feat(mall): authenticate against the live API

Wave 2 of replacing the fixed-data mock adapter: the auth domain joins the live
list, so credentials, roles and tokens belong to the real user.

- session: validate a restored token through /auth/me instead of trusting
  localStorage, clearing it on 401/403 but keeping it when the API is merely
  unreachable; the route guard now uses the validated session
- login: report a 401 as invalid credentials rather than a generic failure, and
  drop the 6-character client rule so the API owns the password policy
- register: raise the rule to the API's 8 characters, remove the
  verification-code field (its button only counted down and the value was never
  sent), and report a duplicate email (409) distinctly
- TopBar and the user profile render their session-dependent branch client-only:
  validating the session before hydration made those localStorage-backed
  branches report hydration mismatches the previous code did not

Verified against the running backend: wrong password rejected, real JWT issued,
/user reachable, a short password refused with no network call, duplicate email
reported, a tampered token cleared and bounced to sign-in, a stale token kept
when the API is down, and the fixed-data rollback still signs in with the
backend stopped.

Also re-cuts docs/TBD-migrate-wave.md: auth is done, and cart, orders,
shipments and invoices must move together, because the mock adapter keeps their
state in one shared object and a partial flip fails at checkout.

OpenSpec change: openspec/changes/replace-mock-api-wave-2

apps/mall/pages/login.vue

@@ -1,4 +1,6 @@
 <script setup lang="ts">
+import { ApiError } from "@vmall/shared";
+
 const { t } = useI18n();
 const { $api } = useNuxtApp();
 const session = useSessionStore();
@@ -18,10 +20,8 @@ function validate(): boolean {
     errorKey.value = "auth.validationEmail";
     return false;
   }
-  if (password.value.length < 6) {
-    errorKey.value = "auth.validationPassword";
-    return false;
-  }
+  // No length rule here: the auth API owns the password policy, and a wrong
+  // password should be reported as rejected credentials, not as bad input.
   return true;
 }
 
@@ -33,8 +33,10 @@ async function submit(): Promise<void> {
     const auth = await $api.login(email.value, password.value);
     session.setAuth(auth);
     await router.push("/");
-  } catch {
-    errorKey.value = "auth.requestFailed";
+  } catch (error) {
+    errorKey.value = error instanceof ApiError && error.status === 401
+      ? "auth.invalidCredentials"
+      : "auth.requestFailed";
   } finally {
     submitting.value = false;
   }