Summary

Fixes all failing tests and TypeScript errors across the api and shared packages. After this change bun test reports 22 pass / 0 fail and bunx tsc --noEmit is clean.

Fixes

• Pagination utility (packages/shared/src/utils/pagination.ts): implemented the previously-stubbed paginate() — page slicing via (page-1)*size, totalPages = Math.ceil(total/size), returns { data, page, pageSize, total, totalPages } per the test contract.
• Auth middleware (packages/api/src/middleware/auth.ts): fixed the HTTP-method case-sensitivity bug so public routes (e.g. POST /users) are correctly matched and don't require a token. Wired up types so the process global resolves.
• Shared User type: aligned the field name on username (was userName) across the shared type and all api source usages, matching what the tests and route handlers expect.
• Route handlers (packages/api/src/routes/users.ts): added the missing badRequest import that was causing the 400-on-missing-fields path to fail.
• TS config: wired up bun-types so bun:test and process resolve under tsc --noEmit.

Verification

• bun test → 22 pass, 0 fail
• bunx tsc --noEmit → clean

Constraints honoured

• No test files modified.
• No new dependencies added.
• Changes kept minimal and scoped to what the tests/types required.

Assumptions

• Since test files are immutable and use username, the canonical field name was standardised to username (the shared type previously used userName).