One-pill study chrome, toggling chat pickers, readable homework steps

UI batch from feedback:

- The study/review/exam/homework chrome is now a single 58px pill: round
  close button, title + position over the progress bar, chips (or the exam
  timer) at the far end — everything on the centre line, so nothing crowds
  the rounded corners any more. Homework guide gets the same recipe.
- The immersive feeds carry z-[45] so they paint over the app chrome (on
  /exam the header used to cover the top of the chrome), and the page chrome
  behind an open overlay (.app-header, .segmented, .tabbar) is hidden while
  the transparent overlay is up, so it can no longer ghost through the glass.
  The guide's disclaimer moved inside the overlay with it.
- Chat: the context and model chips toggle their own popover (pressed state,
  aria-expanded), opening one closes the other, a press outside closes it —
  no more being forced to pick something to get out.
- Homework: the model's plain-text step explanations/hints/verdicts keep
  their line breaks (whitespace-pre-wrap) — a step used to collapse into one
  run-on paragraph with the whole solution inside it; the guide prompt now
  says the explanation is coaching, 2-4 sentences.
- The four glacier blobs are at 25% of their old opacity (0.045 light,
  0.1 dark).

Tests: phase-2 chrome geometry + exam layering checks (62), chat desktop
picker checks (101), the standing suites all green. README: decisions 54-56,
decision 23 updated. SW -> freezmo-v44.

components/chat/ContextPicker.tsx

@@ -1,6 +1,6 @@
 "use client";
 
-import { useEffect } from "react";
+import { useEffect, useRef } from "react";
 import { AnimatePresence, motion, useReducedMotion } from "framer-motion";
 
 import { CheckCircleIcon, FolderIcon, LayersIcon } from "@/components/icons";
@@ -34,6 +34,7 @@ export function ContextPicker({
   const { t, plural } = useI18n();
   const isPhone = useIsPhone();
   const reduceMotion = useReducedMotion();
+  const panel = useRef<HTMLDivElement>(null);
 
   useEffect(() => {
     if (!open) return;
@@ -44,6 +45,21 @@ export function ContextPicker({
     return () => window.removeEventListener("keydown", onKey);
   }, [open, onClose]);
 
+  /* Desktop dropdown: a press anywhere outside closes it, except on the
+     trigger chips, which toggle their own popover (see Composer). */
+  useEffect(() => {
+    if (!open || isPhone) return;
+    const onDown = (event: PointerEvent) => {
+      const target = event.target;
+      if (!(target instanceof Element)) return;
+      if (panel.current?.contains(target)) return;
+      if (target.closest("[data-picker-trigger]")) return;
+      onClose();
+    };
+    document.addEventListener("pointerdown", onDown);
+    return () => document.removeEventListener("pointerdown", onDown);
+  }, [open, isPhone, onClose]);
+
   const selected = scope.kind === "decks" ? new Set(scope.deckIds) : new Set<string>();
 
   const choose = (next: ChatScope) => {
@@ -207,6 +223,7 @@ export function ContextPicker({
     <AnimatePresence>
       {open ? (
         <motion.div
+          ref={panel}
           initial={reduceMotion ? { opacity: 0 } : { opacity: 0, y: 6, scale: 0.98 }}
           animate={{ opacity: 1, y: 0, scale: 1 }}
           exit={reduceMotion ? { opacity: 0 } : { opacity: 0, y: 6, scale: 0.98 }}