Lighthouse

Build a tiny browser game called "Lighthouse".

Game spec
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A fog-bound lighthouse and a rotating beam you control. Ships enter from the bottom of the screen and sail upward toward a safe channel between two rock banks; the fog is thick, so a ship can only see the rocks it is inside the beam of. Steer the beam with the arrow keys, A/D, or a mouse drag (a touch drag works too) and the lamp sweeps with a little angular momentum, so you cannot snap it from one ship to another instantly. A ship lit by the beam corrects its heading toward the channel centreline; a ship left in the dark drifts with the current and will find a rock. Each ship delivered pays by how close it came to the rocks and how little time it spent unlit, and a ship lost on a rock costs a flare, three ending the night. Keep the beam on the rocks too long and the lamp overheats, dimming until it cools, so you cannot just park the light. Waves of traffic grow denser and the fog thickens as the night wears on; a short foghorn (Space) startles a drifting ship back toward the light at the cost of a few seconds. Difficulty sets ship count, fog density, current strength and lamp heat. Warm look: a slate-and-cream tower, a warm amber cone with dust motes, black rock silhouettes with foam collars, a starless grey fog gradient, and ships as small lit hulls.

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.