fix: repair failing tests and type errors across api and shared packages - #111

Summary

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

Changes

• shared/src/utils/pagination.ts — implemented paginate() (was a stub throwing not implemented). Handles page slicing, total/totalPages, out-of-range pages (empty data, truthful metadata), and empty arrays; guards pageSize > 0.
• api/src/middleware/auth.ts — fixed case-sensitivity bug on public HTTP methods (normalises c.req.method.toUpperCase()), so POST /users is correctly public while PUT/DELETE/PATCH still require a token.
• api/src/routes/users.ts — added missing badRequest import (was a runtime ReferenceError on invalid POST /users).
• shared/src/types.ts — aligned User field userName → username to match tests and all consumers.
• tsconfig.json — set types: ["bun-types"] so bun:test and process resolve (bun-types already a devDependency; no new deps).

Assumptions

• Test files not modified (confirmed). Fixes bend source to the tests' expected contract.
• No new dependencies; no lockfile/manifest changes.
• Scope kept to exactly what the tests require.

Verification

• bun test → 22 pass, 0 fail (4 files, 37 assertions)
• bunx tsc --noEmit → clean
• Independent review pass confirmed all four seeded bugs fixed with no regression.