Extensions: Connection Info both eras + io.modelcontextprotocol/ui advertisement (Phase 3 of #1633) (#1744)

* Extensions: Connection Info both eras + io.modelcontextprotocol/ui advertisement (Phase 3 of #1633)

Completes the Extensions surfacing and advertises the MCP Apps UI extension.

core/mcp/extensions.ts:
- Add UI_EXTENSION_KEY ("io.modelcontextprotocol/ui") and MCP_APP_MIME_TYPE
  ("text/html;profile=mcp-app") — mirrors of ext-apps' EXTENSION_ID /
  RESOURCE_MIME_TYPE (hardcoded: the constants live on the package's /server
  subpath or an extensionless re-export that doesn't resolve under NodeNext).
- Registry entries gain an optional `advertisement` value; the builder stamps
  it (default {}). Add the `ui` entry, default-advertised with
  `{ mimeTypes: ["text/html;profile=mcp-app"] }` — the Inspector always renders
  App tools. No ext-apps cast needed: the extensions value type is `object`.

ConnectionInfoContent:
- Source server extensions from the negotiated `capabilities.extensions`
  (getServerCapabilities, era-transparent) instead of the modern-only
  discoverResult, so a LEGACY server advertising extensions renders them too.
- Add a two-column "Server Extensions" / "Advertised Extensions" section (the
  Inspector's own advertised map), mirroring the server/client capability
  columns. Drop the now-redundant Extensions row from the modern Discovery
  section (Supported versions stays).

Tests: extensions.test.ts covers the ui entry + advertisement value and the
per-extension enable/disable; ConnectionInfoContent.test.tsx covers legacy
server extensions (no discovery), the advertised-extensions column, and the
em-dash empty states. npm run ci green (the one InspectorView localStorage
failure is a known flake, passes on re-run).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01XNdjEPKLG637X8YmhDiEk5

* Extensions: address Phase 3 review (mime drift-guard, story, exact em-dashes)

- extensions-mimetype.test.ts (integration/node): assert MCP_APP_MIME_TYPE
  equals ext-apps' real RESOURCE_MIME_TYPE (imported) and that the client
  advertises it — a genuine drift guard, replacing the tautological literal
  check in the unit test (which now just asserts the type).
- ConnectionInfoContent Modern story: carry the tasks extension on
  initializeResult.capabilities too, so the Server Extensions row demonstrates
  it (the row now reads negotiated server capabilities, not discoverResult).
- Tighten the em-dash assertions from ">= N" to exact counts (3/3/2) now that
  the two extension sections em-dash by default.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01XNdjEPKLG637X8YmhDiEk5

* Extensions: drift-guard UI_EXTENSION_KEY against ext-apps EXTENSION_ID (review)

Address the Phase 3 re-review nit: the extension id was the other half of the
same drift risk (a server keys its Apps lookup on it). Import ext-apps'
EXTENSION_ID from the /server subpath in the node integration test (fine there;
the browser build is what avoids /server) and assert UI_EXTENSION_KEY === it.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01XNdjEPKLG637X8YmhDiEk5

* Extensions: clone the registry advertisement so builds never alias it (review)

Address the Phase 3 re-review's optional defensive nit: buildClientExtensions
returned the same object reference as the ADVERTISABLE_EXTENSIONS entry for the
ui advertisement, so an in-place mutation of a stamped value (e.g.
extensions[ui].mimeTypes) could corrupt the registry for later connections.
structuredClone the advertisement on stamp; add a test asserting a mutated
build doesn't leak into the next one. Nothing mutates it today — purely
defensive.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01XNdjEPKLG637X8YmhDiEk5

* Extensions: add #1740 Connection Info proof screenshots

End-to-end smoke test in the web client against the legacy
advertised-extensions-http.json server: the Connection Info modal shows the new
era-transparent Server Extensions / Advertised Extensions section, with
Advertised Extensions listing io.modelcontextprotocol/tasks + io.modelcontextprotocol/ui
(the new ui advertisement), and only ui after toggling Tasks off and reconnecting.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01XNdjEPKLG637X8YmhDiEk5

---------

Co-authored-by: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

clients/web/src/components/groups/ConnectionInfoContent/ConnectionInfoContent.tsx

@@ -128,11 +128,14 @@ function formatSession(
   return isModernEra(era) ? "Sessionless" : "Session-based";
 }
 
-// The `extensions` map the server advertised in its `server/discover`
-// capabilities (SEP-2133). Render the extension identifiers, or an em dash when
-// none were advertised.
-function formatExtensions(discoverResult: DiscoverResult): string {
-  const extensions = discoverResult.capabilities.extensions;
+// Render an `extensions` capability map (SEP-2133) as a comma-separated list of
+// its extension identifiers, or an em dash when none are present. Works for
+// either side's map: the server's negotiated `capabilities.extensions` (present
+// on both eras via `getServerCapabilities()`) or the Inspector's own advertised
+// `clientCapabilities.extensions`. (#1740)
+function formatExtensions(
+  extensions: Record<string, unknown> | undefined,
+): string {
   const keys = extensions ? Object.keys(extensions) : [];
   return keys.length > 0 ? keys.join(", ") : "—";
 }
@@ -220,9 +223,6 @@ export function ConnectionInfoContent({
                 ? discoverResult.supportedVersions.join(", ")
                 : "—"}
             </ValueText>
-
-            <Text size="sm">Extensions</Text>
-            <ValueText>{formatExtensions(discoverResult)}</ValueText>
           </SimpleGrid>
         </Stack>
       )}
@@ -250,6 +250,23 @@ export function ConnectionInfoContent({
         </Stack>
       </SimpleGrid>
 
+      {/* Extensions (SEP-2133) shown for both eras: the server's from its
+          negotiated capabilities, the Inspector's from what it advertised
+          (the Advertised Extensions setting). Mirrors the server/client
+          capability columns above. (#1740) */}
+      <SimpleGrid cols={2}>
+        <Stack gap="xs">
+          <SectionHeading>Server Extensions</SectionHeading>
+          <ValueText>{formatExtensions(capabilities.extensions)}</ValueText>
+        </Stack>
+        <Stack gap="xs">
+          <SectionHeading>Advertised Extensions</SectionHeading>
+          <ValueText>
+            {formatExtensions(clientCapabilities.extensions)}
+          </ValueText>
+        </Stack>
+      </SimpleGrid>
+
       {instructions && (
         <Stack gap="xs">
           <SectionHeading>Server Instructions</SectionHeading>