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/proposals/home-path-expansion.md

@@ -0,0 +1,49 @@
+# Proposal: ~ / $HOME Expansion in Mount Paths
+
+## Context
+
+Users want to write mount target paths like `~/workspace` or `$HOME/workspace` instead of absolute paths like `/home/user/workspace` or `/root/workspace`. Docker Compose does not expand these — they must be resolved before writing the compose file.
+
+## Goal
+
+Resolve `~` and `$HOME` in container target paths to the correct home directory for the target container.
+
+## Direction
+
+**Manifest-aware home directory (Option B)**
+- Extract `user.name` from the tool manifest to compute `/home/{name}`
+- For root-based manifests (no user block), use `/root`
+- For legacy tool types (non-manifest), default to `/root`
+- Apply expansion at compose generation time for both regular mounts and git mount targets
+- Also set `HOME` env var in the Dockerfile for runtime compatibility
+
+## Acceptance Criteria
+
+1. `~` in a mount target path is expanded to the container's home directory
+2. `$HOME` in a mount target path is expanded to the container's home directory
+3. For manifest-based tools with `user.name`, home is `/home/{user.name}`
+4. For manifest-based tools without user block, home is `/root`
+5. For legacy tool types, home is `/root`
+6. Git mount `mapping.target_path` also supports `~` and `$HOME`
+7. `HOME` env var is set in generated Dockerfile
+8. No frontend changes needed (users type `~`, backend resolves it)
+
+## Out of Scope
+
+- `~` expansion in host-side source paths
+- `~` expansion in mount file relative paths
+- `~user` syntax (e.g., `~alice`)
+
+## Risks
+
+| Risk | Mitigation |
+|------|------------|
+| Wrong home for custom containers | Document that manifest should declare `user.name` |
+| `$HOME` env var not set in container | Set it in Dockerfile via `ENV HOME=...` |
+
+## Related Artifacts
+
+- Exploration: `openspec/explorations/home-path-expansion.md`
+- Spec: `openspec/specs/home-path-expansion.md`
+- Design: `openspec/designs/home-path-expansion.md`
+- Tasks: `openspec/tasks/home-path-expansion.md`