LITS

The name is redundant. LITS is named after the four tetrominoes you may shade — L, I, T, S — with the fifth, O, forbidden. But O is exactly the tetromino that is a 2×2 square, and a fully shaded 2×2 square is already illegal on its own. Enumerate every connected 4-subset of a 5×5 board — 228 of them, 16 containing a 2×2 square — and the square-containing ones are precisely the ones that classify as O (a test does exactly this). Drop the "L, I, T, S only" clause, keep the square rule, and nothing changes. The letters in the name are a consequence, not a constraint.

Like Norinori before it, nothing is written on the board — the partition is the puzzle — so the natural variable is the region, whose one decision is which tetromino it holds. Its domain is its tetromino placements, and under that lifting the signature rule — congruent shapes must not stare at each other across a border — collapses into an ordinary binary constraint between adjacent regions: textbook AC-3. What does not decompose is connectivity; it gets a global propagator that prunes placements stranded outside the one component every region can still reach. Three rule sets ship — local (count to four, complete no square), tetro (plus placement arcs plus connectivity), probe (singleton consistency on top). Measured over unique boards generated with no solvability filter, so the last column is not circular:

A 900-position property test says the same thing from the other side: the lifting decided more cells than the local rules in ? of 900 positions, +? cells on average, and the local rules never once beat it.

Density is destiny. A random partition is essentially never a LITS puzzle — but for the opposite reason to Norinori. Norinori's random partitions mostly had no solution at all; LITS random partitions (region sizes 6–10) are about 90% multi-solution, about 10% unsolvable, and exactly-one came up zero times in hundreds of trials. Every region holds exactly four shaded cells, so mean region size is pinned at 4 / shaded fraction, and big regions mean big domains and hopelessly loose boards. Greedy tetromino accretion jams around 55% shading; a randomised backtracking packing reaches ~65% — regions of six and a half cells instead of nine. The unshaded cells then flood into whichever neighbouring region they enlarge the least, measured in placements.

The last rivals standing are slides. Autopsy a nearly-unique board and its rival solutions form a lattice of independent one-region slides — a tetromino pivoting in place while every other region stays put (8 rivals = 2³ slides). Call the number of such alternatives a region's local slack; it is computable per region with no global search. The generator descends total slack to zero by walking white cells between regions — a move that can never hurt the intended solution, since no shaded cell ever changes region — peeling cells away from slacky regions when nothing improves: a smaller region has fewer placements and nowhere left to slide. A raw carve admits a median of ? solutions; slack descent closes that to one.

Three rules, all load-bearing. Take the shipped 6×6 boards and switch one rule off in the validator: dropping the congruence rule leaves ? of 16 boards unique; dropping connectivity leaves ?; dropping the square rule — which also re-admits the O tetromino — leaves ?.

Soundness is pinned by two brute-force counters that share no code with the propagators — one walking the cells, one walking the regions — plus an independent validator that re-derives every shape from scratch; all counters must agree on solution counts, and a disagreement is how an unsound propagator gets caught. A rule set that finishes a board with no search is also a uniqueness certificate. 65 tests.