fix: resolve failing tests and type errors across api and shared packages

[stooit]

Summary

Fixes all failing tests and type errors in the multi-package repo. After this change: 22 tests pass, 0 fail and tsc --noEmit is clean.

Fixes

Area	Bug	Fix
api/src/routes/users.ts	badRequest called but not imported → ReferenceError (500 instead of 400)	Added badRequest to the ../lib/errors import
api/src/middleware/auth.ts	Public-methods allow-list used lowercase ("post") but c.req.method is uppercase, so POST /users wrongly returned 401	Changed allow-list entries to uppercase ("GET", "POST")
shared/src/types.ts	User.userName mismatched the username field used by routes and tests	Renamed userName → username (canonical, per authoritative tests)
shared/src/utils/pagination.ts	paginate() was a throwing stub	Implemented: 1-indexed pages, real slice, total/totalPages/page/pageSize, handles partial last page, out-of-range, and empty input
tsconfig.json	bun:test module and process unresolved	Added `"types": ["bun-types"]` (already installed; pulls in node globals)

Also removed two now-stale code comments (the `// BUG:` note and the pagination `TODO`).

Verification

• `bun test` → 22 pass, 0 fail
• `npx tsc --noEmit` → exit 0, no errors

Constraints honoured

• No test files modified
• No new dependencies added
• Reviewed by the `review` subagent (verdict: APPROVE)

Assumptions

• `username` (lowercase n) is the canonical field name, since the test files are authoritative and cannot be changed.
• Pagination pages are 1-indexed, matching the contract asserted by `shared/test/pagination.test.ts`.