docs: sync CLAUDE.md and READMEs with codex lanes and the tmux run surface

Add the codex-lane facts and doc-sync marker to CLAUDE.md, document the
per-lane tool picker in README/README.vi, and bring server/README.md and
client/README.md's API and routing tables up to date with the tmux-backed
run endpoints and the /workspace route.

CLAUDE.md

@@ -1,5 +1,7 @@
 # Claude Code Working Guide
 
+_Last synced: d884d19 (2026-08-26)._ A change that makes any statement in this file, `AGENTS.md`, or a README wrong — a moved path, a new or changed extension point, a retired rule — updates that file in the same change-set and bumps this marker.
+
 ## Project mission
 - Maintain a reliable local-first dashboard for Claude Code session monitoring.
 - Preserve real-time behavior (hooks -> API -> SQLite -> WebSocket -> UI).
@@ -33,6 +35,7 @@ A **lane** is a durable unit of parallel agent work — one working directory, m
 - Liveness: a silent **watcher** (stage matching `/watch|poll/`) is dead after `LANE_DEAD_SEC` seconds (default 300); a silent **idle** lane is at rest, not dead.
 - **Never `rm -rf` a lane.** Destructive lane operations (`reset`, `remove`, `purge`) go through `server/lib/worktree.js`/`server/lib/lanes.js`, never a raw filesystem delete — that's what keeps the three-check destroy guard and the DB bookkeeping in lockstep.
 - **Never build a git command as a shell string.** Lane git operations use `execFile("git", [...args])` with an explicit argument array (see `server/lib/worktree.js`'s `git()` helper), never a concatenated/interpolated string passed to a shell.
+- **A lane's `tool` is a lane fact, immutable, and never read from a request.** `lanes.tool` (`claude` | `codex`, default `claude`) is set by `createLane` only — it is deliberately absent from `PATCHABLE`, so no PATCH can flip the CLI under a live pane. Every spawn path (`POST /api/lanes/:id/start`, `POST /api/run`) takes the tool from the lane row and ignores whatever the body says; the run panel's Codex-only fields (`sandbox`, `approval`, `resumeLast`) are validated against the fixed allowlists in `server/lib/pty-run.js` before they reach an argv, and the bypass flags (`--dangerously-bypass-approvals-and-sandbox`, `--dangerously-bypass-hook-trust`) are never emitted. `GET /api/run/binary?tool=` resolves the probed binary name from that same allowlist — the query string reaches `which`.
 - **Adopted lanes are not destroyable.** A lane with `kind === "adopted"` may never have its worktree reset or removed; "removing" one only drops the dashboard's own record of it. This is enforced in code (`assertDestroyable`'s first check) — do not add a path that bypasses it.
 
 See `docs/LANES.md` for full guide: stage reporting, custom templates, lane actions, and the five node states.