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.

README.md

@@ -17,10 +17,11 @@ Internal build — all rights reserved.
 - **Stage detection.** The stage is inferred from the tool stream, so a session
   that never calls `ccam stage` still shows progress — rendered dashed amber and
   never as done, because an inference is not evidence.
-- **Run Claude from the browser.** A real terminal (tmux + a real PTY,
-  rendered with xterm.js) attached to a lane's directory — the exact TUI you'd
-  see locally, fully interactive, resumable, and attachable from a real
-  terminal too via `ccam lanes shell`.
+- **Run Claude Code or Codex from the browser.** A real terminal (tmux + a real
+  PTY, rendered with xterm.js) attached to a lane's directory — the exact TUI
+  you'd see locally, fully interactive, resumable, and attachable from a real
+  terminal too via `ccam lanes shell`. Which CLI it starts is the lane's own
+  setting.
 - **Analytics (in-dashboard tab), alerts, and a workflow view**, plus an MCP server and a CLI.
 
 ## Requirements
@@ -101,6 +102,13 @@ ccam lanes add --cwd /path/to/repo --title "My feature"    # adopt
 ccam lanes add --repo /path/to/repo --slug my-feature      # managed worktree
 ```
 
+Every lane also carries the CLI that drives it — `claude` (the default) or
+`codex` — picked in the Add-lane modal and fixed for the life of the lane. A
+Codex lane's run panel offers Codex's own sandbox, approval and reasoning-effort
+choices, and resumes with `codex resume --last`. Codex emits none of this
+dashboard's hooks, so a Codex lane records its runs but stays absent from
+Sessions, Analytics and stage detection.
+
 The declared stage comes from `ccam stage`. The inferred stage comes from tool
 events and expires after `DETECTION_TTL_MS` (default 5 minutes), so a lane can
 move backwards between work sessions. Detection never writes the declared stage,