Conversation sharing is supported

Shared conversations are one of the highest-leverage organic growth loops an AI product has — every shared link is both a demo of capability and a referral. ChatGPT's share feature, Claude's share feature, and Poe's share feature all exist because shared chats convert viewers to signups at rates that paid channels cannot match. Without a native share path, users screenshot their conversations (which loses formatting and the ability to continue) or switch to a competitor whose share link works.

Add a share button that POSTs to /api/conversations/[id]/share, flips an is_public flag on the row, returns a slug, and copies ${origin}/share/${slug} to the clipboard. Gate the public read route on the flag and strip sensitive metadata on serialization. Implement the route at app/api/conversations/[id]/share/route.ts.

await navigator.clipboard.writeText(`${location.origin}/share/${slug}`)