[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>

server/src/__tests__/heartbeat-issue-liveness-escalation.test.ts

@@ -103,6 +103,7 @@ describeEmbeddedPostgres("heartbeat issue graph liveness escalation", () => {
     await instanceSettingsService(db).updateExperimental({
       enableIssueGraphLivenessAutoRecovery: false,
       enableIsolatedWorkspaces: false,
+      issueGraphLivenessAutoRecoveryLookbackHours: 24,
     });
   });
 
@@ -116,7 +117,7 @@ describeEmbeddedPostgres("heartbeat issue graph liveness escalation", () => {
     });
   }
 
-  async function seedBlockedChain(opts: { stale?: boolean } = {}) {
+  async function seedBlockedChain(opts: { outsideLookback?: boolean } = {}) {
     const companyId = randomUUID();
     const managerId = randomUUID();
     const coderId = randomUUID();
@@ -157,9 +158,9 @@ describeEmbeddedPostgres("heartbeat issue graph liveness escalation", () => {
       },
     ]);
 
-    const issueTimestamp = opts.stale === false
-      ? new Date()
-      : new Date(Date.now() - 25 * 60 * 60 * 1000);
+    const issueTimestamp = opts.outsideLookback === true
+      ? new Date(Date.now() - 25 * 60 * 60 * 1000)
+      : new Date(Date.now() - 60 * 60 * 1000);
     await db.insert(issues).values([
       {
         id: blockedIssueId,
@@ -197,6 +198,9 @@ describeEmbeddedPostgres("heartbeat issue graph liveness escalation", () => {
   }
 
   it("keeps liveness findings advisory when auto recovery is disabled", async () => {
+    await instanceSettingsService(db).updateExperimental({
+      enableIssueGraphLivenessAutoRecovery: false,
+    });
     const { companyId } = await seedBlockedChain();
     const heartbeat = heartbeatService(db);
 
@@ -214,16 +218,16 @@ describeEmbeddedPostgres("heartbeat issue graph liveness escalation", () => {
     expect(escalations).toHaveLength(0);
   });
 
-  it("does not create recovery issues until the dependency path is stale for 24 hours", async () => {
+  it("does not create recovery issues outside the configured lookback window", async () => {
     await enableAutoRecovery();
-    const { companyId } = await seedBlockedChain({ stale: false });
+    const { companyId } = await seedBlockedChain({ outsideLookback: true });
     const heartbeat = heartbeatService(db);
 
     const result = await heartbeat.reconcileIssueGraphLiveness();
 
     expect(result.findings).toBe(1);
     expect(result.escalationsCreated).toBe(0);
-    expect(result.skippedAutoRecoveryTooYoung).toBe(1);
+    expect(result.skippedOutsideLookback).toBe(1);
 
     const escalations = await db
       .select()
@@ -424,7 +428,7 @@ describeEmbeddedPostgres("heartbeat issue graph liveness escalation", () => {
     const dependentExecutionWorkspaceId = randomUUID();
     const blockerExecutionWorkspaceId = randomUUID();
     const issuePrefix = `T${companyId.replace(/-/g, "").slice(0, 6).toUpperCase()}`;
-    const issueTimestamp = new Date(Date.now() - 25 * 60 * 60 * 1000);
+    const issueTimestamp = new Date(Date.now() - 60 * 60 * 1000);
 
     await db.insert(companies).values({
       id: companyId,
@@ -559,7 +563,7 @@ describeEmbeddedPostgres("heartbeat issue graph liveness escalation", () => {
     const { companyId, blockedIssueId, blockerIssueId } = await seedBlockedChain();
     const secondBlockedIssueId = randomUUID();
     const issuePrefix = `T${companyId.replace(/-/g, "").slice(0, 6).toUpperCase()}`;
-    const issueTimestamp = new Date(Date.now() - 25 * 60 * 60 * 1000);
+    const issueTimestamp = new Date(Date.now() - 60 * 60 * 1000);
     await db.insert(issues).values({
       id: secondBlockedIssueId,
       companyId,