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

benchmarks/run.py

@@ -79,11 +79,20 @@ def environment(binary):
     }
 
 
-def run_once(binary, order, mode, search_policy, search_route, timeout):
+def run_once(
+    binary, order, mode, search_policy, search_route, symmetry, timeout
+):
     started = time.monotonic()
     try:
         process = subprocess.run(
-            [str(binary), str(order), mode, search_policy, search_route],
+            [
+                str(binary),
+                str(order),
+                mode,
+                search_policy,
+                search_route,
+                symmetry,
+            ],
             check=False,
             capture_output=True,
             text=True,
@@ -223,6 +232,11 @@ def main():
         default="ascending",
     )
     parser.add_argument("--direct-search", action="store_true")
+    parser.add_argument(
+        "--no-symmetry",
+        action="store_true",
+        help="disable unique-unit-square D4 symmetry breaking",
+    )
     parser.add_argument(
         "--measure-overhead",
         action="store_true",
@@ -254,6 +268,7 @@ def main():
                     mode,
                     args.search_policy,
                     "direct" if args.direct_search else "public",
+                    "none" if args.no_symmetry else "d4",
                     args.timeout,
                 )
         for trial in range(args.repetitions):
@@ -265,6 +280,7 @@ def main():
                         mode,
                         args.search_policy,
                         "direct" if args.direct_search else "public",
+                        "none" if args.no_symmetry else "d4",
                         args.timeout,
                     )
                 )
@@ -293,6 +309,7 @@ def main():
             "per_run_timeout_seconds": args.timeout,
             "candidate_order": args.search_policy,
             "search_route": "direct" if args.direct_search else "public",
+            "symmetry_breaking": "none" if args.no_symmetry else "d4",
             "stdout": "captured; rendered grid suppressed by probe",
         },
         "cases": cases,