreement]") {
+    // Every ordered (a, b, c, d) of distinct nodes with a, b, c strictly
+    // counter-clockwise: all ties, all near misses and all clear cases that
+    // fit. At the origin and at the far corner of the benchmark's grid, in
+    // four exact frames (dx = 10 is the benchmark's).
+    const double dx = GENERATE(1.0, 10.0, 0.5, 0.375);
+    const std::int64_t base = GENERATE(std::int64_t{0}, std::int64_t{5046});
+    CAPTURE(dx, base);
+    const LatticeFrame f = lattice_frame(dx, dx, kBench, kBench);
+    std::vector<MeshVertex> nodes;
+    for (std::int64_t r = 0; r < 5; ++r)
+        for (std::int64_t c = 0; c < 5; ++c)
+            nodes.push_back(node(base + c, base + r));
+    Agreement agree;
+    for (const auto& a : nodes)
+        for (const auto& b : nodes)
+            for (const auto& c : nodes) {
+                if (iorient(a, b, c) <= 0)
+                    continue;
+                for (const auto& d : nodes)
+                    if (d != a && d != b && d != c)
+                        agree.check(f, dx, a, b, c, d);
+            }
+    agree.require_all_agree();
+    // All three answers occur, ties included, so a constant answer cannot pass.
+    REQUIRE(agree.by_sign[0] > 1000);
+    REQUIRE(agree.by_sign[1] > 1000);
+    REQUIRE(agree.by_sign[2] > 1000);
+}
+
+TEST_CASE("lattice_incircle answers Cocircular on the tie shapes 21b-ties found, in every rotation",
+          "[lattice_incircle][agreement][ties]") {
+    // docs/benchmarks/2026-09-27/21b-ties/README.md, "The shapes of the
+    // exact-path quads": axis-aligned squares and rectangles (1x1, 1x2, 2x2,
+    // 1x3, 2x3, and legalise_all's 40x40 and 10x40), rotated squares and
+    // rectangles, isosceles trapezoids with parallel sides on rows or columns
+    // and in another direction, and other cyclic quads. Each is a tie: the
+    // answer must be Cocircular, and that must be the oracle's answer too.
+    const double dx = GENERATE(1.0, 10.0);
+    CAPTURE(dx);
+    const LatticeFrame f = lattice_frame(dx, dx, kBench, kBench);
+    const auto rect = [](std::int64_t c, std::int64_t r, std::int64_t w, std::int64_t h) {
+        return ccw_cycle({node(c, r), node(c + w, r), node(c + w, r + h), node(c, r + h)});
+    };
+    const auto quad = [](std::array<std::array<std::int64_t, 2>, 4> p) {
+        return ccw_cycle({node(p[0][0], p[0][1]), node(p[1][0], p[1][1]), node(p[2][0], p[2][1]),
+                          node(p[3][0], p[3][1])});
+    };
+    const std::vector<std::pair<const char*, std::array<MeshVertex, 4>>> shapes{
+        {"axis square 1x1", rect(7, 9, 1, 1)},
+        {"axis square 2x2", rect(7, 9, 2, 2)},
+        {"axis square 40x40", rect(40, 80, 40, 40)},
+        {"axis rectangle 1x2", rect(7, 9, 1, 2)},
+        {"axis rectangle 2x1", rect(7, 9, 2, 1)},
+        {"axis rectangle 1x3", rect(7, 9, 1, 3)},
+        {"axis rectangle 2x3", rect(7, 9, 2, 3)},
+        {"axis rectangle 10x40", rect(40, 80, 10, 40)},
+        {"rotated square", quad({{{10, 11}, {11, 10}, {12, 11}, {11, 12}}})},
+        {"rotated square, side (3, 2)", quad({{{10, 12}, {13, 10}, {15, 13}, {12, 15}}})},
+        {"rotated rectangle", quad({{{10, 11}, {11, 10}, {13, 12}, {12, 13}}})},
+        {"isosceles trapezoid, parallel sides on rows", quad({{{10, 10}, {14, 10}, {13, 12}, {11, 12}}})},
+        {"isosceles trapezoid, parallel sides on columns", quad({{{10, 10}, {12, 11}, {12, 13}, {10, 14}}})},
+        // On the radius-5 circle about (20, 20): chords (5,0)-(0,5) and
+        // (4,-3)-(-3,4) are both along (-1, 1).
+        {"isosceles trapezoid, parallel sides along a diagonal",
+         quad({{{25, 20}, {20, 25}, {17, 24}, {24, 17}}})},
+        // Radius 5 again: no two sides parallel.
+        {"other cyclic quad", quad({{{25, 20}, {23, 24}, {15, 20}, {16, 17}}})},
+    };
+    for (const auto& [label, q] : shapes) {
+        CAPTURE(std::string{label});
+        for_each_rotation(q, [&](MeshVertex a, MeshVertex b, MeshVertex c, MeshVertex d) {
+            CAPTURE(describe(a), describe(b), describe(c), describe(d));
+            REQUIRE(iorient(a, b, c) > 0);
+            REQUIRE(oracle(dx, dx, a, b, c, d) == Incircle::Cocircular);
+            REQUIRE(lattice_incircle(a, b, c, d, f) == Incircle::Cocircular);
+        });
+    }
+}
+
+TEST_CASE("lattice_incircle agrees with DetriaExact on every quad of lattice circles, and one node off them",
+          "[lattice_incircle][agreement][ties]") {
+    // Every four of the 12 lattice points on the radius-5 circle and of the 16
+    // on the radius-sqrt(65) circle, in every rotation: all ties. Then d moved
+    // one node in each axis direction, which puts it strictly inside or outside.
+    // The scaled radius-5 circle (x 1617 = 3 * 7^2 * 11, radius 8085, still 12
+    // lattice points) has spreads up to 16170 nodes, just under 2^14: there a
+    // one-node move of d changes the determinant by about 2^40 in terms near
+    // 2^56, beyond what plain double arithmetic on the differences resolves.
+    const double dx = GENERATE(1.0, 10.0);
+    const std::int64_t r2 = GENERATE(std::int64_t{25}, std::int64_t{65}, std::int64_t{25} * 1617 * 1617);
+    CAPTURE(dx, r2);
+    const std::int64_t centre = 9000;  // keeps every point at col, row >= 0
+    const auto ring = circle(centre, centre, r2);
+    REQUIRE(ring.size() == (r2 == 65 ? 16u : 12u));
+    const LatticeFrame f = lattice_frame(dx, dx, 20000, 20000);
+    Agreement ties, moved;
+    std::size_t cocircular = 0;
+    for (const auto& q : quads_of(ring))
+        for_each_rotation(q, [&](MeshVertex a, MeshVertex b, MeshVertex c, MeshVertex d) {
+            ties.check(f, dx, a, b, c, d);
+            cocircular += lattice_incircle(a, b, c, d, f) == Incircle::Cocircular;
+            for (const auto& [mc, mr] : {std::pair{1, 0}, std::pair{-1, 0}, std::pair{0, 1}, std::pair{0, -1}}) {
+                const MeshVertex e{d.col + mc, d.row + mr};
+                if (e != a && e != b && e != c)
+                    moved.check(f, dx, a, b, c, e);
+            }
+        });
+    ties.require_all_agree();
+    moved.require_all_agree();
+    REQUIRE(cocircular == ties.quads);
+    REQUIRE(ties.by_sign[1] == ties.quads);
+    REQUIRE(moved.by_sign[0] > 0);
+    REQUIRE(moved.by_sign[2] > 0);
+}
+
+TEST_CASE("lattice_incircle agrees with DetriaExact on random quads up to the spread bound",
+          "[lattice_incircle][agreement][random]") {
+    // Two families, spreads drawn log-uniformly from 1 to 2^14 nodes:
+    //   - uniform: a, b, c anywhere within the spread of d;
+    //   - near-circle: four points rounded from one circle, so the
+    //     determinant is small against its terms, where rounding would show.
+    const std::uint32_t seed = GENERATE(range(1u, 5u));
+    const double dx = GENERATE(1.0, 10.0, 0.25, 3.0);
+    CAPTURE(seed, dx);
+    const LatticeFrame f = lattice_frame(dx, dx, 1u << 16, 1u << 16);
+    std::mt19937_64 rng{seed};
+    std::uniform_real_distribution<double> unit{0.0, 1.0};
+    const std::int64_t mid = std::int64_t{1} << 15;
+    Agreement uniform, near;
+    const auto within = [](MeshVertex p, MeshVertex d) {
+        return std::fabs(p.col - d.col) <= static_cast<double>(kBound)
+            && std::fabs(p.row - d.row) <= static_cast<double>(kBound);
+    };
+    for (int i = 0; i < 4000; ++i) {
+        const auto s = static_cast<std::int64_t>(std::exp2(14.0 * unit(rng)));
+        const auto off = [&] { return static_cast<std::int64_t>(std::llround((2.0 * unit(rng) - 1.0) * s)); };
+        // One draw per statement: argument evaluation order is unspecified.
+        std::array<std::int64_t, 8> o{};
+        for (auto& x : o)
+            x = off();
+        const MeshVertex d = node(mid + o[0], mid + o[1]);
+        MeshVertex a = node(mid + o[0] + o[2], mid + o[1] + o[3]);
+        MeshVertex b = node(mid + o[0] + o[4], mid + o[1] + o[5]);
+        const MeshVertex c = node(mid + o[0] + o[6], mid + o[1] + o[7]);
+        if (iorient(a, b, c) == 0 || d == a || d == b || d == c)
+            continue;
+        if (iorient(a, b, c) < 0)
+            std::swap(a, b);
+        uniform.check(f, dx, a, b, c, d);
+    }
+    for (int i = 0; i < 4000; ++i) {
+        const double radius = std::exp2(1.0 + 12.9 * unit(rng));  // diameter <= 2^14
+        const double cx = static_cast<double>(mid) + 1000.0 * unit(rng);
+        const double cy = static_cast<double>(mid) + 1000.0 * unit(rng);
+        std::array<MeshVertex, 4> p{};
+        for (auto& v : p) {
+            const double t = 2.0 * std::numbers::pi * unit(rng);
+            v = MeshVertex{std::round(cx + radius * std::cos(t)), std::round(cy + radius * std::sin(t))};
+        }
+        auto [a, b, c, d] = p;
+        if (iorient(a, b, c) == 0 || d == a || d == b || d == c || !within(a, d) || !within(b, d)
+            || !within(c, d))
+            continue;
+        if (iorient(a, b, c) < 0)
+            std::swap(a, b);
+        near.check(f, dx, a, b, c, d);
+    }
+    uniform.require_all_agree();
+    near.require_all_agree();
+    REQUIRE(uniform.quads > 3000);
+    REQUIRE(near.quads > 3000);
+    REQUIRE(near.by_sign[0] > 100);
+    REQUIRE(near.by_sign[2] > 100);
+}
+
+// ---------------------------------------------------------------- the spread bound
+
+namespace {
+
+// a, b, c with one corner `far` from d along one axis, placed at `pos` in the
+// counter-clockwise triple; the other two close to d.
+std::array<MeshVertex, 3> far_triple(MeshVertex d, std::int64_t dc, std::int64_t dr, unsigned pos) {
+    const auto dcol = static_cast<std::int64_t>(d.col), drow = static_cast<std::int64_t>(d.row);
+    std::array<MeshVertex, 3> t{node(dcol + dc, drow + dr), node(dcol + 1, drow + 2), node(dcol - 2, drow + 1)};
+    if (iorient(t[0], t[1], t[2]) < 0)
+        std::swap(t[1], t[2]);
+    std::rotate(t.begin(), t.begin() + (3 - pos) % 3, t.end());
+    return t;
+}
+
+}  // namespace
+
+TEST_CASE("lattice_incircle answers at a spread of exactly 2^14 nodes and refuses at 2^14 + 1",
+          "[lattice_incircle][bound]") {
+    // QW2: every coordinate difference from d at most 2^14 nodes, in absolute
+    // value. For each corner position, each axis and each direction: at 2^14
+    // it answers, with the oracle's sign; one node further it refuses.
+    const std::int64_t spread = GENERATE(kBound, kBound + 1);
+    const unsigned pos = GENERATE(0u, 1u, 2u);
+    const auto [dc, dr] = GENERATE(std::pair{1, 0}, std::pair{-1, 0}, std::pair{0, 1}, std::pair{0, -1});
+    CAPTURE(spread, pos, dc, dr);
+    const double dx = 10.0;
+    const LatticeFrame f = lattice_frame(dx, dx, 1u << 16, 1u << 16);
+    const MeshVertex d = node(1 << 15, 1 << 15);
+    const auto [a, b, c] = far_triple(d, dc * spread, dr * spread, pos);
+    REQUIRE(iorient(a, b, c) > 0);
+    const auto got = lattice_incircle(a, b, c, d, f);
+    if (spread == kBound) {
+        REQUIRE(got.has_value());
+        REQUIRE(got == oracle(dx, dx, a, b, c, d));
+    } else {
+        REQUIRE_FALSE(got.has_value());
+    }
+}
+
+TEST_CASE("lattice_incircle measures the spread from d, not between the other corners",
+          "[lattice_incircle][bound]") {
+    // a and b are 2^14 from d on either side, so 2^15 apart; the bound is on
+    // differences from d (QW2), and this quad is within it.
+    const LatticeFrame f = lattice_frame(1.0, 1.0, 1u << 16, 1u << 16);
+    const MeshVertex d = node(1 << 15, 1 << 15);
+    MeshVertex a = node((1 << 15) + kBound, (1 << 15) + 3), b = node((1 << 15) - kBound, (1 << 15) + 1),
+               c = node(1 << 15, (1 << 15) - kBound);
+    if (iorient(a, b, c) < 0)
+        std::swap(a, b);
+    const auto got = lattice_incircle(a, b, c, d, f);
+    REQUIRE(got.has_value());
+    REQUIRE(got == oracle(1.0, 1.0, a, b, c, d));
+}
+
+// ---------------------------------------------------------------- orientation
+
+TEST_CASE("lattice_incircle takes the determinant on (col, -row), as the frame is oriented",
+          "[lattice_incircle][orientation]") {
+    // In the frame (col, -row): a (0, -2), b (2, -2), c (1, 0) turn
+    // counter-clockwise; their circle has centre (1, -1.25) and radius 1.25,
+    // and d (1, -1) is 0.25 from the centre, strictly inside. On (col, row)
+    // the four points are the mirror image: a, b, c turn clockwise, so the
+    // same determinant formula has the opposite sign and says Outside. That is
+    // the answer an implementation taking (col, row) gives here.
+    const LatticeFrame f = lattice_frame(1.0, 1.0, 16, 16);
+    const MeshVertex a = node(0, 2), b = node(2, 2), c = node(1, 0), d = node(1, 1);
+    REQUIRE(iorient(a, b, c) > 0);
+    REQUIRE(oracle(1.0, 1.0, a, b, c, d) == Incircle::Inside);
+    REQUIRE(lattice_incircle(a, b, c, d, f) == Incircle::Inside);
+    // And a point clearly outside, which (col, row) would call Inside.
+    const MeshVertex e = node(5, 5);
+    REQUIRE(oracle(1.0, 1.0, a, b, c, e) == Incircle::Outside);
+    REQUIRE(lattice_incircle(a, b, c, e, f) == Incircle::Outside);
+}
+
+// ---------------------------------------------------------------- refusals
+
+TEST_CASE("lattice_incircle refuses a quad with an off-node corner, in each position",
+          "[lattice_incircle][refuse]") {
+    // QW2 answers only when all four corners are nodes (MeshVertex::is_node).
+    // Off-node corners are the domain's arc vertices (16 R2) and 20b's feet.
+    const LatticeFrame f = lattice_frame(10.0, 10.0, kBench, kBench);
+    std::array<MeshVertex, 4> q{node(10, 12), node(12, 12), node(11, 10), node(11, 11)};
+    REQUIRE(iorient(q[0], q[1], q[2]) > 0);
+    REQUIRE(lattice_incircle(q[0], q[1], q[2], q[3], f).has_value());  // all nodes: answers
+    const std::size_t which = GENERATE(0u, 1u, 2u, 3u);
+    const auto [dc, dr] = GENERATE(std::pair{0.5, 0.0}, std::pair{0.0, 0.25},
+                                   std::pair{std::ldexp(1.0, -40), 0.0}, std::pair{0.0, std::ldexp(1.0, -40)});
+    CAPTURE(which, dc, dr);
+    q[which] = MeshVertex{q[which].col + dc, q[which].row + dr};
+    REQUIRE_FALSE(q[which].is_node());
+    REQUIRE(terrain::mesh::orient_sign(q[0], q[1], q[2]) > 0);  // still counter-clockwise
+    REQUIRE_FALSE(lattice_incircle(q[0], q[1], q[2], q[3], f).has_value());
+}
+
+TEST_CASE("lattice_incircle refuses every frame QW2 excludes", "[lattice_incircle][refuse]") {
+    // The quad is a plain all-node quad the benchmark frame answers; only the
+    // frame changes.
+    const MeshVertex a = node(3, 7), b = node(6, 7), c = node(4, 3), d = node(4, 5);
+    REQUIRE(iorient(a, b, c) > 0);
+    REQUIRE(lattice_incircle(a, b, c, d, lattice_frame(10.0, 10.0, 100, 100)).has_value());
+
+    const double inf = std::numeric_limits<double>::infinity();
+    const double nan = std::numeric_limits<double>::quiet_NaN();
+    SECTION("dx != dy: the frame is not the lattice times one constant") {
+        REQUIRE_FALSE(lattice_incircle(a, b, c, d, lattice_frame(1.0, 2.0, 100, 100)).has_value());
+        REQUIRE_FALSE(lattice_incircle(a, b, c, d, lattice_frame(10.0, 5.0, 100, 100)).has_value());
+        REQUIRE_FALSE(
+            lattice_incircle(a, b, c, d, lattice_frame(1.0, std::nextafter(1.0, 2.0), 100, 100)).has_value());
+    }
+    SECTION("dx not finite and positive") {
+        REQUIRE_FALSE(lattice_incircle(a, b, c, d, lattice_frame(0.0, 0.0, 100, 100)).has_value());
+        REQUIRE_FALSE(lattice_incircle(a, b, c, d, lattice_frame(-1.0, -1.0, 100, 100)).has_value());
+        REQUIRE_FALSE(lattice_incircle(a, b, c, d, lattice_frame(inf, inf, 100, 100)).has_value());
+        REQUIRE_FALSE(lattice_incircle(a, b, c, d, lattice_frame(nan, nan, 100, 100)).has_value());
+    }
+    SECTION("an inexact frame: some col * dx or row * dy of the grid rounds") {
+        // 3 * 0.1 != 0.3 exactly; the grid holds col and row 3.
+        REQUIRE(std::fma(3.0, 0.1, -(3.0 * 0.1)) != 0.0);
+        REQUIRE_FALSE(lattice_incircle(a, b, c, d, lattice_frame(0.1, 0.1, 100, 100)).has_value());
+        // 0.1 has 52 significant bits, so a grid two nodes wide is exact on
+        // that axis; the other axis is long enough to round. Rows and cols
+        // must both count. The quads lie inside those grids.
+        const MeshVertex ta = node(0, 5), tb = node(1, 5), tc = node(0, 3), td = node(1, 4);  // cols 0..1
+        const MeshVertex wa = node(3, 1), wb = node(6, 1), wc = node(4, 0), wd = node(5, 0);  // rows 0..1
+        REQUIRE(iorient(ta, tb, tc) > 0);
+        REQUIRE(iorient(wa, wb, wc) > 0);
+        REQUIRE_FALSE(lattice_incircle(ta, tb, tc, td, lattice_frame(0.1, 0.1, 100, 2)).has_value());
+        REQUIRE_FALSE(lattice_incircle(wa, wb, wc, wd, lattice_frame(0.1, 0.1, 2, 100)).has_value());
+        // A 53-bit dx: 3 * dx needs 54 bits.
+        const double full = 1.0 + std::ldexp(1.0, -52);
+        REQUIRE(std::fma(3.0, full, -(3.0 * full)) != 0.0);
+        REQUIRE_FALSE(lattice_incircle(a, b, c, d, lattice_frame(full, full, 100, 100)).has_value());
+    }
+    SECTION("a LatticeFrame built directly never enables the integer path") {
+        REQUIRE_FALSE(lattice_incircle(a, b, c, d, LatticeFrame{10.0, 10.0}).has_value());
+        REQUIRE_FALSE(lattice_incircle(a, b, c, d, LatticeFrame{1.0, 1.0}).has_value());
+    }
+}
+
+TEST_CASE("lattice_frame enables the integer path wherever QW2's sufficient condition holds",
+          "[lattice_incircle][frame]") {
+    // QW2: the frame is exact when the significant bits of dx plus
+    // bit_width(max(rows, cols) - 1) are at most 53. Inside that, it must
+    // answer (that is the speed-up, and must_flip's zero-kernel-call test
+    // below depends on it).
+    const MeshVertex a = node(0, 1), b = node(1, 1), c = node(0, 0), d = node(1, 0);  // unit square
+    REQUIRE(iorient(a, b, c) > 0);
+    // The benchmark: 3 + 13 = 16.
+    REQUIRE(lattice_incircle(a, b, c, d, lattice_frame(10.0, 10.0, kBench, kBench)) == Incircle::Cocircular);
+    // Exactly 53: a 52-bit dx on a 2 x 2 grid (bit_width(1) = 1).
+    const double dx52 = 1.0 + std::ldexp(1.0, -51);
+    REQUIRE(lattice_incircle(a, b, c, d, lattice_frame(dx52, dx52, 2, 2)) == Incircle::Cocircular);
+    // A power of two on a 2^20 grid: 1 + 20.
+    const double tiny = std::ldexp(1.0, -30);
+    REQUIRE(lattice_incircle(a, b, c, d, lattice_frame(tiny, tiny, 1u << 20, 1u << 20)) == Incircle::Cocircular);
+    // 0.1 on a 2 x 2 grid (52 + 1 = 53): col and row are 0 or 1, exact.
+    REQUIRE(lattice_incircle(a, b, c, d, lattice_frame(0.1, 0.1, 2, 2)) == Incircle::Cocircular);
+}
+
+TEST_CASE("lattice_frame refuses one bit past QW2's limit and answers at it, on both axes",
+          "[lattice_incircle][frame][limit]") {
+    // Both sides of "significant bits of dx + bit_width(max(rows, cols) - 1)
+    // <= 53". One bit past it the frame can be inexact, and is for these dx:
+    // a grid with col or row 3 and a 52-bit dx whose 3 * dx needs 54 bits.
+    // A limit taken one bit loose (e.g. bit_width(top - 1) - 1) accepts them.
+    const double dx51 = 1.5 + std::ldexp(1.0, -50);  // 51 significant bits
+    // 0.1 and 1.5 + 2^-51 both have 52 significant bits.
+    const double dx = GENERATE(0.1, 1.5 + std::ldexp(1.0, -51));
+    CAPTURE(dx);
+    REQUIRE(std::fma(3.0, dx, -(3.0 * dx)) != 0.0);  // col or row 3 rounds
+    // 52 + bit_width(3) = 54: refused, whichever axis carries the 4.
+    REQUIRE_FALSE(lattice_frame(dx, dx, 4, 4).integer());
+    REQUIRE_FALSE(lattice_frame(dx, dx, 4, 2).integer());
+    REQUIRE_FALSE(lattice_frame(dx, dx, 2, 4).integer());
+    // 52 + bit_width(1) = 53: at the limit, it must answer.
+    REQUIRE(lattice_frame(dx, dx, 2, 2).integer());
+    // 51 + bit_width(3) = 53 on the 4 x 4 grid itself: must answer.
+    REQUIRE(std::fma(3.0, dx51, -(3.0 * dx51)) == 0.0);
+    REQUIRE(lattice_frame(dx51, dx51, 4, 4).integer());
+    REQUIRE(lattice_frame(dx51, dx51, 4, 2).integer());
+    REQUIRE(lattice_frame(dx51, dx51, 2, 4).integer());
+}
+
+TEST_CASE("lattice_incircle never disagrees with DetriaExact on a frame one bit past the limit",
+          "[lattice_incircle][frame][limit]") {
+    // Every ordered quad of the 4 x 4 grid, a, b, c counter-clockwise, under
+    // lattice_frame(dx, dx, 4, 4) with dx one bit past the limit. Wherever it
+    // answers, it must give the oracle's sign; and the grid holds quads whose
+    // lattice sign differs from the frame's, so an answer there is wrong.
+    const double dx = GENERATE(0.1, 1.5 + std::ldexp(1.0, -51));
+    CAPTURE(dx);
+    const LatticeFrame f = lattice_frame(dx, dx, 4, 4);
+    const LatticeFrame exact = lattice_frame(1.0, 1.0, 4, 4);  // the lattice sign
+    std::vector<MeshVertex> nodes;
+    for (std::int64_t r = 0; r < 4; ++r)
+        for (std::int64_t c = 0; c < 4; ++c)
+            nodes.push_back(node(c, r));
+    std::size_t quads = 0, sign_differs = 0, wrong = 0;
+    std::string first;
+    for (const auto& a : nodes)
+        for (const auto& b : nodes)
+            for (const auto& c : nodes) {
+                if (iorient(a, b, c) <= 0)
+                    continue;
+                for (const auto& d : nodes) {
+                    if (d == a || d == b || d == c)
+                        continue;
+                    ++quads;
+                    const auto want = oracle(dx, dx, a, b, c, d);
+                    const auto lattice = lattice_incircle(a, b, c, d, exact);
+                    REQUIRE(lattice.has_value());
+                    sign_differs += want != lattice;
+                    const auto got = lattice_incircle(a, b, c, d, f);
+                    if (got && got != want && wrong++ == 0)
+                        first = describe(a) + " " + describe(b) + " " + describe(c) + " d " + describe(d);
+                }
+            }
+    CAPTURE(quads, sign_differs, wrong, first);
+    REQUIRE(sign_differs > 0);
+    REQUIRE(wrong == 0);
+}
+
+TEST_CASE("lattice_incircle refuses the reviewer's quad on 0.1 at 4 x 4, a lattice tie the frame calls Inside",
+          "[lattice_incircle][frame][limit]") {
+    // In (col, -row): a (0, 0), b (0, -1), c (3, 0), d (1, -2). The circle
+    // through a, b, c has centre (1.5, -0.5) and squared radius 2.5, and d is
+    // on it; at dx = 0.1 the rounded frame puts d strictly inside.
+    const MeshVertex a = node(0, 0), b = node(0, 1), c = node(3, 0), d = node(1, 2);
+    REQUIRE(iorient(a, b, c) > 0);
+    REQUIRE(lattice_incircle(a, b, c, d, lattice_frame(1.0, 1.0, 4, 4)) == Incircle::Cocircular);
+    REQUIRE(oracle(0.1, 0.1, a, b, c, d) == Incircle::Inside);
+    REQUIRE_FALSE(lattice_incircle(a, b, c, d, lattice_frame(0.1, 0.1, 4, 4)).has_value());
+}
+
+TEST_CASE("lattice_frame refuses a frame whose extent overflows, and answers where it does not",
+          "[lattice_incircle][frame][extent]") {
+    // dx = 2^1023 has one significant bit, so the bit count passes on any
+    // small grid; but (top - 1) * dx is infinite once top - 1 >= 2, and an
+    // infinite frame coordinate is not exact.
+    const double huge = std::ldexp(1.0, 1023);
+    REQUIRE(std::isinf(2.0 * huge));
+    REQUIRE(std::isfinite(1.0 * huge));
+    REQUIRE_FALSE(lattice_frame(huge, huge, 3, 3).integer());
+    REQUIRE_FALSE(lattice_frame(huge, huge, 3, 2).integer());
+    REQUIRE_FALSE(lattice_frame(huge, huge, 2, 3).integer());
+    const MeshVertex a = node(0, 1), b = node(1, 1), c = node(0, 0), d = node(1, 0);  // unit square
+    REQUIRE(iorient(a, b, c) > 0);
+    REQUIRE_FALSE(lattice_incircle(a, b, c, d, lattice_frame(huge, huge, 3, 3)).has_value());
+    // On a 2 x 2 grid every product is 0 or 2^1023, finite and exact.
+    REQUIRE(lattice_frame(huge, huge, 2, 2).integer());
+    REQUIRE(lattice_incircle(a, b, c, d, lattice_frame(huge, huge, 2, 2)) == Incircle::Cocircular);
+}
+
+TEST_CASE("lattice_incircle refuses a corner with a non-finite coordinate", "[lattice_incircle][refuse][nonfinite]") {
+    // MeshVertex::is_node is true for +-inf (floor(inf) == inf), so the node
+    // check does not stop them. A corner infinite alone gives an infinite
+    // difference from d, which the spread bound refuses; but a corner and d
+    // on the same infinity give inf - inf = NaN, which passes `fabs > bound`
+    // and reaches the int64 cast, undefined for NaN. When a, b, c and d all
+    // share it, no difference is refused and a value comes back. NaN is not a
+    // node (NaN != floor(NaN)). lattice_incircle is public, so it must refuse
+    // all of these. Unreachable from refine's grid mesh. The precondition
+    // (a, b, c counter-clockwise) has no meaning here; the refusal comes first.
+    const LatticeFrame f = lattice_frame(10.0, 10.0, kBench, kBench);
+    const std::array<MeshVertex, 4> base{node(10, 12), node(12, 12), node(11, 10), node(11, 11)};
+    REQUIRE(iorient(base[0], base[1], base[2]) > 0);
+    REQUIRE(lattice_incircle(base[0], base[1], base[2], base[3], f).has_value());
+    const double bad = GENERATE(std::numeric_limits<double>::infinity(), -std::numeric_limits<double>::infinity(),
+                                std::numeric_limits<double>::quiet_NaN());
+    const bool on_col = GENERATE(true, false);
+    const auto spoil = [&](MeshVertex v) { return on_col ? MeshVertex{bad, v.row} : MeshVertex{v.col, bad}; };
+    SECTION("one corner non-finite, in each position") {
+        for (std::size_t which = 0; which < 4; ++which) {
+            auto q = base;
+            q[which] = spoil(q[which]);
+            CAPTURE(which, bad, on_col);
+            REQUIRE_FALSE(lattice_incircle(q[0], q[1], q[2], q[3], f).has_value());
+        }
+    }
+    SECTION("a corner and d on the same non-finite value; the other two corners infinitely far") {
+        for (std::size_t which = 0; which < 3; ++which) {
+            auto q = base;
+            q[which] = spoil(q[which]);
+            q[3] = spoil(q[3]);
+            CAPTURE(which, bad, on_col);
+            REQUIRE_FALSE(lattice_incircle(q[0], q[1], q[2], q[3], f).has_value());
+        }
+    }
+    SECTION("every corner on the same non-finite value, so every difference from d is NaN") {
+        auto q = base;
+        for (auto& v : q)
+            v = spoil(v);
+        CAPTURE(bad, on_col);
+        REQUIRE_FALSE(lattice_incircle(q[0], q[1], q[2], q[3], f).has_value());
+    }
+}
+
+TEST_CASE("an inexact frame changes the answer on some lattice tie, so its refusal is load-bearing",
+          "[lattice_incircle][refuse]") {
+    // QW2's determinism argument: without the exact-frame condition the
+    // integer answer is the true lattice answer, and the rounded frame's
+    // answer can differ on a tie. Find one: a cocircular lattice quad on which
+    // DetriaExact at dx = dy = 0.1 does not say Cocircular. lattice_incircle
+    // must refuse it (it would otherwise say Cocircular and change a flip).
+    const double dx = 0.1;
+    const LatticeFrame f = lattice_frame(dx, dx, 100, 100);
+    std::optional<std::array<MeshVertex, 4>> found;
+    for (const std::int64_t r2 : {25, 50, 65, 85, 125})
+        for (const auto& q : quads_of(circle(40, 40, r2)))
+            for_each_rotation(q, [&](MeshVertex a, MeshVertex b, MeshVertex c, MeshVertex d) {
+                const auto o = oracle(dx, dx, a, b, c, d);
+                if (!found && o && *o != Incircle::Cocircular)
+                    found = std::array<MeshVertex, 4>{a, b, c, d};
+            });
+    REQUIRE(found.has_value());
+    const auto [a, b, c, d] = *found;
+    CAPTURE(describe(a), describe(b), describe(c), describe(d));
+    REQUIRE_FALSE(lattice_incircle(a, b, c, d, f).has_value());
+}
+
+// ---------------------------------------------------------------- must_flip
+
+namespace {
+
+// must_flip as of 93a8066, the oracle for decisions.
+bool reference_must_flip(const LatticeMesh& m, std::uint32_t t, unsigned e, const LatticeFrame& f) {
+    const auto u = m.neighbours(t)[e];
+    if (u == kNoNeighbour || m.is_constrained(t, e))
+        return false;
+    const auto& tri = m.triangles()[t];
+    unsigned j = 0;
+    while (m.triangles()[u][j] != tri[(e + 1) % 3])
+        ++j;
+    const auto v = m.vertices();
+    const Point2 a = fp(f.dx, f.dy, v[tri[e]]), b = fp(f.dx, f.dy, v[tri[(e + 1) % 3]]),
+                 c = fp(f.dx, f.dy, v[tri[(e + 2) % 3]]), d = fp(f.dx, f.dy, v[m.triangles()[u][(j + 2) % 3]]);
+    if (DefaultKernel::orient2d(a, b, c) == Orientation::CounterClockwise)
+        return DefaultKernel::incircle(a, b, c, d) == Incircle::Inside;
+    return DefaultKernel::orient2d(b, a, d) == Orientation::CounterClockwise
+        && DefaultKernel::incircle(b, a, d, c) == Incircle::Inside;
+}
+
+// DefaultKernel, counting its calls: must_flip asks it nothing when the
+// integer path answers.
+struct CountingKernel {
+    static inline std::size_t calls = 0;
+    static Orientation orient2d(const Point2& a, const Point2& b, const Point2& c) {
+        ++calls;
+        return DefaultKernel::orient2d(a, b, c);
+    }
+    static Incircle incircle(const Point2& a, const Point2& b, const Point2& c, const Point2& d) {
+        ++calls;
+        return DefaultKernel::incircle(a, b, c, d);
+    }
+};
+static_assert(terrain::pred::GeometryKernel<CountingKernel>);
+
+// An (n+1) x (n+1) grid of nodes `step` apart from (base, base), two triangles
+// per cell, the ring constrained. Mixed diagonals give ties both ways.
+LatticeMesh grid(std::uint32_t n, std::uint32_t step, std::uint32_t base) {
+    std::vector<LatticeVertex> v;
+    for (std::uint32_t r = 0; r <= n; ++r)
+        for (std::uint32_t c = 0; c <= n; ++c)
+            v.push_back(LatticeVertex{base + r * step, base + c * step});
+    const auto at = [n](std::uint32_t r, std::uint32_t c) { return r * (n + 1) + c; };
+    std::vector<TriangleIndices> t;
+    for (std::uint32_t r = 0; r < n; ++r)
+        for (std::uint32_t c = 0; c < n; ++c) {
+            const auto tl = at(r, c), tr = at(r, c + 1), bl = at(r + 1, c), br = at(r + 1, c + 1);
+            if ((r + c) % 2 == 0) {
+                t.push_back({tl, bl, br});
+                t.push_back({tl, br, tr});
+            } else {
+                t.push_back({tl, bl, tr});
+                t.push_back({bl, br, tr});
+            }
+        }
+    std::vector<std::uint8_t> bits;
+    std::vector<std::array<std::uint32_t, 3>> masks;
+    const std::uint32_t lo = base, hi = base + n * step;
+    for (const auto& tri : t) {
+        std::uint8_t b = 0;
+        for (unsigned k = 0; k < 3; ++k) {
+            const auto p = v[tri[k]], q = v[tri[(k + 1) % 3]];
+            if ((p.row == q.row && (p.row == lo || p.row == hi)) || (p.col == q.col && (p.col == lo || p.col == hi)))
+                b = static_cast<std::uint8_t>(b | (1u << k));
+        }
+        bits.push_back(b);
+        masks.push_back({b & 1u, (b >> 1) & 1u, (b >> 2) & 1u});
+    }
+    auto m = LatticeMesh::build(std::move(v), std::move(t), std::move(bits), std::move(masks));
+    REQUIRE(m.has_value());
+    return *m;
+}
+
+// Insert node p strictly inside a triangle or on an edge; false when p is a
+// vertex. Exact orientation on (col, -row), so off-node vertices are fine.
+bool insert_node(LatticeMesh& m, LatticeVertex p) {
+    const MeshVertex x{p};
+    for (std::uint32_t t = 0; t < m.triangle_count(); ++t) {
+        const auto& tri = m.triangles()[t];
+        const auto v = m.vertices();
+        std::array<int, 3> o{};
+        for (unsigned k = 0; k < 3; ++k)
+            o[k] = terrain::mesh::orient_sign(v[tri[k]], v[tri[(k + 1) % 3]], x);
+        if (o[0] < 0 || o[1] < 0 || o[2] < 0)
+            continue;
+        const int zeros = (o[0] == 0) + (o[1] == 0) + (o[2] == 0);
+        if (zeros >= 2)
+            return false;
+        if (zeros == 0)
+            m.split_inside(t, p);
+        else
+            m.split_edge(t, o[0] == 0 ? 0u : o[1] == 0 ? 1u : 2u, x);
+        return true;
+    }
+    return false;
+}
+
+// Split a constrained ring edge at an off-node point strictly inside it (a
+// foot, as 20b inserts them). false if no ring edge is long enough.
+bool insert_foot(LatticeMesh& m, std::mt19937& rng) {
+    const auto n = m.triangle_count();
+    const auto start = static_cast<std::uint32_t>(rng() % n);
+    for (std::uint32_t i = 0; i < n; ++i) {
+        const auto t = (start + i) % n;
+        for (unsigned k = 0; k < 3; ++k) {
+            if (!m.is_constrained(t, k))
+                continue;
+            const auto p = m.vertices()[m.triangles()[t][k]], q = m.vertices()[m.triangles()[t][(k + 1) % 3]];
+            const double len = std::fabs(p.col - q.col) + std::fabs(p.row - q.row);  // axis-aligned
+            if (len < 2.0)
+                continue;
+            const double s = (std::floor(len / 2.0) + 0.5) / len;  // off-node, strictly inside
+            m.split_edge(t, k, MeshVertex{p.col + s * (q.col - p.col), p.row + s * (q.row - p.row)});
+            return true;
+        }
+    }
+    return false;
+}
+
+struct Decisions {
+    std::size_t edges = 0, flips = 0, ties = 0, differ = 0;
+    std::string first;
+};
+
+// Every (t, e): must_flip under the lattice frame against today's must_flip.
+void compare_decisions(const LatticeMesh& m, double dx, double dy, std::size_t grid_n, Decisions& out) {
+    const LatticeFrame with = lattice_frame(dx, dy, grid_n, grid_n);
+    const LatticeFrame without{dx, dy};
+    for (std::uint32_t t = 0; t < m.triangle_count(); ++t)
+        for (unsigned e = 0; e < 3; ++e) {
+            if (m.neighbours(t)[e] == kNoNeighbour || m.is_constrained(t, e))
+                continue;
+            ++out.edges;
+            const bool want = reference_must_flip(m, t, e, without);
+            const bool got = terrain::mesh::detail::must_flip<DefaultKernel>(m, t, e, with);
+            out.flips += want;
+            const auto& tri = m.triangles()[t];
+            const auto u = m.neighbours(t)[e];
+            unsigned j = 0;
+            while (m.triangles()[u][j] != tri[(e + 1) % 3])
+                ++j;
+            const auto v = m.vertices();
+            out.ties += DefaultKernel::incircle(fp(dx, dy, v[tri[0]]), fp(dx, dy, v[tri[1]]), fp(dx, dy, v[tri[2]]),
+                                                fp(dx, dy, v[m.triangles()[u][(j + 2) % 3]]))
+                     == Incircle::Cocircular;
+            if (got != want && out.differ++ == 0)
+                out.first = "t " + std::to_string(t) + " e " + std::to_string(e) + ": got "
+                          + std::to_string(got) + ", today " + std::to_string(want);
+        }
+}
+
+struct FrameCase {
+    double dx, dy;
+    bool integer;  // whether lattice_frame enables the integer path on it
+};
+const std::array<FrameCase, 5> kFrameCases{FrameCase{1.0, 1.0, true}, FrameCase{10.0, 10.0, true},
+                                           FrameCase{0.5, 0.5, true}, FrameCase{0.1, 0.1, false},
+                                           FrameCase{10.0, 5.0, false}};
+
+}  // namespace
+
+TEST_CASE("must_flip decides as today on random lattice meshes, with ties and off-node feet",
+          "[lattice_incircle][must_flip]") {
+    // Random nodes inserted into a mixed-diagonal grid without legalising, so
+    // many edges must flip and many are ties; every few insertions a foot
+    // splits a ring edge off the lattice. After every insertion, every
+    // interior unconstrained edge is decided by must_flip under
+    // lattice_frame(dx, dy, ...) and by a copy of today's must_flip under
+    // LatticeFrame{dx, dy}: the decisions must be equal. Near the origin and
+    // near the benchmark's far corner.
+    const std::uint32_t seed = GENERATE(range(1u, 7u));
+    const std::size_t frame = GENERATE(range(std::size_t{0}, kFrameCases.size()));
+    const std::uint32_t base = GENERATE(0u, 5000u);
+    const auto [dx, dy, integer] = kFrameCases[frame];
+    CAPTURE(seed, dx, dy, base);
+    auto m = grid(4, 6, base);  // nodes base .. base + 24
+    std::mt19937 rng{seed};
+    Decisions dec;
+    std::size_t inserted = 0, feet = 0;
+    for (int attempt = 0; attempt < 120; ++attempt) {
+        if (attempt % 15 == 7 && insert_foot(m, rng))
+            ++feet;
+        const LatticeVertex p{base + static_cast<std::uint32_t>(rng() % 25),
+                              base + static_cast<std::uint32_t>(rng() % 25)};
+        if (!insert_node(m, p))
+            continue;
+        ++inserted;
+        compare_decisions(m, dx, dy, kBench, dec);
+    }
+    CAPTURE(dec.edges, dec.flips, dec.ties, dec.first);
+    REQUIRE(dec.differ == 0);
+    REQUIRE(inserted >= 40);
+    REQUIRE(feet >= 3);
+    REQUIRE(dec.flips >= 100);
+    REQUIRE(dec.ties >= 100);
+    REQUIRE(dec.edges - dec.flips - dec.ties >= 100);
+}
+
+TEST_CASE("legalise_all under lattice_frame makes the same flips, writes and mesh as under LatticeFrame{dx, dy}",
+          "[lattice_incircle][must_flip]") {
+    // The mesh-level form of bit-identity: the whole Lawson run, not one
+    // decision at a time. LatticeFrame{dx, dy} never enables the integer path,
+    // so it is today's run.
+    const std::uint32_t seed = GENERATE(range(1u, 7u));
+    const std::size_t frame = GENERATE(range(std::size_t{0}, kFrameCases.size()));
+    const auto [dx, dy, integer] = kFrameCases[frame];
+    CAPTURE(seed, dx, dy);
+    auto m = grid(5, 5, 0);
+    std::mt19937 rng{seed};
+    for (int attempt = 0; attempt < 150; ++attempt) {
+        if (attempt % 20 == 3)
+            insert_foot(m, rng);
+        insert_node(m, LatticeVertex{static_cast<std::uint32_t>(rng() % 26), static_cast<std::uint32_t>(rng() % 26)});
+    }
+    LatticeMesh today = m;
+    std::vector<std::uint32_t> w_today, w_new;
+    const auto f_today = terrain::mesh::legalise_all<DefaultKernel>(
+        today, LatticeFrame{dx, dy}, [&](std::uint32_t s) { w_today.push_back(s); });
+    const auto f_new = terrain::mesh::legalise_all<DefaultKernel>(
+        m, lattice_frame(dx, dy, 26, 26), [&](std::uint32_t s) { w_new.push_back(s); });
+    REQUIRE(f_today >= 20);
+    REQUIRE(f_new == f_today);
+    REQUIRE(w_new == w_today);
+    REQUIRE(m.triangle_count() == today.triangle_count());
+    for (std::uint32_t t = 0; t < m.triangle_count(); ++t) {
+        CAPTURE(t);
+        REQUIRE(m.triangles()[t] == today.triangles()[t]);
+        REQUIRE(m.neighbours(t) == today.neighbours(t));
+    }
+}
+
+TEST_CASE("must_flip asks the kernel nothing when lattice_incircle answers, and today's questions when it refuses",
+          "[lattice_incircle][must_flip]") {
+    // QW2: lattice_incircle is called at the top of must_flip, and when it
+    // answers, must_flip also skips the frame orient2d (the mesh triangle is
+    // counter-clockwise in integers, and so in an exact frame). That is the
+    // whole speed-up, so it is pinned: zero kernel calls on an all-node quad
+    // under an enabling frame; kernel calls under a direct LatticeFrame, an
+    // excluded frame, or with an off-node corner.
+    auto m = grid(4, 3, 0);
+    std::mt19937 rng{7};
+    for (int i = 0; i < 40; ++i)
+        insert_node(m, LatticeVertex{static_cast<std::uint32_t>(rng() % 13), static_cast<std::uint32_t>(rng() % 13)});
+    const auto run = [&m](const LatticeFrame& f, bool node_quads_only, std::size_t min_edges) {
+        std::size_t asked = 0, edges = 0;
+        for (std::uint32_t t = 0; t < m.triangle_count(); ++t)
+            for (unsigned e = 0; e < 3; ++e) {
+                const auto u = m.neighbours(t)[e];
+                if (u == kNoNeighbour || m.is_constrained(t, e))
+                    continue;
+                bool nodes = true;
+                for (const auto i : m.triangles()[t])
+                    nodes = nodes && m.vertices()[i].is_node();
+                for (const auto i : m.triangles()[u])
+                    nodes = nodes && m.vertices()[i].is_node();
+                if (nodes != node_quads_only)
+                    continue;
+                ++edges;
+                CountingKernel::calls = 0;
+                const bool got = terrain::mesh::detail::must_flip<CountingKernel>(m, t, e, f);
+                REQUIRE(got == reference_must_flip(m, t, e, LatticeFrame{f.dx, f.dy}));
+                asked += CountingKernel::calls;
+            }
+        REQUIRE(edges >= min_edges);
+        return asked;
+    };
+    REQUIRE(run(lattice_frame(10.0, 10.0, kBench, kBench), true, 20) == 0);
+    REQUIRE(run(LatticeFrame{10.0, 10.0}, true, 20) > 0);
+    REQUIRE(run(lattice_frame(0.1, 0.1, 100, 100), true, 20) > 0);
+    REQUIRE(run(lattice_frame(10.0, 5.0, kBench, kBench), true, 20) > 0);
+    for (int i = 0; i < 6; ++i)
+        REQUIRE(insert_foot(m, rng));
+    REQUIRE(run(lattice_frame(10.0, 10.0, kBench, kBench), false, 3) > 0);
+}