[codex] Add configurable liveness auto-recovery controls (#4587)

## Thinking Path

> - Paperclip orchestrates AI agents for zero-human companies.
> - Heartbeat liveness recovery decides when stalled issue trees need
manager-visible follow-up.
> - Automatic recovery issue creation is useful, but operators need
instance-level controls for how aggressive it is.
> - Without controls, recovery behavior is harder to tune for local
development, production operations, and noisy edge cases.
> - This pull request adds configurable liveness auto-recovery settings
across shared contracts, API routes, services, and the instance
experimental settings UI.
> - The benefit is that operators can keep liveness findings advisory or
enable bounded recovery automation with explicit intervals and lookback
windows.

## What Changed

- Added shared types and validators for liveness auto-recovery settings.
- Extended instance settings routes and services to persist and validate
the new controls.
- Wired heartbeat/recovery services to honor enablement, minimum
interval, and lookback settings.
- Added UI controls for liveness recovery under instance experimental
settings.
- Covered the new server behavior with instance settings and liveness
escalation tests.

## Verification

- `pnpm exec vitest run --project @paperclipai/server
server/src/__tests__/heartbeat-issue-liveness-escalation.test.ts
server/src/__tests__/instance-settings-routes.test.ts --pool=forks
--poolOptions.forks.isolate=true`
- `pnpm --filter @paperclipai/shared typecheck`
- `pnpm --filter @paperclipai/server typecheck`
- `pnpm --filter @paperclipai/ui typecheck`

## Risks

- Moderate behavioral risk because recovery automation timing changes
when enabled; defaults keep existing advisory behavior unless the
setting is turned on.
- No database migration in this PR; settings are stored through the
existing instance settings path.

> 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`, coding model with tool use and local command
execution; context window 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 run tests locally and they pass
- [x] I have added or updated tests where applicable
- [ ] If this change affects the UI, I have included before/after
screenshots
- [x] I have updated relevant documentation to reflect my changes
- [x] I have considered and documented any risks above
- [x] I will address all Greptile and reviewer comments before
requesting merge

---------

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

ui/src/api/instanceSettings.ts

@@ -1,6 +1,7 @@
 import type {
   InstanceExperimentalSettings,
   InstanceGeneralSettings,
+  IssueGraphLivenessAutoRecoveryPreview,
   PatchInstanceGeneralSettings,
   PatchInstanceExperimentalSettings,
 } from "@paperclipai/shared";
@@ -15,4 +16,25 @@ export const instanceSettingsApi = {
     api.get<InstanceExperimentalSettings>("/instance/settings/experimental"),
   updateExperimental: (patch: PatchInstanceExperimentalSettings) =>
     api.patch<InstanceExperimentalSettings>("/instance/settings/experimental", patch),
+  previewIssueGraphLivenessAutoRecovery: (input: { lookbackHours?: number }) =>
+    api.post<IssueGraphLivenessAutoRecoveryPreview>(
+      "/instance/settings/experimental/issue-graph-liveness-auto-recovery/preview",
+      input,
+    ),
+  runIssueGraphLivenessAutoRecovery: (input: { lookbackHours?: number }) =>
+    api.post<{
+      findings: number;
+      autoRecoveryEnabled: boolean;
+      lookbackHours: number;
+      cutoff: string;
+      escalationsCreated: number;
+      existingEscalations: number;
+      skipped: number;
+      skippedAutoRecoveryDisabled: number;
+      skippedOutsideLookback: number;
+      escalationIssueIds: string[];
+    }>(
+      "/instance/settings/experimental/issue-graph-liveness-auto-recovery/run",
+      input,
+    ),
 };