Accept story 39: Persistent Claude Code Sessions in Web UI

Use --resume <session_id> with claude -p so the web UI claude-code-pty
provider maintains full conversation context across messages, identical
to a long-running terminal Claude Code session.

Changes:
- Capture session_id from claude -p stream-json system event
- Pass --resume on subsequent messages in same chat session
- Thread session_id through ProviderConfig, ChatResult, WsResponse
- Frontend stores sessionId per chat, clears on New Session
- Unset CLAUDECODE env to allow nested spawning from server
- Wait for clean process exit to ensure transcript flush to disk

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

frontend/src/api/client.ts

@@ -11,6 +11,7 @@ export type WsRequest =
 export type WsResponse =
 	| { type: "token"; content: string }
 	| { type: "update"; messages: Message[] }
+	| { type: "session_id"; session_id: string }
 	| { type: "error"; message: string };
 
 export interface ProviderConfig {
@@ -18,6 +19,7 @@ export interface ProviderConfig {
 	model: string;
 	base_url?: string;
 	enable_tools?: boolean;
+	session_id?: string;
 }
 
 export type Role = "system" | "user" | "assistant" | "tool";
@@ -212,6 +214,7 @@ export class ChatWebSocket {
 	private socket?: WebSocket;
 	private onToken?: (content: string) => void;
 	private onUpdate?: (messages: Message[]) => void;
+	private onSessionId?: (sessionId: string) => void;
 	private onError?: (message: string) => void;
 	private connected = false;
 	private closeTimer?: number;
@@ -220,12 +223,14 @@ export class ChatWebSocket {
 		handlers: {
 			onToken?: (content: string) => void;
 			onUpdate?: (messages: Message[]) => void;
+			onSessionId?: (sessionId: string) => void;
 			onError?: (message: string) => void;
 		},
 		wsPath = DEFAULT_WS_PATH,
 	) {
 		this.onToken = handlers.onToken;
 		this.onUpdate = handlers.onUpdate;
+		this.onSessionId = handlers.onSessionId;
 		this.onError = handlers.onError;
 
 		if (this.connected) {
@@ -256,6 +261,7 @@ export class ChatWebSocket {
 				const data = JSON.parse(event.data) as WsResponse;
 				if (data.type === "token") this.onToken?.(data.content);
 				if (data.type === "update") this.onUpdate?.(data.messages);
+				if (data.type === "session_id") this.onSessionId?.(data.session_id);
 				if (data.type === "error") this.onError?.(data.message);
 			} catch (err) {
 				this.onError?.(String(err));