feat: expand ~ and $HOME in mount target paths

- Add expand_container_path() helper that resolves ~/ and $HOME/ prefixes
- Add get_manifest_home_dir() to compute /home/{user.name} or /root from manifest
- Set ENV HOME=... and ENV USER=... in generated Dockerfile for runtime compatibility
- Pass home_dir through instance creation and startup pipeline
- Expand mount targets in apply_resolved_profile() for regular profile mounts
- Expand mapping targets in _resolve_git_mount_mappings() for git mounts
- Expand working_directory and volume targets in _modify_compose_file()
- Update _prepare_manifest_instance to return home_dir alongside image tag
- Fetch tool_type early in start_instance to determine home_dir before profile application

Quality gates: pytest 188 passed, frontend typecheck clean

Addresses: home-path-expansion

openspec/designs/home-path-expansion.md

@@ -0,0 +1,102 @@
+# Design: ~ / $HOME Expansion in Mount Paths
+
+## Architecture
+
+### New Helpers
+
+#### `expand_container_path(path: str, home_dir: str) -> str`
+Located in `config_profile_resolver.py` (or new shared module).
+
+```python
+def expand_container_path(path: str, home_dir: str) -> str:
+    if path.startswith("~/"):
+        return os.path.join(home_dir, path[2:])
+    if path == "~":
+        return home_dir
+    path = path.replace("$HOME/", home_dir + "/")
+    path = path.replace("$HOME", home_dir)
+    return path
+```
+
+#### `get_manifest_home_dir(manifest: dict) -> str`
+Located in `manifest_compiler.py`.
+
+```python
+def get_manifest_home_dir(manifest: dict) -> str:
+    user = manifest.get("user")
+    if user and user.get("name"):
+        return f"/home/{user['name']}"
+    return "/root"
+```
+
+#### `get_tool_home_dir(tool_type: ToolType, manifest: dict | None) -> str`
+Located in `tool_instances.py` or `manifest_compiler.py`.
+
+```python
+def get_tool_home_dir(tool_type: ToolType, manifest: dict | None = None) -> str:
+    if tool_type.definition_type == "manifest" and manifest:
+        return get_manifest_home_dir(manifest)
+    return "/root"
+```
+
+### Pipeline Changes
+
+#### `create_instance` flow
+
+1. Determine `home_dir` from tool type + manifest (if manifest-based)
+2. Pass `home_dir` to `_modify_compose_file()` — expand mount targets in compose
+
+#### `start_instance` flow
+
+1. Determine `home_dir` from tool type + resolved manifest
+2. Pass `home_dir` to `apply_resolved_profile()` — expand profile mount targets
+3. Pass `home_dir` to `_resolve_git_mounts()` — expand git mount mapping targets
+
+#### `apply_resolved_profile()`
+
+```python
+def apply_resolved_profile(
+    instance_dir: str,
+    resolved: ResolvedProfile,
+    home_dir: str = "/root",
+) -> tuple[...]:
+    ...
+    for mount in resolved.mounts.values():
+        target = expand_container_path(mount.target, home_dir)
+        ...
+```
+
+#### `_resolve_git_mount_mappings()`
+
+```python
+def _resolve_git_mount_mappings(
+    repo_path: str,
+    mappings: list[dict],
+    working_directory: str | None,
+    home_dir: str = "/root",
+) -> list[dict]:
+    ...
+    final_target = expand_container_path(target_path, home_dir)
+    ...
+```
+
+### Dockerfile Change
+
+In `compile_dockerfile()`, after user creation, set `HOME`:
+
+```python
+if user:
+    home = f"/home/{user['name']}"
+    lines.append(f"ENV HOME={home}")
+    lines.append(f"ENV USER={user['name']}")
+```
+
+### File Changes
+
+| File | Change |
+|------|--------|
+| `apps/api/src/services/config_profile_resolver.py` | Add `expand_container_path()`, apply in `apply_resolved_profile()` |
+| `apps/api/src/services/manifest_compiler.py` | Add `get_manifest_home_dir()`, set `HOME`/`USER` env in Dockerfile |
+| `apps/api/src/api/tool_instances.py` | Determine `home_dir`, pass through all mount resolution functions |
+| `apps/api/tests/unit/test_home_path_expansion.py` | New unit tests |
+| `apps/api/tests/unit/test_manifest_compiler.py` | Add tests for `get_manifest_home_dir` and Dockerfile `HOME` env |