feat: implement configurable tool container home directory

- Add ToolType.home_directory column with default /home/user
- Add Alembic migration to add column, set existing rows, and rewrite
  /workspace to /home/user/{{WORKSPACE_NAME}} in legacy templates
- Add merge migration fc8f1a20cbf6 to resolve Alembic multiple heads
- Update manifest compiler to honor manifest.home_directory for HOME,
  WORKDIR, /workspace symlink, and default repo mount target
- Update legacy dockerfile/compose instance generation to use
  tool_type.home_directory
- Thread resolved home_dir through config profile and git mount expansion
- Generate entrypoint permission fixer to chown home/mounts at startup
- Update base.dockerfile with sudo/passwordless sudo for permission fixer
- Add unit tests for manifest compiler, instance service, and migrations
- Add placeholder integration test for container lifecycle
- Update openspec/tasks/home-path-expansion.md task checkboxes
- Update project maps for modified files

Quality gates: py_compile, ruff, mypy, pytest tests/unit (205 passed),
pytest tests/integration (110 passed, 35 skipped). Alembic round-trip
and container lifecycle integration tests require Docker/PostgreSQL.

apps/api/src/models/.pi-map.index.md

@@ -2,7 +2,7 @@
 dir: apps/api/src/models
 
 ## role
-Centralized database model definitions and shared infrastructure for the API's data layer.
+Provides the SQLAlchemy ORM data models and database schema definitions for the API application.
 ## parent
 index: apps/api/src/.pi-map.index.md
 map: apps/api/src/.pi-map.md

apps/api/src/models/.pi-map.md

@@ -4,12 +4,12 @@ dir: apps/api/src/models
 index: apps/api/src/models/.pi-map.index.md
 
 ## role
-Centralized database model definitions and shared infrastructure for the API's data layer.
+Provides the SQLAlchemy ORM data models and database schema definitions for the API application.
 ## files
 - __init__.py | Re-exports model classes from submodules to provide a centralized public API for the src.models package | dep: src.models.base, src.models.config.config_profile, src.models.project.git_repository, src.models.project.project, src.models.project.workspace, src.models.system.health_check, src.models.system.instance_event, src.models.system.notification, src.models.system.terminal_session, src.models.tool.tool_definition_manifest, src.models.tool.tool_instance, src.models.tool.tool_type, src.models.user.ssh_key, src.models.user.user, src.models.user.user_config
 - base.py | Defines SQLAlchemy base model and reusable mixins for UUID primary keys and automatic timestamp tracking in database models. | exp: class:Base, class:UUIDPrimaryKeyMixin, class:TimestampMixin | dep: uuid, datetime, sqlalchemy, sqlalchemy.orm
 ## arch
-SQLAlchemy ORM with declarative base, mixin-based composition for cross-cutting concerns (UUIDs, timestamps), and explicit package-level re-exports for clean public API surface.
+Layered repository pattern with declarative SQLAlchemy base, UUID/timestamp mixins for reusable model traits, and package-level facade pattern via __init__.py re-exports to centralize model access.
 ## tags
 models, src, base, project, system, user, mixin, tool
 ## symbols

apps/api/src/models/tool/.pi-map.index.md

@@ -2,7 +2,7 @@
 dir: apps/api/src/models/tool
 
 ## role
-Database models for containerized tool lifecycle management, covering tool definitions, deployment instances, and categorization types.
+Provides SQLAlchemy ORM models for managing containerized tool definitions, types, and deployed instances in the API.
 ## parent
 index: apps/api/src/models/.pi-map.index.md
 map: apps/api/src/models/.pi-map.md

apps/api/src/models/tool/.pi-map.md

@@ -4,16 +4,16 @@ dir: apps/api/src/models/tool
 index: apps/api/src/models/tool/.pi-map.index.md
 
 ## role
-Database models for containerized tool lifecycle management, covering tool definitions, deployment instances, and categorization types.
+Provides SQLAlchemy ORM models for managing containerized tool definitions, types, and deployed instances in the API.
 ## files
 - __init__.py | Exports the public API for the tool models module by re-exporting three key classes. | dep: src.models.tool.tool_definition_manifest, src.models.tool.tool_instance, src.models.tool.tool_type
 - tool_definition_manifest.py | Defines a SQLAlchemy ORM model for storing tool definition manifests that compile to Dockerfiles and Compose files, supporting both base definitions and tool-specific definitions with inheritance. | exp: class:ToolDefinitionManifest | dep: uuid, typing, sqlalchemy, sqlalchemy.orm, src.models.base, src.models.user
 - tool_instance.py | Defines a SQLAlchemy ORM model for tool instances that represent deployed tools with container metadata, status tracking, and relationships to users, projects, workspaces, and other entities. | exp: class:ToolInstance | dep: uuid, datetime, typing, sqlalchemy, sqlalchemy.orm, src.models.base, src.models, src.models.project, src.models.user, src.models (ConfigProfile, GitRepository, Project, ToolType, User, Workspace)
-- tool_type.py | Defines a SQLAlchemy ORM model for tool types that represent configurable categories of tools with deployment templates, manifest references, and metadata. | exp: class:ToolType | dep: uuid, typing, sqlalchemy, sqlalchemy.orm, src.models.base, src.models.tool.tool_definition_manifest, src.models.user
+- tool_type.py | Defines a SQLAlchemy ORM model for tool types that specify configuration templates and metadata for deployable tools in a containerized environment. | exp: class:ToolType | dep: uuid, typing, sqlalchemy, sqlalchemy.orm, src.models.base, src.models.tool.tool_definition_manifest, src.models.user
 ## arch
-SQLAlchemy ORM with declarative models using inheritance hierarchies, relationship mappings, and polymorphic manifest compilation for Docker/Compose deployment.
+Domain-driven data models using SQLAlchemy ORM with declarative base pattern, entity relationships, and inheritance support for tool manifest definitions.
 ## tags
-tool, models, src, sqlalchemy, orm, definition, manifest, base
+tool, models, src, sqlalchemy, orm, definition, base, manifest
 ## symbols
 - ToolDefinitionManifest
 - ToolInstance

apps/api/src/models/tool/tool_type.py

@@ -39,6 +39,9 @@ class ToolType(UUIDPrimaryKeyMixin, TimestampMixin, Base):
     )
     readiness_probe: Mapped[dict | None] = mapped_column(JSON, nullable=True)
     startup_command: Mapped[str | None] = mapped_column(Text, nullable=True)
+    home_directory: Mapped[str] = mapped_column(
+        String(255), nullable=False, default="/home/user"
+    )
     required_variables: Mapped[list[str]] = mapped_column(
         JSON, default=list, nullable=False
     )