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

[stooit]

Summary

Fixes all failing tests and type errors in the multi-package TypeScript HTTP API. After the change, bun test reports 22 pass / 0 fail and tsc --noEmit exits clean with zero errors.

Changes

• packages/shared/src/utils/pagination.ts — implemented the paginate<T>() stub: 1-indexed page slicing, total from items.length, totalPages via Math.ceil(total / pageSize), correct page/pageSize fields. Handles partial last pages, out-of-range pages, and empty arrays.
• packages/api/src/middleware/auth.ts — fixed a case-sensitivity bug: public-method matching used lowercase while Hono normalises c.req.method to uppercase, so the intentionally public POST /users was treated as protected. Protected methods (e.g. DELETE) remain protected.
• packages/shared/src/types.ts + packages/api/src/routes/users.ts — reconciled the inconsistent User field name to username, matching the route handler and test assertions.
• packages/api/src/routes/users.ts — added the missing badRequest import.
• tsconfig.json — added bun-types to compilerOptions.types so tsc recognises Bun globals, eliminating spurious type errors without adding any dependency.

Verification

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

Assumptions / constraints honoured

• No test files modified.
• No new dependencies added (bun-types was already present; only referenced via tsconfig).
• Changes are minimal and targeted to what the tests/typecheck require.
• username (lowercase) chosen because the test suite asserts that field.
• Public POST confirmed intentional by the test suite.

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