Stream the docs index docbot generates over the WebSocket

[DavertMik]

Docbot was already half-wired to the --ws stream: its bin calls remote.registerOption, and it inherits every frame the shared modules emit — hello, log, activity, state (state-manager), screenshot (action.ts), research (researcher cache) and ask through the execution controller. But every tag('data') call site lives in src/; nothing under boat/ emitted a frame of its own, so docs/index.md — the artifact the whole run produces — never reached a listener.

Changes

• saveIndex() emits a docs frame (boat/doc-collector/src/docbot.ts) — keeps the rendered markdown in a local, writes it, then tag('data').log('docs', { path, content }), mirroring writeReport() in src/ai/session-analyst.ts. Same {path, content} payload as report, but a new kind rather than reusing it: a UI keeps the last frame per type, and the analyst's report and the spec index are unrelated artifacts.
• The exit path closes the remote (boat/doc-collector/src/cli.ts) — docs collect ended on a bare process.exit(0), while the main CLI routes exits through showStatsAndExit → remote.close(code), which sends result and flushes. Docbot did neither, so a listener never learned the run ended. This is a prerequisite, not a separate nit: saveIndex() fires at the very last moment before exit, so the new frame was the one most likely to die in the socket buffer. close() early-returns when unattached, so no guard is needed.
• docs/reference/websocket.md — added the docs row to the frame table, which is the contract listeners actually read.

Scope is index.md only. Per-page pages/*.md and state-diagram.mmd stay unstreamed, matching the one-shot-at-end shape of report.

Not fixed here

boat/api-tester has the same missing remote.close() on exit, so apibot runs never send result either. Left alone to keep this PR to its stated purpose, and because that boat has unrelated work in flight.

Testing

bun run format, bun run lint, and bun test tests/unit/doc-collector.test.ts (45 pass) are green.

🤖 Generated with Claude Code

https://claude.ai/code/session_01Twc8quqSCEZNzPn4sK1o55