I'll guide you through TDD for your auth module, from test definition through red-green-refactor cycles.

RED-GREEN-REFACTOR WORKFLOW
RED PHASE (Write failing test first)
Identify a single behavior: "authenticate should return a token for valid credentials"
Write a test that calls your auth function with known-valid inputs and asserts the output:
it("returns JWT token for valid username/password") {
  const token = authenticate("user@example.com", "password123");
  expect(token).toBeDefined();
  expect(token.startsWith("eyJ")).toBe(true);
}
Run test -> RED (fails because authenticate() doesn't exist yet).

GREEN PHASE (Write minimal code to pass)
Implement only what's needed to pass:
function authenticate(email, password) {
  return "eyJhbGciOiJIUzI1NiJ9...";
}
Run test -> GREEN (passes).

REFACTOR PHASE (Improve without breaking tests)
Strengthen the implementation: add credential validation, real JWT signing, error handling. Tests confirm each change doesn't break behavior.

TEST CASES FOR AUTH MODULE
Valid credentials: token returned, token contains email claim, token has expiry
Invalid password: AuthError thrown with "invalid credentials" message
User not found: AuthError thrown with "user not found" message
Expired token: verify() rejects with "token expired"
Malformed token: verify() rejects
Token revocation: revoked tokens don't authenticate

MOCKING STRATEGY
Mock database: pass a mock DB to auth() instead of real queries. Test returns correct error when DB fails.
Mock clock: test token expiry by advancing time (jest.useFakeTimers).

DELIVERABLE
Test file with 8-10 test cases covering happy path + all error scenarios. Implementation grows incrementally to pass each test.