solver: prune unplaceable large squares

A skyline may retain enough total empty area while no longer containing a box for its largest remaining square. Scan for the required consecutive low columns and reject such monotonic dead states.

Keep each pruning combination independently measurable and record the small public-path gain, the direct-order-9 regression, and the rejected periodic schedule.

Tests: Debug CTest (13 passed)

Tests: ASan+UBSan CTest (13 passed)

Refs: #15

BENCHMARKING.md

@@ -21,8 +21,9 @@ and `--candidate-order`.  The choices are `ascending`, `descending`, and
 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.
-The production default also applies the valley-capacity rule described below.
-Pass `--no-pruning` to obtain an otherwise identical unpruned search.
+The production default also applies the pruning rules described below.
+Use `--pruning valley-capacity`, `--pruning large-square`, or
+`--no-pruning` to measure each rule alone or obtain an unpruned search.
 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:
 
@@ -107,6 +108,55 @@ The rule reduced nodes by 7.8% for order 7 and 7.1% for order 8.  Its low
 per-node cost also reduced median counted solve time by 10.3% and 9.9%,
 respectively, so it remains enabled by default.
 
+## Remaining-large-square pruning
+
+For a remaining side `k`, a skyline can contain an empty `k`-by-`k` box only
+if it has `k` consecutive columns whose filled heights are no greater than the
+board height minus `k`.  A single linear scan tracks qualifying consecutive
+columns.  If no such run exists, the square cannot be placed and the state is
+infeasible even when its total empty area is sufficient.
+
+Empty-box feasibility is monotonic in the side length: a box which fits the
+largest remaining square also fits every smaller remaining size.  The solver
+therefore invokes only one `O(board width)` scan per surviving node, after the
+cheaper valley-capacity check.  It does not attempt an unproven multiplicity
+bound.  `--pruning large-square` measures the rule independently;
+`--pruning valley-capacity` provides the production baseline without it.
+Instrumented output records `large_square_checks` and
+`large_square_prunes`.
+
+Measurements used the issue #15 working tree based on commit `e27427d`, Apple
+Clang 21.0.0, `-O3 -DNDEBUG`, macOS arm64, one worker, one warm-up, and seven
+measured repetitions.  Counter-free and counted runs were interleaved; the
+table reports the counter-free production instantiation.  The baseline keeps
+valley-capacity pruning enabled, so it isolates the new rule.  All results
+passed independent validation and counters were stable:
+
+| Search | Large-square check | Median solve | Nodes | Checks | Prunes |
+| --- | --- | ---: | ---: | ---: | ---: |
+| Order 7 exhaustive | enabled | 0.946 s | 13,221,239 | 6,163,390 | 227,321 |
+| Order 7 exhaustive | disabled | 0.947 s | 13,833,048 | 0 | 0 |
+| Order 8 first solution | enabled | 0.195 s | 2,597,678 | 1,063,472 | 38,177 |
+| Order 8 first solution | disabled | 0.197 s | 2,724,096 | 0 | 0 |
+| Order 9 direct | enabled | 1.225 s | 14,840,146 | 5,592,843 | 54,900 |
+| Order 9 direct | disabled | 1.208 s | 14,995,127 | 0 | 0 |
+
+The rule reduced nodes by 4.4% for order 7, 4.6% for order 8, and 1.0% for
+direct order 9.  Counter-free median time improved by 0.12% and 0.89% on the
+two distinct searches in the public benchmark set.  Public order 9 constructs
+from the improving order-8 search.  Direct order 9, which bypasses that public
+route, regressed by 1.39%; it remains documented as a caution for future
+policy tuning.
+
+Checking only every fourth placement depth was also measured.  It retained
+fewer prunes and was slower than checking every surviving node by 2.2% for
+order 7, 2.3% for order 8, and 0.6% for direct order 9, so the periodic
+schedule was rejected.  Incremental maintenance would need additional
+per-size window state and undo logic for an `O(W)` scan whose public net cost
+is already recovered; it was not added without evidence that the complexity
+would improve elapsed time.  The simple every-node scan remains enabled by
+default for the measured public benchmark benefit.
+
 ## D4 board symmetry
 
 Every solution contains exactly one 1-by-1 square.  Rotations and reflections