lazy-auth: address review nits

- Build the well-known rewrite regex from LAZY_AUTH_SLUG so it can't drift
  from the mount path if the slug changes.
- Clarify in the docstring that an explicit PUBLIC_URL must include the
  /lazy-auth mount path (the package advertises it verbatim).
- Tests: clear/restore ambient PUBLIC_URL around the integration server
  (was a latent flake); add unit coverage for the PUBLIC_URL derivation
  (trailing-slash strip, no-trailing-slash, explicit-wins, unset cases);
  add a public tool-call success case and an invalid-token 401 case.

src/modules/example-apps/lazy-auth.integration.test.ts

@@ -51,8 +51,15 @@ function callTool(base: string, name: string, accessToken?: string): Promise<Htt
 describe('Lazy Auth example mount', () => {
   let server: http.Server;
   let base: string;
+  let savedPublicUrl: string | undefined;
 
   beforeAll(async () => {
+    // These tests rely on the package's per-request Host resolution (no baseUri
+    // passed below), so an ambient PUBLIC_URL would pin advertised URLs and
+    // break the assertions. Clear it for the duration and restore afterwards.
+    savedPublicUrl = process.env.PUBLIC_URL;
+    delete process.env.PUBLIC_URL;
+
     const app = express();
     // Mirror src/index.ts: the example mounts before the host's middleware
     // and routes.
@@ -75,6 +82,8 @@ describe('Lazy Auth example mount', () => {
 
   afterAll(async () => {
     await new Promise((resolve) => server.close(resolve));
+    if (savedPublicUrl === undefined) delete process.env.PUBLIC_URL;
+    else process.env.PUBLIC_URL = savedPublicUrl;
   });
 
   it('serves public MCP requests without auth', async () => {
@@ -95,12 +104,24 @@ describe('Lazy Auth example mount', () => {
     expect(res.body).toContain('Lazy Auth');
   });
 
+  it('serves a public tool call without auth', async () => {
+    const res = await callTool(base, 'show_auth_button');
+    expect(res.status).toBe(200);
+    expect(res.headers['www-authenticate']).toBeUndefined();
+  });
+
   it('answers 401 with resource_metadata under the mount path for protected tools', async () => {
     const res = await callTool(base, 'get_secret');
     expect(res.status).toBe(401);
     expect(res.headers['www-authenticate']).toContain(`resource_metadata="${base}/lazy-auth/auth/prm"`);
   });
 
+  it('answers 401 with invalid_token for a bad bearer token', async () => {
+    const res = await callTool(base, 'get_secret', 'not-a-real-token');
+    expect(res.status).toBe(401);
+    expect(res.headers['www-authenticate']).toContain('invalid_token');
+  });
+
   it('advertises mount-prefixed URLs in PRM and AS metadata', async () => {
     const prm = JSON.parse((await request(`${base}/lazy-auth/auth/prm`)).body);
     expect(prm.resource).toBe(`${base}/lazy-auth/mcp`);
@@ -172,3 +193,38 @@ describe('Lazy Auth example mount', () => {
     expect(secretRes.body).toContain('the-answer-is-42');
   });
 });
+
+describe('mountLazyAuthExample PUBLIC_URL derivation', () => {
+  let savedPublicUrl: string | undefined;
+
+  beforeEach(() => {
+    savedPublicUrl = process.env.PUBLIC_URL;
+    delete process.env.PUBLIC_URL;
+  });
+
+  afterEach(() => {
+    if (savedPublicUrl === undefined) delete process.env.PUBLIC_URL;
+    else process.env.PUBLIC_URL = savedPublicUrl;
+  });
+
+  it('derives PUBLIC_URL from baseUri, stripping trailing slashes and appending the mount path', () => {
+    mountLazyAuthExample(express(), 'https://example.test/');
+    expect(process.env.PUBLIC_URL).toBe('https://example.test/lazy-auth');
+  });
+
+  it('handles a baseUri with no trailing slash', () => {
+    mountLazyAuthExample(express(), 'https://example.test');
+    expect(process.env.PUBLIC_URL).toBe('https://example.test/lazy-auth');
+  });
+
+  it('does not override an explicitly set PUBLIC_URL', () => {
+    process.env.PUBLIC_URL = 'https://tunnel.example/lazy-auth';
+    mountLazyAuthExample(express(), 'https://example.test/');
+    expect(process.env.PUBLIC_URL).toBe('https://tunnel.example/lazy-auth');
+  });
+
+  it('leaves PUBLIC_URL unset when no baseUri is provided', () => {
+    mountLazyAuthExample(express());
+    expect(process.env.PUBLIC_URL).toBeUndefined();
+  });
+});

src/modules/example-apps/lazy-auth.ts

@@ -17,30 +17,35 @@
  * them from PUBLIC_URL, falling back to the request Host header for loopback
  * hosts only. We derive PUBLIC_URL from this server's own public base URI -
  * already the source of truth for the host's OAuth issuer - so deployments
- * need no separate env var. An explicit PUBLIC_URL still wins (e.g. a tunnel);
- * omitting baseUri (e.g. tests on an ephemeral port) keeps the package's
- * per-request Host resolution.
+ * need no separate env var. An explicit PUBLIC_URL still wins (e.g. a tunnel) -
+ * note it must include the /lazy-auth mount path, since the package advertises
+ * it verbatim. Omitting baseUri (e.g. tests on an ephemeral port) keeps the
+ * package's per-request Host resolution.
  */
 import { Express, Request, Response, NextFunction } from 'express';
 import { createApp as createLazyAuthApp } from '@modelcontextprotocol/server-lazy-auth';
 
 export const LAZY_AUTH_SLUG = 'lazy-auth';
 
+// RFC 8414/9728 place well-known discovery documents at the origin root, with
+// the resource path inserted after the well-known prefix
+// (e.g. /.well-known/oauth-authorization-server/lazy-auth). Built from
+// LAZY_AUTH_SLUG so it can't drift from the mount path.
+const WELL_KNOWN_INSERTION = new RegExp(
+  `^/\\.well-known/(oauth-authorization-server|oauth-protected-resource)/${LAZY_AUTH_SLUG}(/.*)?$`
+);
+
 export function mountLazyAuthExample(app: Express, baseUri?: string): void {
   if (baseUri && !process.env.PUBLIC_URL) {
     process.env.PUBLIC_URL = `${baseUri.replace(/\/+$/, '')}/${LAZY_AUTH_SLUG}`;
   }
 
-  // RFC 8414/9728 place well-known discovery documents at the origin root,
-  // with the resource path inserted after the well-known prefix
-  // (e.g. /.well-known/oauth-authorization-server/lazy-auth), and MCP SDK
-  // clients only try that insertion form. Rewrite those root paths into the
-  // mount - rather than dispatching to the sub-app directly - so req.baseUrl
-  // (and therefore every URL the example advertises) stays consistent.
+  // MCP SDK clients only try the path-insertion form above for discovery.
+  // Rewrite those root paths into the mount - rather than dispatching to the
+  // sub-app directly - so req.baseUrl (and therefore every URL the example
+  // advertises) stays consistent.
   app.use((req: Request, _res: Response, next: NextFunction) => {
-    const wellKnown = req.url.match(
-      /^\/\.well-known\/(oauth-authorization-server|oauth-protected-resource)\/lazy-auth(\/.*)?$/
-    );
+    const wellKnown = req.url.match(WELL_KNOWN_INSERTION);
     if (wellKnown) {
       req.url = `/${LAZY_AUTH_SLUG}/.well-known/${wellKnown[1]}${wellKnown[2] ?? ''}`;
     }