fix(inbox): hide external object summaries (#10020)

## Thinking Path

> - Paperclip is the open source app people use to manage AI agents for
work
> - The inbox helps operators scan and act on tasks that need attention
> - Inbox task rows currently include external-object summary markers
alongside the core task information
> - Those markers add a column of visual noise that is not needed for
inbox triage
> - External-object data must remain available to the inbox filters even
when the row marker is removed
> - This pull request stops passing external-object summaries into inbox
rows and adds regression coverage
> - The benefit is a cleaner inbox while preserving external-object
filtering behavior

## Linked Issues or Issue Description

- Refs #4556
- **Problem:** Inbox task rows display external-object summary markers
that operators do not need for triage.
- **Expected behavior:** Inbox rows omit the external-object marker,
while filters that depend on external-object summaries continue to work.

## What Changed

- Removed the external-object summary prop from inbox task rows.
- Added a regression test that provides external-object summary data and
confirms the inbox row does not render its marker.
- Kept external-object summary loading intact for inbox filtering.

## Verification

- `pnpm exec vitest run ui/src/pages/Inbox.test.tsx` — 18 tests passed.
- `pnpm check:token-gates` — reproduces five pre-existing `#9627`
color-literal violations; this PR adds no token values or new gate
violations.

## Risks

- Low risk: the change removes one optional presentation prop from the
inbox row call site and leaves filtering data flow unchanged.
- Regression coverage verifies summary data no longer produces the
removed inbox marker.

> For core feature work, check [`ROADMAP.md`](ROADMAP.md) first and
discuss it in `#dev` before opening the PR. Feature PRs that overlap
with planned core work may need to be redirected — check the roadmap
first. See `CONTRIBUTING.md`.

## Model Used

- OpenAI Codex, GPT-5.4, tool-enabled coding agent with shell and code
execution; reasoning enabled; context-window size not exposed by the
runtime.

## Checklist

- [x] I have included a thinking path that traces from project context
to this change
- [x] I have specified the model used (with version and capability
details)
- [x] I have checked ROADMAP.md and confirmed this PR does not duplicate
planned core work
- [x] I have searched GitHub for duplicate or related PRs and linked
them above
- [x] I have either (a) linked existing issues with `Fixes: #` / `Closes
#` / `Refs #` OR (b) described the issue in-PR following the relevant
issue template
- [x] I have not referenced internal/instance-local Paperclip issues or
links (only public GitHub `#NNN` / `github.com/paperclipai/paperclip`
URLs)
- [x] My branch name describes the change (e.g. `docs/...`, `fix/...`)
and contains no internal Paperclip ticket id or instance-derived details
- [x] I have run tests locally and they pass
- [x] I have added or updated tests where applicable
- [x] I have updated relevant documentation to reflect my changes
- [x] I have considered and documented any risks above
- [x] All Paperclip CI gates are green
- [x] Greptile is 5/5 with no open P2s, recommendations, or follow-ups
- [x] I will address all Greptile and reviewer comments before
requesting merge

Co-authored-by: Paperclip <noreply@paperclip.ing>

ui/src/pages/Inbox.test.tsx

@@ -17,6 +17,10 @@ const routerMock = vi.hoisted(() => ({
   navigate: vi.fn(),
 }));
 
+const externalObjectMocks = vi.hoisted(() => ({
+  summaries: new Map(),
+}));
+
 const apiMocks = vi.hoisted(() => ({
   approvalsList: vi.fn(),
   joinRequestsList: vi.fn(),
@@ -126,6 +130,14 @@ vi.mock("../hooks/useInboxBadge", () => ({
   }),
 }));
 
+vi.mock("../hooks/useIssueExternalObjects", () => ({
+  useIssueExternalObjectSummaries: () => ({
+    summaries: externalObjectMocks.summaries,
+    isLoading: false,
+    isReady: true,
+  }),
+}));
+
 import {
   FailedRunInboxRow,
   Inbox,
@@ -255,6 +267,7 @@ function createJoinRequest(
 
 function resetInboxApiMocks() {
   for (const mock of Object.values(apiMocks)) mock.mockReset();
+  externalObjectMocks.summaries.clear();
   routerMock.location.pathname = "/";
   routerMock.location.search = "";
   routerMock.location.hash = "";
@@ -299,6 +312,38 @@ describe("Inbox toolbar", () => {
     container.remove();
   });
 
+  it("does not render external-object summaries in inbox rows", async () => {
+    routerMock.location.pathname = "/inbox/mine";
+    const issue = createIssue({ title: "Inbox row without external object column" });
+    apiMocks.issuesList.mockResolvedValue([issue]);
+    externalObjectMocks.summaries.set(issue.id, {
+      total: 1,
+      highestSeverity: "failed",
+      byStatusCategory: { failed: 1 },
+      objects: [],
+    });
+
+    const queryClient = new QueryClient({
+      defaultOptions: { queries: { retry: false, staleTime: 0, gcTime: 0 } },
+    });
+    const root = createRoot(container);
+
+    await act(async () => {
+      root.render(
+        <QueryClientProvider client={queryClient}>
+          <Inbox />
+        </QueryClientProvider>,
+      );
+    });
+    await vi.waitFor(() => {
+      expect(container.textContent).toContain(issue.title);
+    });
+
+    expect(container.querySelector('[aria-label^="External objects:"]')).toBeNull();
+
+    act(() => root.unmount());
+  });
+
   it("shows blocked toolbar controls on the Blocked tab", async () => {
     routerMock.location.pathname = "/inbox/blocked";
     const queryClient = new QueryClient({

ui/src/pages/Inbox.tsx

@@ -2599,7 +2599,6 @@ export function Inbox() {
                       issueLinkState={issueLinkState}
                       treeGuides={depth}
                       hideDivider={hasChildren && isExpanded}
-                      externalObjectSummary={externalObjectSummaryByIssueId.get(issue.id) ?? null}
                       selected={selected}
                       className={
                         isArchiving