Add “Make my homework”, and finish the mobile polish pass

Homework (/homework — entry points in the drawer and on the create screen)

- Assignment intake reuses the study pipeline: pasted text, photos
  (camera/picker, compressed in the browser) and documents (PDF/DOCX/PPTX/
  ODT/text) through lib/image + lib/documents, the same vision step and the
  same language guard. Raw files never leave the device — only text and JPEG
  data URLs reach the server.
- Two large choice cards: “Just make it” returns the finished homework
  (answer only, plain text, copy + make it again); “Explain how to make this
  homework” builds an interactive guide.
- The guide is one step per screen in the study-feed snap layout:
  explanation, three progressive hints (nudge -> hint -> the answer) and an
  optional checkpoint rendered by the existing activity components (graded
  through /api/grade). The last screen assembles the solution from the step
  answers, with a copy button. Checkpoints carry xp 0 so the guide stays free
  of XP/streak chrome.
- Results live in freezmo.homework.v1 with a small history (open/delete);
  “Make it again” re-runs from the stored assignment text (the vision
  transcription when photos were used) — no original files are stored.
- The answer and the guide are written in the assignment's language, never
  translated into the UI language.

Server

- POST /api/homework: NDJSON stage events like /api/generate, the same
  http-guard, auth checked before parsing, and its own RATE_LIMIT_HOMEWORK
  (documented in .env.example + the README table).
- New prompts in lib/prompts.ts and zod schemas in lib/schemas.ts with the
  same single repair retry; TEXT_MODEL for content, VISION_MODEL only for
  photos. New shared client pieces: useNotesIntake, lib/homework store,
  lib/clipboard, and a shared NDJSON reader in lib/client-api.

Also finishing the mobile polish pass from the previous request

- Phone ice scene: the top-left orb moved behind the glass header (it sat
  behind the page heading/subtitle). The settings sheet's scroll cue is now a
  bare chevron inside a content fade (mask) instead of a chip over the status
  banner, and the API key card's note/disclosure/actions are tightened.
- theme-color now follows the in-app theme (SSR metas still fall back to the
  OS preference); apple-mobile-web-app-status-bar-style stays “default”
  because iOS fixes the status-bar text colour at launch.

Verified: 78/78 homework checks (text/photo/PDF, both modes, EN+CS, dark+light,
360/390px, copy, history persistence, storage privacy, API language, axe), plus
the existing suites re-run (mobile 47/47, pixel contrast 0 below AA, i18n 27/27,
desktop regression 72/72). Lint + build clean; service worker freezmo-v6.

lib/schemas.ts

@@ -135,6 +135,95 @@ export const gradeResultSchema = z.object({
   missing: z.array(z.string().max(200)).max(6).default([]),
 });export type GradeResult = z.infer<typeof gradeResultSchema>;
 
+/* ------------------------------------------------------------------ */
+/* Make my homework                                                    */
+/* ------------------------------------------------------------------ */
+
+export const homeworkModeSchema = z.enum(["make", "guide"]);
+export type HomeworkMode = z.infer<typeof homeworkModeSchema>;
+
+/** Mode 1 — the finished homework, nothing else. */
+export const homeworkMakeSchema = z.object({
+  title: z.string().min(1).max(200),
+  /** English name of the language the assignment was written in. */
+  language: z.string().max(60).optional(),
+  answer: z.string().min(1).max(20_000),
+});
+export type HomeworkMake = z.infer<typeof homeworkMakeSchema>;
+
+/**
+ * One step of the interactive guide, as the model returns it. `checkpoint` is
+ * a normal study activity (validated by the same schema as the feed) that the
+ * student answers to check the step landed.
+ */
+export const homeworkStepSchema = z.object({
+  title: z.string().min(1).max(200),
+  explanation: z.string().min(1).max(1500),
+  /** Progressive help: nudge -> real hint -> the full answer for this step. */
+  hints: z.array(z.string().min(1).max(1500)).min(1).max(4),
+  /** The full answer for this step; the guide assembles the solution from these. */
+  answer: z.string().min(1).max(5000),
+  checkpoint: generatedActivitySchema.optional(),
+});
+
+export const homeworkGuideSchema = z.object({
+  title: z.string().min(1).max(200),
+  language: z.string().max(60).optional(),
+  steps: z.array(homeworkStepSchema).min(2).max(10),
+});
+export type HomeworkGuide = z.infer<typeof homeworkGuideSchema>;
+
+/* ------------------------------------------------------------------ */
+/* Client-side homework types (checkpoints carry id/xp, plus storage)  */
+/* ------------------------------------------------------------------ */
+
+export const homeworkCheckpointSchema = generatedActivitySchema.and(
+  z.object({ id: z.string(), xp: z.number() }),
+);
+export type HomeworkCheckpoint = z.infer<typeof homeworkCheckpointSchema>;
+
+export type HomeworkStep = {
+  title: string;
+  explanation: string;
+  hints: string[];
+  answer: string;
+  checkpoint?: HomeworkCheckpoint;
+};
+
+export const homeworkStepStoredSchema: z.ZodType<HomeworkStep> = z.object({
+  title: z.string().min(1).max(200),
+  explanation: z.string().min(1).max(1500),
+  hints: z.array(z.string().min(1).max(1500)).min(1).max(4),
+  answer: z.string().min(1).max(5000),
+  checkpoint: homeworkCheckpointSchema.optional(),
+});
+
+const homeworkBaseFields = {
+  id: z.string().min(1).max(80),
+  title: z.string().min(1).max(200),
+  language: z.string().max(60).optional(),
+  createdAt: z.number(),
+  updatedAt: z.number(),
+  /** The assignment text (photos/documents were reduced to text client-side). */
+  sourceText: z.string().max(40_000),
+};
+
+export const homeworkRecordSchema = z.discriminatedUnion("mode", [
+  z.object({
+    ...homeworkBaseFields,
+    mode: z.literal("make"),
+    answer: z.string().min(1).max(20_000),
+  }),
+  z.object({
+    ...homeworkBaseFields,
+    mode: z.literal("guide"),
+    steps: z.array(homeworkStepStoredSchema).min(2).max(10),
+  }),
+]);
+export type HomeworkRecord = z.infer<typeof homeworkRecordSchema>;
+export type HomeworkMakeRecord = Extract<HomeworkRecord, { mode: "make" }>;
+export type HomeworkGuideRecord = Extract<HomeworkRecord, { mode: "guide" }>;
+
 /* ------------------------------------------------------------------ */
 /* Client-side deck types (activity + id/xp, plus storage metadata)     */
 /* ------------------------------------------------------------------ */