Implement PWA shell (task 50)

Add web manifest, placeholder icons, app-shell service worker, and
install prompt so famapp is installable on iOS and Android home screens.

- public/manifest.webmanifest — name, short_name, standalone display,
  start_url /, all four icon sizes (192/384/512 + maskable)
- public/icon-{180,192,384,512,512-maskable}.png — indigo placeholder
  squares generated by scripts/generate-icons.mjs (pnpm gen:icons)
- public/icon.svg — house-icon SVG source for future branded export
- public/sw.js — hand-rolled app-shell SW: precaches offline.html,
  cache-first for /_next/static/, network-first navigation with offline
  fallback, network-only for /api/*
- public/offline.html — branded offline fallback page
- src/components/pwa-register.tsx — client SW registration
- src/components/install-prompt.tsx — dismissible install banner:
  beforeinstallprompt on Android, "Add to Home Screen" hint on iOS
- Root layout: viewport themeColor, manifest/appleWebApp metadata,
  apple-touch-icon, mounts PwaRegister + InstallPrompt

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

scripts/generate-icons.mjs

@@ -0,0 +1,101 @@
+/**
+ * Generates placeholder PNG icons for the famapp PWA.
+ *
+ * Usage: node scripts/generate-icons.mjs
+ *
+ * Produces:
+ *   public/icon-192.png
+ *   public/icon-384.png
+ *   public/icon-512.png
+ *   public/icon-512-maskable.png  (same image; OS applies its own mask)
+ *   public/icon-180.png           (apple-touch-icon)
+ *
+ * All images are solid indigo (#4F46E5) squares — replace with a real
+ * branded export from icon.svg when assets are finalised.
+ */
+
+import { deflateSync } from "zlib";
+import { writeFileSync } from "fs";
+import { resolve, dirname } from "path";
+import { fileURLToPath } from "url";
+
+const __dir = dirname(fileURLToPath(import.meta.url));
+const publicDir = resolve(__dir, "../public");
+
+// Build CRC-32 lookup table once.
+const CRC_TABLE = new Uint32Array(256);
+for (let i = 0; i < 256; i++) {
+  let c = i;
+  for (let k = 0; k < 8; k++) c = c & 1 ? 0xedb88320 ^ (c >>> 1) : c >>> 1;
+  CRC_TABLE[i] = c;
+}
+
+function crc32(buf) {
+  let crc = 0xffffffff;
+  for (let i = 0; i < buf.length; i++)
+    crc = CRC_TABLE[(crc ^ buf[i]) & 0xff] ^ (crc >>> 8);
+  return (crc ^ 0xffffffff) >>> 0;
+}
+
+function u32(n) {
+  const b = Buffer.alloc(4);
+  b.writeUInt32BE(n, 0);
+  return b;
+}
+
+function pngChunk(type, data) {
+  const typeBytes = Buffer.from(type, "ascii");
+  const crc = u32(crc32(Buffer.concat([typeBytes, data])));
+  return Buffer.concat([u32(data.length), typeBytes, data, crc]);
+}
+
+/**
+ * Returns a Buffer containing a valid PNG of size×size filled with (r, g, b).
+ */
+function solidPNG(size, r, g, b) {
+  const PNG_SIG = Buffer.from([137, 80, 78, 71, 13, 10, 26, 10]);
+
+  const ihdr = pngChunk(
+    "IHDR",
+    Buffer.concat([
+      u32(size),
+      u32(size),
+      Buffer.from([8, 2, 0, 0, 0]), // 8-bit RGB, no interlace
+    ])
+  );
+
+  // One filter byte (0 = None) followed by size×3 RGB bytes per row.
+  const rowLen = 1 + size * 3;
+  const raw = Buffer.alloc(size * rowLen);
+  for (let y = 0; y < size; y++) {
+    const base = y * rowLen;
+    raw[base] = 0;
+    for (let x = 0; x < size; x++) {
+      raw[base + 1 + x * 3] = r;
+      raw[base + 2 + x * 3] = g;
+      raw[base + 3 + x * 3] = b;
+    }
+  }
+
+  const idat = pngChunk("IDAT", deflateSync(raw));
+  const iend = pngChunk("IEND", Buffer.alloc(0));
+
+  return Buffer.concat([PNG_SIG, ihdr, idat, iend]);
+}
+
+// Indigo-600 (#4F46E5)
+const [R, G, B] = [0x4f, 0x46, 0xe5];
+
+const icons = [
+  { name: "icon-192.png", size: 192 },
+  { name: "icon-384.png", size: 384 },
+  { name: "icon-512.png", size: 512 },
+  { name: "icon-512-maskable.png", size: 512 },
+  { name: "icon-180.png", size: 180 },
+];
+
+for (const { name, size } of icons) {
+  const out = resolve(publicDir, name);
+  writeFileSync(out, solidPNG(size, R, G, B));
+  console.log(`  ✓  public/${name}  (${size}×${size})`);
+}