feat: flow runtime on LFX with tests

- LfxClient (upload/run/info/delete, typed errors, serve-key auth)
- Flows API: CRUD, versions, deploy, sync run, SSE stream proxy, runs
- Approvals callback endpoint (flow terminal-state contract)
- Runner: envelope in, status from approval side-effect (no interrupts)
- Fixture generator (lfx serializers) + simple_rule/approval_handoff
  fixtures verified against real lfx serve
- 14 tests green (real LFX subprocess + live uvicorn callback server)

backend/app/flows/schemas.py

@@ -0,0 +1,86 @@
+"""Pydantic contracts for the flows API."""
+
+from datetime import datetime
+from uuid import UUID
+
+from pydantic import BaseModel, Field
+
+
+class FlowDefinitionCreate(BaseModel):
+    name: str = Field(min_length=1, max_length=200)
+    description: str = Field(default="", max_length=2000)
+    continuation_flow_id: UUID | None = None
+
+
+class FlowDefinitionRead(BaseModel):
+    id: UUID
+    name: str
+    description: str
+    status: str
+    continuation_flow_id: UUID | None
+    created_at: datetime
+
+
+class FlowVersionCreate(BaseModel):
+    langflow_json: dict
+    author: str = Field(default="", max_length=200)
+
+
+class FlowVersionRead(BaseModel):
+    id: UUID
+    flow_id: UUID
+    version_no: int
+    langflow_json: dict
+    lfx_flow_id: str | None
+    deployed_at: datetime | None
+    author: str
+    created_at: datetime
+
+
+class RunRequest(BaseModel):
+    """Start a flow run. `input` is the domain payload handed to the flow."""
+
+    input: dict = Field(default_factory=dict)
+    version_no: int | None = None
+    tenant_id: str = Field(default="default", max_length=200)
+    session_id: str | None = None
+    # Base URL the flow uses for callbacks (e.g. approval tasks).
+    # Defaults to settings.callback_base_url; tests point it at a live server.
+    callback_base_url: str | None = None
+    continuation_flow_id: UUID | None = None
+
+
+class RunRead(BaseModel):
+    id: UUID
+    version_id: UUID
+    tenant_id: str
+    status: str
+    input: dict | None
+    output: dict | None
+    session_id: str | None
+    created_at: datetime
+    finished_at: datetime | None
+
+
+class ApprovalCreate(BaseModel):
+    """Callback contract: a flow reports a terminal approval state."""
+
+    run_id: UUID
+    title: str = Field(default="", max_length=500)
+    payload: dict = Field(default_factory=dict)
+    continuation_flow_id: UUID | None = None
+
+
+class ApprovalRead(BaseModel):
+    id: UUID
+    run_id: UUID
+    version_id: UUID
+    tenant_id: str
+    title: str
+    payload: dict
+    continuation_flow_id: UUID | None
+    approver: str | None
+    status: str
+    decision_note: str
+    created_at: datetime
+    decided_at: datetime | None