solver: break board dihedral symmetry

Constrain the unique unit square to a closed D4 fundamental region once its placement is known. This preserves one representative of every board-orientation orbit without assigning identities to repeated squares.

Keep a symmetry-disabled benchmark path, document the proof and measurements, and cover generic, diagonal, midline, corner, and centre orbits.

Tests: Release, Debug, ASan, and UBSan CTest (11 passed each)

Refs: #3

BENCHMARKING.md

@@ -18,6 +18,9 @@ the probe renders into an in-memory stream so grids do not perturb terminal I/O.
 Override the policy with `--orders`, `--warmup`, `--repetitions`, `--timeout`,
 and `--candidate-order`.  The choices are `ascending`, `descending`, and
 `best-fit`; ascending candidate sizes are the production default.
+The production default also removes equivalent D4 board orientations by
+constraining the unique unit square.  Pass `--no-symmetry` to obtain an
+otherwise identical unconstrained baseline.
 Order 9 uses the constructive odd-order path, searching order 8 and then tiling
 the enlarged border, so it is suitable for normal local benchmarking:
 
@@ -69,6 +72,50 @@ Do not use wall-clock thresholds as correctness checks.  Keep the generated
 JSON outside version control unless it is being deliberately added as a named
 comparison baseline.
 
+## D4 board symmetry
+
+Every solution contains exactly one 1-by-1 square.  Rotations and reflections
+of the whole board preserve square sizes, multiplicities, and coverage, so the
+unit square can select the orientation without assigning identities to any of
+the repeated larger squares.  For a board of width `W`, the solver accepts the
+unit square only in the closed fundamental triangle
+`x <= y <= floor((W - 1) / 2)`.  Reflecting a cell toward the left edge,
+swapping its coordinates if necessary, and reflecting toward the top edge
+maps every D4 orbit into this triangle.  Closed diagonal and midline
+boundaries retain the smaller orbits of symmetric cells.
+
+The check is made only when the skyline search is ready to place the unit
+square, so it never rejects a partial state before the square's position is
+decidable.  The implementation remains compile-time counter-free in normal
+solver calls and keeps the single-threaded deterministic search policy.
+Instrumented runs count examined and rejected unit-square placements as prune
+checks and hits.
+
+Measurements used the issue #3 dirty working tree based on commit `74bde26`,
+Apple Clang 21.0.0, `-O3 -DNDEBUG`, macOS arm64, one worker, one warm-up, and
+five sequential measured repetitions.  All results passed the benchmark's
+independent cell-coverage validator and counts were stable:
+
+| Route | D4 constraint | Median solve (range) | Nodes | Prune hits |
+| --- | --- | ---: | ---: | ---: |
+| Order 8 public | enabled | 0.245 s (0.244-0.248 s) | 2,931,203 | 1,329,567 |
+| Order 8 public | disabled | 0.679 s (0.659-0.709 s) | 7,735,369 | 0 |
+| Order 9 direct | enabled | 1.705 s (1.671-1.740 s) | 16,231,918 | 6,755,171 |
+| Order 9 direct | disabled | 4.542 s (4.482-4.662 s) | 45,840,266 | 0 |
+
+Thus the constraint reduced nodes by 62% for order 8 and 65% for direct order
+9; counted median solve time fell by 64% and 62%, respectively.
+
+A public order-10 probe with the D4 constraint, ascending policy, and a
+45-second per-process bound did not complete.  Public order 11 first performs
+that identical order-10 core search and only then adds its inexpensive odd
+border, so adding either order to routine correctness tests would duplicate
+the same unresolved bottleneck.  They remain useful opt-in heavyweight
+benchmark targets with explicit timeouts.  A direct order-11 benchmark is a
+different experiment: it bypasses the public odd construction and searches
+the larger core itself, so it must not be presented as public order-11
+performance.
+
 ## Smallest-valley skyline
 
 The solver stores one filled height per board column instead of one value per