refactor: system prompt sections + docs/tests

docs/web/control-ui.md

@@ -28,7 +28,7 @@ The dashboard settings panel lets you store a token; passwords are not persisted
 The onboarding wizard generates a gateway token by default, so paste it here on first connect.
 
 ## What it can do (today)
-- Chat with the model via Gateway WS (`chat.history`, `chat.send`, `chat.abort`)
+- Chat with the model via Gateway WS (`chat.history`, `chat.send`, `chat.abort`, `chat.inject`)
 - Stream tool calls + live tool output cards in Chat (agent events)
 - Connections: WhatsApp/Telegram status + QR login + Telegram config (`channels.status`, `web.login.*`, `config.patch`)
 - Instances: presence list + refresh (`system-presence`)
@@ -60,6 +60,7 @@ Notes:
 
 - `chat.send` is **non-blocking**: it acks immediately with `{ runId, status: "started" }` and the response streams via `chat` events.
 - Re-sending with the same `idempotencyKey` returns `{ status: "in_flight" }` while running, and `{ status: "ok" }` after completion.
+- `chat.inject` appends an assistant note to the session transcript and broadcasts a `chat` event for UI-only updates (no agent run, no channel delivery).
 - Stop:
   - Click **Stop** (calls `chat.abort`)
   - Type `/stop` (or `stop|esc|abort|wait|exit|interrupt`) to abort out-of-band

docs/web/webchat.md

@@ -19,7 +19,8 @@ Status: the macOS/iOS SwiftUI chat UI talks directly to the Gateway WebSocket.
 3) Ensure gateway auth is configured if you are not on loopback.
 
 ## How it works (behavior)
-- The UI connects to the Gateway WebSocket and uses `chat.history` + `chat.send`.
+- The UI connects to the Gateway WebSocket and uses `chat.history`, `chat.send`, and `chat.inject`.
+- `chat.inject` appends an assistant note directly to the transcript and broadcasts it to the UI (no agent run).
 - History is always fetched from the gateway (no local file watching).
 - If the gateway is unreachable, WebChat is read-only.