Paper Boat

Build a tiny browser game called "Paper Boat".

Game spec
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A paper boat folded from a single sheet, and the boat has to float. The fold is the puzzle: the sheet is shown creased with numbered valley and mountain folds, and you fold them in an order that keeps the shape physically possible — a fold is only legal if the paper occupying that crease is currently flat and unobstructed, so folding the wrong crease first makes later folds refuse. Drag a crease to fold it (or click it, or use the arrows plus Enter), and the paper folds in real time with the flaps moving, so a bad order is visible immediately rather than hidden behind a button. Once the boat exists you set it in the water and it must hold a payload of pebbles: a hull folded with an uneven crease list rides with a tilted gunwale and swamps early, while a true fold carries more. A boat pays by payload carried, folds used and time, then the next sheet has more creases and a tighter water; capsizing three times ends the run, and difficulty sets crease count, water roughness and the payload step. Boathouse look: a pale sheet with a blue crease pattern and warm paper fibre, a green dockside pond with lily pads and ripples, a jar of pebbles, a wooden folding bench with a bone folder, and water lapping plus a paper crease in WebAudio.

Hard constraints — the game will be embedded in a sandboxed iframe on a static gallery site, and a headless browser will screenshot it:

1. Output exactly ONE self-contained HTML file. All CSS and JS inline. Zero external requests: no CDNs, no external fonts, images, or sounds, no fetch/XHR/WebSocket. Any sound must be synthesized with WebAudio.
2. The iframe sandbox is "allow-scripts allow-pointer-lock" (plus an autoplay permission). So: no localStorage, cookies, or indexedDB (opaque origin — access throws); no popups, window.open, or navigation; no access to the parent page; alert()/prompt()/confirm() are silently blocked.
3. The playfield must be visible within 800 ms of load — that is when the screenshot is taken. No dead white screen. If there is a start screen, make it visually rich.
4. Controls: keyboard AND mouse must both work, plus on-screen touch buttons for phones. Call preventDefault() on game keys (arrows, space) so the page never scrolls.
5. The view is 16:9 and must scale to fill the viewport with no overflow or scrollbars.
6. Show a visible score. If the genre has a losing state, show a game-over screen with a restart (button and a key). Idle/clicker games skip game over but still need clear numeric feedback.
7. If you track a score, report it to the arcade shell so the player's best is remembered between visits: on every score change, call parent.postMessage({ source: "ai-arcade", type: "score", score: <number> }, "*"). It is safe to call it no matter what the parent does with it.
8. Keep it tiny: aim for under 400 lines, 60 fps, no frameworks. That 400-line target does not apply to 3D games — true 3D (a camera and perspective-projected world) or pseudo-3D (projected road/track, depth-sorted sprites) — which need their own software renderer and typically land at 800–1400 lines. The rest of this constraint still binds every game: 60 fps, no frameworks, no external assets, and no padding to reach a length.
9. Difficulty: the arcade shell passes the player's chosen difficulty as a URL query parameter. Read it with new URLSearchParams(location.search).get("difficulty") — the value is "easy", "medium", or "hard" (missing or unknown means "medium"). Make the three levels meaningfully different — speed, lives, enemy pressure, reaction time, whatever fits the genre; clicker/idle games with no losing state use starting resources and upgrade cost curves instead. Show the active level in the HUD. Without the parameter the game must be fully playable at its default (medium) tuning.

Return only the complete HTML file, no commentary.