Build new Next.js web frontend (web/) with Google SSO + API Gateway

# Why?

**As a** user of rms-app

**I want** a working web interface to browse, borrow, and return items

**So that** I'm not limited to the SMS router or direct API calls — the previous frontend (React Native/Expo, `frontend/`) has been untouched since 2021 and isn't built or deployed by CI; it's abandoned reference material, not living code.

# What?

Build a new Node.js web frontend using Next.js, living under `web/` (new top-level directory, doesn't exist yet):

- **Auth**: Google Workspace SSO — Cognito federated with Google, restricted to the Acts2 Network org. Uses the Gen 2 Cognito User Pool already live in `alpha` (`amplify/auth/resource.ts`) as the identity provider, extended to support Google federation.
- **Backend access**: currently no HTTP API exists at all — the 10 Gen 2 Lambda functions (`amplify/functions/*/resource.ts`) are only reachable via direct SDK invoke (see `ts-code/__tests__/integration/Amplify.test.ts` for the pattern) or the SMS router. This needs a new **API Gateway REST layer** in front of those Lambdas so the web frontend has something to call over HTTP.
- **Golden path**: login → browse/search items → borrow → return → view reservations.

# Additional context

- Replaces the abandoned `frontend/` (React Native/Expo, last touched 2021) — that app is not being migrated from, just superseded.
- Depends on nothing further from the Gen 1→Gen 2 migration (#292, complete) or the schema epic (#244, complete) — both are done, so this can build on the current `MainSchema`/`ItemsSchema` shape directly.
- Per root `CLAUDE.md`'s testing policy, this needs its own test setup once `web/` exists: Jest + React Testing Library for components/utilities, Playwright e2e tests under `web/e2e/` for user-facing flows (starting with the golden path above).
- This is a large, multi-phase build (new API Gateway layer + auth + full frontend) — expect this to be broken into sub-issues (API Gateway/auth infra first, then the Next.js app itself) during planning, similar to how the Gen 2 migration was split into subtasks.