Swap the server engine to pi-ai/pi-agent-core (Band B+C of docs/pi-engine-swap.md)

The Vercel AI SDK survives only as Band D (useChat client + UIMessage
record types); @openrouter/ai-sdk-provider is uninstalled.

- src/agent/models.ts: PiEngine {model, streamFn} via createModels +
  openrouterProvider subpath; thinkingLevelFor maps config effort none->off
- src/agent/lh-tool.ts: neutral LhTool (zod authoring, optional execute,
  clientQuestion flag); toPiTool converts via z.toJSONSchema and re-runs
  zod inside execute so refinements ride the error-result retry path
- src/server/pi/record-adapter.ts: UIMessage records -> pi Messages
  (dangling tool calls get synthesized interrupted results)
- src/server/pi/ui-stream.ts: SSE bridge emitting AI-SDK-shaped chunks so
  the useChat client parses unchanged (continuation id rule, question
  tools never emit output, tool-input-error for bad authoring)
- teacher-agent.ts: stateless per-turn pi Agents; shouldStopAfterTurn
  stops only when every question-tool call resolved cleanly (errors retry)
- chat-handler.ts: buildAgent(outgoing) factory seam (stateless doctrine);
  assertValidUiMessages owns the retired SDK validation gate
- sub-agents via shared runSubagent (pi-run.ts); extraction stages stop
  after a clean submit; visual-maker keeps its hand-rolled loop over
  direct streamSimple calls; aborted model calls bill no usage
- tests: all model seams ported to pi-ai faux provider (scripted
  FauxResponseSteps, agent factories); handleChatRequest second param
  is now an agent factory

Gates: tsc, lint, build, 305/305. Live GLM smoke (disposable driver,
deleted): prober streaming + persistence, question seam, turn-2
continuation same message id, per-turn real token usage - ALL PASS.
Data wiped at the pivot (chats/graph/jobs, .gitkeeps kept).

tests/question-tools.test.ts

@@ -50,10 +50,10 @@ const validQuiz = {
 };
 
 interface QuizToolSpec {
-	inputSchema: { safeParse: (v: unknown) => { success: boolean; error?: { issues: Array<{ message: string }> } } };
+	schema: { safeParse: (v: unknown) => { success: boolean; error?: { issues: Array<{ message: string }> } } };
 }
 /** teachQuizToolFor returns the single quiz tool; createProberTools a registry. */
-const schemaOf = (toolLike: unknown) => (toolLike as QuizToolSpec).inputSchema;
+const schemaOf = (toolLike: unknown) => (toolLike as QuizToolSpec).schema;
 const quizSchemaOf = (tools: Record<string, unknown>) => schemaOf(tools.quiz);
 
 describe("teachQuizToolFor: the dynamic plan-nodeId validation (position signal)", () => {