Partial merge leaves stale stack base and Rebase stack replays merged commits

## Reproduction

1. Create a stack with a merge queue on `main`.
2. Merge a bottom prefix of the stack, leaving eight pull requests open.
3. Rebase the eight remaining branches locally onto the current `main`, then force-push them as one atomic ref update.
4. Verify that the lowest open branch is one commit ahead of `main`, zero commits behind, and that every higher branch contains the branch below it.
5. Verify that every open pull request is approved and has passing required checks.
6. Enqueue the remaining stack from its top pull request.

All eight pull requests entered the queue within one second. Eighteen seconds later, the lowest open pull request was removed with:

```text
github-merge-queue removed this pull request from the merge queue due to invalid changes in the merge commit
```

Every higher pull request was then removed because an earlier pull request in the stack was removed. No `merge_group` workflow was created.

The stack merge box reported that the stack was out of date with `main`. The pull request API reported `stack.base.ref = main`, but `stack.base.sha` still identified the last layer of the already-merged prefix rather than the current `main` commit.

7. Click **Rebase stack** in the GitHub UI.

The UI reported that all eight branches were rebased and that the stack had a linear history. The actual lowest branch changed from one commit ahead and zero behind to 16 commits ahead and 15 behind. Commits already present on `main` reappeared in the pull request.

## Expected

After a partial stack merge, GitHub updates the remaining stack's base to the current `main` commit and retargets the lowest open pull request to `main`, as described in [Merging stacked pull requests](https://docs.github.com/en/pull-requests/how-tos/merge-and-close-pull-requests/merging-stacked-pull-requests).

A locally verified linear suffix based on current `main` should enter the merge queue. **Rebase stack** must use the current base commit and must not replay commits that already landed.

## Actual

The remaining stack retains a stale base SHA after the partial merge. Merge-queue admission rejects a correct branch graph before creating a merge group. **Rebase stack** then rewrites the branches against the stale anchor and reintroduces already-merged commits.

Recovery requires restoring every branch ref, removing the stale Stack object, retargeting the lowest open pull request to `main`, and linking the same pull requests into a new stack.

## Environment

- GitHub Stacked Pull Requests public preview.
- GitHub merge queue.
- Original stack: 26 pull requests, with 18 merged and eight open.
- All open pull requests approved and required checks passing.
- `gh stack` v0.1.0.
- `gh` 2.96.0.
- git 2.50.1.
- macOS arm64.

## Related

- #174 concerns approval invalidation after a partial merge. Approvals remained valid in this reproduction.
- #444 concerns a pull request being closed at a merge-group boundary. The remaining pull requests stayed open in this reproduction, but the stack base remained stale.

Start by tracing the merge-queue admission and Rebase stack paths, focusing on how the Stack object's base SHA is updated after a partial merge. Reproduce the eight-PR scenario and inspect the reported base ref and SHA before and after Rebase stack. Done means the remaining stack targets the current main commit, admits a verified suffix, and does not replay merged commits.

Written by the indexing model from the issue text.