fix(ui): hide empty inbox search sections (#10700)

## Thinking Path

> - Paperclip is the open source app people use to manage AI agents for
work.
> - The Inbox helps operators scan issues that need attention.
> - Inbox search can add supplemental sections for archived matches and
other matches.
> - The supplemental search section builder still sent empty sections
into the grouped render path.
> - That made the Archived and Other results dividers appear even when
those sections had no rows.
> - This pull request drops empty supplemental sections before
rendering.
> - The benefit is a cleaner near-empty inbox search view.

## Linked Issues or Issue Description

No public GitHub issue exists for this report. Public GitHub search
found no duplicate or related open issues or pull requests for this
inbox search behavior.

**What happened?**

Inbox search could show Archived and Other results divider headers even
when those supplemental sections had no rows.

**Expected behavior**

Empty supplemental search sections should not render divider headers.

**Steps to reproduce**

1. Open the Inbox.
2. Search in a near-empty inbox with no archived matches and no
outside-inbox matches.
3. Observe that empty supplemental divider headers can appear.

**Paperclip version or commit**

`master` before this change.

**Deployment mode**

Built from source.

## What Changed

- Dropped empty supplemental inbox search sections before they reach the
grouped inbox render path.
- Added a unit regression test for empty Archived and Other results
sections.
- Refreshed the branch against current `master` to clear the merge
conflict.

## Verification

- `git diff --check origin/master...HEAD` passed.
- Public diff is limited to `ui/src/lib/inbox.ts` and
`ui/src/lib/inbox.test.ts`.
- Local focused Vitest could not run in this execution checkout because
dependencies are not installed and `corepack pnpm exec vitest ...`
reports `Command "vitest" not found`.
- Pull request CI is green for typecheck, build, server tests, e2e,
security checks, policy checks, canary dry run, and aggregate verify.
- Greptile Review passed on commit `dc2e224` with confidence score 5/5
and no comments.

## Risks

Low risk. The Inbox change only filters empty supplemental search
sections. Normal inbox sections and non-empty archived or other search
results keep their current behavior.

> 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. See `CONTRIBUTING.md`.

## Model Used

OpenAI Codex, GPT-5, reasoning-enabled with terminal tool use and code
execution. The runtime context-window size is not exposed.

## 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
- [ ] 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/lib/inbox.test.ts

@@ -1054,6 +1054,25 @@ describe("inbox helpers", () => {
     ).toEqual(["inbox", "archived", "other"]);
   });
 
+  it("omits empty archived and other ungrouped search sections", () => {
+    expect(
+      buildGroupedInboxSections(
+        [],
+        "none",
+        {},
+        { keyPrefix: "archived-search:", searchSection: "archived" },
+      ),
+    ).toEqual([]);
+    expect(
+      buildGroupedInboxSections(
+        [],
+        "none",
+        {},
+        { keyPrefix: "other-search:", searchSection: "other" },
+      ),
+    ).toEqual([]);
+  });
+
   it("defaults the remembered inbox tab to mine and persists all", () => {
     localStorage.clear();
     expect(loadLastInboxTab()).toBe("mine");

ui/src/lib/inbox.ts

@@ -1126,6 +1126,9 @@ export function buildGroupedInboxSections(
   const keyPrefix = options?.keyPrefix ?? "";
   const searchSection = options?.searchSection ?? "none";
   const nestingEnabled = options?.nestingEnabled ?? false;
+  if (searchSection !== "none" && items.length === 0) {
+    return [];
+  }
 
   return groupInboxWorkItems(items, groupBy, workspaceGrouping).map((group) => {
     const nestedGroup = nestingEnabled && group.items.some((item) => item.kind === "issue")