[codex] Recover productive terminal continuations (#4956)

## Thinking Path

> - Paperclip orchestrates AI agents through issue-scoped heartbeat runs
> - Recovery logic decides whether in-progress work still has a live
path after a terminal run
> - A productive terminal continuation can still leave an issue stranded
when no active run or wake remains
> - Treating that state as healthy leaves work stuck despite evidence
that more action is needed
> - This pull request re-enqueues recovery for productive terminal
continuations that left no live path
> - The benefit is fewer silently stranded in-progress issues after
agents make partial progress

## What Changed

- Reclassified successful-but-productive terminal continuations as
recoverable when no live path remains.
- Enqueue a follow-up recovery wake with the original run id and
continuation metadata.
- Added regression tests covering productive terminal continuation
recovery and advanced liveness handoff.

## Verification

- `pnpm exec vitest run
server/src/__tests__/heartbeat-process-recovery.test.ts
server/src/__tests__/run-continuations.test.ts`

## Risks

- Medium risk: recovery may schedule one more follow-up where Paperclip
previously considered the work observed. The existing uniqueness,
budget, and escalation checks still constrain retry loops.

> 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 agent, tool use and local command
execution. Exact context window was not exposed in 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
- [x] 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__/run-continuations.test.ts

@@ -106,6 +106,24 @@ describe("run liveness continuations", () => {
     expect(decision.nextAttempt).toBe(2);
   });
 
+  it("leaves advanced terminal runs to stranded issue recovery instead of bounded liveness continuation", () => {
+    const decision = decideRunLivenessContinuation({
+      run: run(),
+      issue: issue(),
+      agent: agent(),
+      livenessState: "advanced",
+      livenessReason: "Run produced concrete action evidence: created an issue comment",
+      nextAction: "Resume the implementation from the remaining acceptance criteria.",
+      budgetBlocked: false,
+      idempotentWakeExists: false,
+    });
+
+    expect(decision).toEqual({
+      kind: "skip",
+      reason: "liveness state is not actionable for continuation",
+    });
+  });
+
   it("does not enqueue a third continuation and returns an exhaustion comment", () => {
     const decision = decideRunLivenessContinuation({
       run: run({ continuationAttempt: 2 }),