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

[stooit]

Summary

Fixes all failing tests and TypeScript type errors in the Bun + Hono monorepo. 22/22 tests pass and tsc --noEmit exits cleanly.

Fixes (5 files, source + tsconfig only)

File	Fix
packages/shared/src/utils/pagination.ts	Implemented paginate<T> (was a stub throwing not implemented). 1-indexed page slicing via (page-1)*size, totalPages = Math.ceil(total/size), graceful empty-array and out-of-range handling.
packages/api/src/middleware/auth.ts	Case-sensitivity bug: public-method check used "post" but Hono c.req.method is uppercase, so every POST wrongly required a token. Changed to "POST".
packages/shared/src/types.ts	User field userName → username for consistency with route handlers and tests.
packages/api/src/routes/users.ts	Added missing badRequest import (was a runtime ReferenceError on invalid POST /users bodies).
tsconfig.json	Wired the already-installed bun-types into compilerOptions.types so bun:test/process resolve. strict preserved.

Verification

• bun test → 22 pass, 0 fail
• bunx tsc --noEmit → exit 0, no errors
• Independently re-run after fixes; reviewed by the review agent (verdict: approve).

Constraints honoured

• No test files modified.
• No new dependencies added (bun-types was already installed; only tsconfig wiring changed).
• Only the code required by the failing tests was changed.

Assumptions

• The canonical User field name is username (chosen to match the tests, which cannot be modified).