feat(sagi): spawn a real claude session for Sagi's Ask handoff

Unmatched Ask questions now adopt a dedicated "Sagi" lane (POST
/api/lanes/ensure, cwd defaults to $HOME) and spawn a tmux-backed claude
run there instead of just prefilling the Run page composer.

ARCHITECTURE.md

@@ -2138,7 +2138,9 @@ The detection layer carries all of the signal value: the dashboard tells the use
 
 ## Sagi Companion Subsystem
 
-Sagi is a **client-only** floating mascot companion that reacts to live session activity. It is purely additive UI: there is **no server/backend code**, **no new API routes**, **no new WebSocket message types**, and **no database changes**. Sagi reuses the existing real-time event stream (the same `eventBus` every page already consumes) and the existing **Run** page for its "ask a real question" path. The entire subsystem lives under `client/src/components/Sagi/`.
+Sagi is a **client-only** floating mascot companion that reacts to live session activity. It is purely additive UI: there are **no new server routes**, **no new WebSocket message types**, and **no database schema changes**. Sagi reuses the existing real-time event stream (the same `eventBus` every page already consumes) and the existing **Run** lane/run APIs for its "ask a real question" path. The entire subsystem lives under `client/src/components/Sagi/`.
+
+The only server-side accommodation is that `POST /api/lanes/ensure` now defaults a missing `cwd` to the user's home directory, so the mascot can adopt a stable "Sagi" lane (titled, fixed cwd) without forcing the client to know any absolute path. The behavior is gated by the ensure route; the underlying `createLane` still rejects empty cwds.
 
 The design follows a strict **pure-core / hook / presentational** split: a framework-free brain (a `WSMessage` reducer plus a mood state machine with an injected clock and zero side effects) is fully unit-tested in isolation, a single React hook is the only consumer of the global `eventBus` and the only owner of timers and side effects, and the SVG/markup components are pure presentational views driven by props.
 
@@ -2192,9 +2194,13 @@ flowchart LR
     DERIVED --> AVATAR["SagiAvatar"]
     DERIVED --> BUBBLE["SpeechBubble"]
     DERIVED --> PANEL["SagiPanel"]
-    PANEL -->|"unmatched Ask"| RUN["/run?prompt=…<br/>(existing Run page)"]
+    PANEL -->|"unmatched Ask"| ENSURE["POST /api/lanes/ensure<br/>(adopt 'Sagi' lane, cwd = home)"]
+    ENSURE --> START["POST /api/run<br/>(spawn tmux-backed claude<br/>in Sagi lane, initialPrompt=…)"]
+    START --> RUN["/run?laneId=…<br/>(existing Run page picks up<br/>the live run)"]
 
     style PUB fill:#f59e0b,stroke:#fbbf24,color:#000
+    style ENSURE fill:#6366f1,stroke:#818cf8,color:#fff
+    style START fill:#6366f1,stroke:#818cf8,color:#fff
     style RUN fill:#10b981,stroke:#34d399,color:#fff
 ```
 
@@ -2210,7 +2216,7 @@ The mood state machine in `deriveMood` resolves to a single expression using a f
 | **`SpeechBubble.tsx`** | Pure presentational speech bubble. |
 | **`SagiPanel.tsx`** | Pure presentational panel: status readout + quick actions + the Ask box. |
 | **`Sagi.tsx`** | Shell component. Mounted once in `client/src/components/Layout.tsx` as a sibling of `UpdateNotifier`. Owns open/closed state, the `⌘B` / `Esc` shortcuts, reduced-motion detection, and navigation. |
-| **`intents.ts`** | Pure local Q&A over the cached status snapshot. Queries that don't match a local intent become a handoff to the existing Run page via `/run?prompt=…`. |
+| **`intents.ts`** | Pure local Q&A over the cached status snapshot. Queries that don't match a local intent become a handoff: `Sagi.tsx` calls `POST /api/lanes/ensure` (adopting the "Sagi" lane whose cwd defaults to `$HOME`), then `POST /api/run` to spawn a fresh tmux-backed `claude` session in that lane with the prompt as its `initialPrompt`, then navigates to `/run?laneId=…&autostart=1` so the live run is observable. |
 | **`quips.ts`** | Pure mood → phrase pools. |
 | **`prefs.ts`** | `localStorage`-backed enabled / muted preferences with cross-tab sync. |
 | **`sagi.css`** | Keyframes and per-mood expressions; selected via the `data-mood` attribute. |
@@ -2223,7 +2229,8 @@ Sagi's only contact with the rest of the app is four light, additive touchpoints
 | --- | --- |
 | `client/src/components/Layout.tsx` | Mounts `<Sagi />` once, as a sibling of `<UpdateNotifier />`. |
 | `client/src/pages/Settings.tsx` | On/off toggle wired to `sagiPrefs` (`localStorage`). |
-| `client/src/pages/Workspace.tsx` | Reads `?prompt=` to prefill the prompt box for Sagi's Ask handoff. |
+| `client/src/pages/Run.tsx` | Consumes `?laneId=` to attach the page to the lane whose run Sagi just spawned. |
+| `server/routes/lanes.js` (`POST /api/lanes/ensure`) | Accepts a missing `cwd` and resolves it to `os.homedir()` so the mascot can adopt a stable "Sagi" lane. |
 | `client/src/i18n/locales/{en,vi}/settings.json` | `sagi.*` strings for the Settings toggle (zh / ko fall back to the hardcoded English defaults). |
 
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