play whole islands, make home the path itself, draw tutor diagrams, add tedi

- Home is now the island path of every unit with one clear Start button on the
  current island; the six-card dashboard is gone.
- Start opens /nodes/{id}/quiz (4 questions) instead of one-question lessons, so
  an island takes one round instead of ~11. Lessons stay one tap away in the sheet.
- Tutor answers render the backend's validated diagram (plane, number line,
  fraction) as React SVG; function expressions go through a small parser, no eval.
- Tedi, the mobile-preview mascot, joins as the fourth companion with all six moods
  and an onboarding slide.
- Light polish: 3D buttons, header gradients, warmer page background.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

src/api/index.ts

@@ -75,6 +75,15 @@ export const api = {
   home: () => http.get<T.HomeResponse>('/home'),
   nodeLessons: (nodeId: string) => http.get<T.NodeLessons>(`/nodes/${nodeId}/lessons`),
 
+  /**
+   * آزمونِ جزیره: از هر درسِ جزیره حداکثر یک سوال (پیش‌فرض ۴). ترتیب برای هر
+   * کاربر ثابت است، پس رفرش همان آزمون را می‌دهد.
+   */
+  nodeQuiz: (nodeId: string) => http.get<T.NodeQuiz>(`/nodes/${nodeId}/quiz`),
+  /** پایانِ آزمون، جزیره را با هر نمره‌ای تمام می‌کند و جزیره‌ی بعدی را باز. */
+  nodeQuizComplete: (nodeId: string, result: { correctCount: number; totalQuestions: number }, idempotencyKey?: string) =>
+    http.create<T.NodeQuizComplete>(`/nodes/${nodeId}/quiz/complete`, result, { idempotencyKey }),
+
   lesson: {
     intro: (lessonId: number) => http.get<T.LessonIntro>(`/lessons/${lessonId}/intro`),
     questions: (lessonId: number) =>