feat(ai): zod due_date field + prompt {{TODAY_KST}} injection

AiResponse extends with dueDate: string|null. zod regex
^\d{4}-\d{2}-\d{2}$, follow-up roundtrip check coerces invalid
dates (2026-13-99 etc.) to null. PROMPT_VERSION → 2: prompt now
takes todayKst arg, asks model to extract due_date as ISO or null.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

tests/unit/ai-schema.test.ts

@@ -52,4 +52,49 @@ describe('parseAiResponse', () => {
   it('rejects non-object input', () => {
     expect(() => parseAiResponse('nope')).toThrow();
   });
+
+  it('parses note with valid due_date', () => {
+    const r = parseAiResponse({
+      title: '내일 회의',
+      summary: 'a\nb\nc',
+      tags: [],
+      due_date: '2026-04-27'
+    });
+    expect(r.dueDate).toBe('2026-04-27');
+  });
+
+  it('null due_date passes through', () => {
+    const r = parseAiResponse({
+      title: '내일 회의',
+      summary: 'a\nb\nc',
+      tags: []
+    });
+    expect(r.dueDate).toBeNull();
+  });
+
+  it('explicit null due_date passes through', () => {
+    const r = parseAiResponse({
+      title: '내일 회의',
+      summary: 'a\nb\nc',
+      tags: [],
+      due_date: null
+    });
+    expect(r.dueDate).toBeNull();
+  });
+
+  it('rejects malformed due_date string', () => {
+    expect(() =>
+      parseAiResponse({ title: '내일', summary: 'a\nb\nc', tags: [], due_date: 'tomorrow' })
+    ).toThrow();
+  });
+
+  it('coerces invalid date that passes regex (e.g. 2026-13-99) to null', () => {
+    const r = parseAiResponse({
+      title: '내일 회의',
+      summary: 'a\nb\nc',
+      tags: [],
+      due_date: '2026-13-99'
+    });
+    expect(r.dueDate).toBeNull();
+  });
 });