feat(FN-4222): add experiment-finalize CLI command and API route

Implements the `experiment-finalize` CLI command (FN-4222) and its companion dashboard API route, wiring the feature through the pi extension as a new callable tool. Includes the command implementation, extension tooling, API integration, and corresponding test coverage.

Fusion-Task-Id: FN-4222

docs/research/experiment-finalize.md

@@ -0,0 +1,56 @@
+# Experiment Session Finalize Workflow
+
+## Motivation
+FN-4136 identified a gap versus upstream `autoresearch-finalize`: Fusion had no workflow to split kept experiment commits into reviewable branches from a shared merge-base.
+
+## Plan model
+Finalize builds groups of kept run records, each with ordered commits and a suggested branch name. Branch names default to:
+
+`experiment/<session-id>/<slug>-<index>`
+
+Override schema:
+
+```json
+{
+  "groups": [
+    {
+      "id": "segment:1",
+      "title": "Latency fixes",
+      "suggestedBranchName": "experiment/exp-1/latency-fixes-1",
+      "runRecordIds": ["RUN-2", "RUN-5"]
+    }
+  ]
+}
+```
+
+## Algorithm
+1. Resolve merge-base from session baseline ref and integration branch.
+2. Build default plan from kept runs.
+3. Apply optional override and validate run IDs / branch uniqueness.
+4. For each group: create branch at merge-base, checkout, cherry-pick commits in order.
+5. Restore original ref.
+6. Append `finalize` session record and move session to `finalized`.
+7. On failure, rollback created branches and restore original ref; session remains `finalizing`.
+
+## Error taxonomy
+
+| Error code | CLI exit | HTTP status |
+|---|---:|---:|
+| `state_error` | 2 | 409 |
+| `no_kept_runs` | 3 | 409 |
+| `plan_error` | 4 | 400 |
+| `merge_base_error` | 5 | 422 |
+| `cherry_pick_conflict` | 6 | 422 |
+| `branch_exists` | 7 | 409 |
+| unknown/untyped | 1 | 500 |
+
+## Recovery
+If finalize fails after session status moves to `finalizing`:
+1. Inspect and delete leftover branches (`git branch -D <name>`).
+2. Restore session status manually via store update to `active`.
+3. Re-run finalize (optionally with a plan override).
+
+## Follow-ups
+- FN-4221: dashboard finalize UI wiring.
+- FN-4223: naming clarifications around research/experiment command surfaces.
+- Upstream reference: `autoresearch-finalize` branch-split semantics.