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

[stooit]

Summary

Repairs all failing tests and eliminates all type errors. bun test → 22 pass / 0 fail; bunx tsc --noEmit → clean (exit 0).

The bugs spanned both packages and matched the areas called out in the task brief.

Changes

• Auth middleware (packages/api/src/middleware/auth.ts) — Fixed a case-sensitivity bug in the public-method allow-list ("post" → "POST") so POST routes are correctly treated as public. Method comparison is now normalised to uppercase to prevent this class of access-control bug from recurring.
• Shared types (packages/shared/src/types.ts) — Renamed User.userName → username. The unmodifiable test files (users.test.ts, auth.test.ts) use username and are the contract, so the type was wrong, not the call sites. The rename is complete across both packages.
• Route handler (packages/api/src/routes/users.ts) — Added the missing badRequest import (already exported from ../lib/errors). This was a genuine runtime ReferenceError on the invalid-POST path, which is why "returns 400 for missing fields" failed.
• Pagination utility (packages/shared/src/utils/pagination.ts) — Implemented the previously-stubbed paginate() against the full test contract (page slicing, total/totalPages, page/pageSize, out-of-range → empty data, empty-array handling). Also guarded against non-finite page/size via Number.isFinite so user-supplied query params can't corrupt the response (Infinity previously produced pageSize: null and zero items).
• tsconfig.json — Referenced the already-installed bun-types ("types": ["bun-types"]) so the process global and bun:test module resolve. No new dependencies added — bun-types was already a declared devDependency.

Constraints honoured

• No test files modified
• No new dependencies / lockfile changes
• Only the files the tests required were changed

Assumptions

• username (not userName) is the canonical field name, inferred from the unmodifiable test files being the source of truth.
• The non-finite pagination guard is a defensive hardening of the documented "safe for user-supplied params" contract; it does not change behaviour for any tested (finite) input.

Follow-ups (out of scope)

• packages/api/src/middleware/validate.ts exports an unused requireFields() middleware duplicating inline route validation — could be consolidated.
• The auth policy leaves POST unauthenticated (anonymous record creation). Pre-existing, deliberate, and asserted by the tests — flagged only for a future access-control decision.
• bun-types is pinned to "latest" in package.json (pre-existing); worth pinning for reproducible builds.