Fix the mode-choice redundancy: standardized names, deep links, compact switch

Terminology (lib/i18n-dict.ts)
- One name per mode everywhere: "Study Deck" / "Studijní balíček" and
  "Homework Solver" / "Řešitel úkolů". The library empty-state cards and the
  /create switcher now read the same, so the choice no longer changes name
  between screens.

Deep links
- Library quick actions carry the mode: /create?mode=deck and
  /homework?mode=homework.
- Both pages canonicalize: /create?mode=homework redirects to the homework
  flow and /homework?mode=deck redirects to /create, so a mode link always
  opens that mode directly.

/create
- The two bulky square cards (106px tall, wrapping/clipping "Study Deck &
  Qu…" on phones) are replaced by a compact segmented control (52px) placed
  directly above "1. Add your notes": ice pill, snowflake/sparkles icons,
  44px segments and one-line labels at 360px. The pill slides to the chosen
  segment and the route follows ~190ms later, so toggling reads as one smooth
  transition (immediate under prefers-reduced-motion).

Service worker: freezmo-v9.

Verified: mode-switch suite 26/26 (deep links, cross-mode redirects, 52px
control, 44px segments, no clipping/wrapping at 360px, "1. Add your notes" at
y=394 of 740 with the upload controls above the fold, pill slide, axe contrast,
en+cs, phone+desktop), mobile 49/49, i18n 27/27, desktop regression 73/73.
Lint + build clean.

components/ModeSwitch.tsx

@@ -0,0 +1,80 @@
+"use client";
+
+import { useRouter } from "next/navigation";
+import { useEffect, useRef, useState } from "react";
+
+import { SnowflakeIcon, SparklesIcon } from "@/components/icons";
+import { useI18n } from "@/lib/i18n";
+import { cn } from "@/lib/utils";
+
+export type StudyMode = "deck" | "homework";
+
+/** Canonical URL per mode, so a deep link always lands on the right screen. */
+export const MODE_HREF: Record<StudyMode, string> = {
+  deck: "/create?mode=deck",
+  homework: "/homework?mode=homework",
+};
+
+/** How long the ice pill takes to slide before the route follows. */
+const SLIDE_MS = 180;
+
+/**
+ * Compact two-segment mode switch (Study Deck / Homework Solver). The ice pill
+ * slides to the chosen segment and the router follows a moment later, so the
+ * toggle reads as one smooth transition instead of a hard jump. The current
+ * mode stays pressed; the other segment navigates.
+ */
+export function ModeSwitch({ current }: { current: StudyMode }) {
+  const { t } = useI18n();
+  const router = useRouter();
+  const [pending, setPending] = useState<StudyMode | null>(null);
+  const timer = useRef<number | null>(null);
+
+  useEffect(
+    () => () => {
+      if (timer.current !== null) window.clearTimeout(timer.current);
+    },
+    [],
+  );
+
+  const choose = (mode: StudyMode) => {
+    if (mode === current || pending !== null) return;
+    setPending(mode);
+    const reduce = window.matchMedia("(prefers-reduced-motion: reduce)").matches;
+    timer.current = window.setTimeout(() => router.push(MODE_HREF[mode]), reduce ? 0 : SLIDE_MS + 10);
+  };
+
+  const shown = pending ?? current;
+
+  const segment = (mode: StudyMode, icon: React.ReactNode, label: string) => (
+    <button
+      type="button"
+      aria-pressed={current === mode}
+      onClick={() => choose(mode)}
+      className={cn(
+        "relative z-10 inline-flex min-h-11 items-center justify-center gap-1.5 rounded-xl px-2 text-[13.5px] font-semibold transition-colors",
+        shown === mode ? "text-[#06111f]" : "text-frost-700 hover:text-[var(--text)] dark:text-frost-300",
+      )}
+    >
+      {icon}
+      <span className="whitespace-nowrap">{label}</span>
+    </button>
+  );
+
+  return (
+    <div
+      data-testid="mode-switch"
+      className="relative grid grid-cols-2 rounded-[0.95rem] border border-[var(--hairline)] bg-[var(--surface-sunken)] p-[3px] backdrop-blur-sm"
+    >
+      <span
+        aria-hidden
+        className={cn(
+          "grad-ice pointer-events-none absolute inset-y-[3px] left-[3px] w-[calc(50%-3px)] rounded-xl shadow-[0_6px_16px_-8px_rgb(56_189_248/0.7)] transition-transform duration-[180ms] ease-out motion-reduce:transition-none",
+          shown === "homework" && "translate-x-full",
+        )}
+      />
+      {segment("deck", <SnowflakeIcon className="size-4" />, t("actionStudyDeck"))}
+      {segment("homework", <SparklesIcon className="size-4" />, t("actionHomework"))}
+    </div>
+  );
+}