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/nuxt.config.ts

@@ -8,8 +8,8 @@ export default defineNuxtConfig({
       apiBase: "http://localhost:8080/api",
       // Domains served by the live backend; every other domain stays on the
       // fixed-data adapter. Override with NUXT_PUBLIC_LIVE_DOMAINS='["catalog"]'.
-      // See openspec/changes/replace-mock-api-wave-1/design.md.
-      liveDomains: ["catalog", "currency"],
+      // See openspec/changes/replace-mock-api-wave-1/design.md and wave 2.
+      liveDomains: ["catalog", "currency", "auth"],
       appName: "mall",
     },
   },