practice entry points: sidebar filter, practice draft, graph picker

- Sidebar: the three-way kind filter (Teaching/All/Practice, persisted
  in localStorage) above the search; two new-session doors (Teaching →
  /chat/new, Practice → /chat/new?mode=practice); practice rows carry
  the dumbbell badge and 'Practicing…' busy meta.
- Draft: ?mode=practice swaps the copy (title, empty state, 'What do
  you want to practice?') and startDraftSession posts kind:'practice'.
- Graph entry (home composer): teach|practice switch on the stats row's
  left edge (StatsBar leadingSlot); practice mode shows the propositions
  chip that opens the chain table as a picker (its floating bar becomes
  'Practice these' — no delete fence); the picks seed the opening prompt
  ('Propositions from the map:…') while the title stays the typed text.

app/chat/new/page.tsx

@@ -15,8 +15,16 @@ export const dynamic = "force-dynamic";
  * default); the session the first send creates is stamped from it
  * server-side (per-session models, owner 2026-09-28).
  */
-export default function NewSessionPage() {
+export default async function NewSessionPage({
+	searchParams,
+}: {
+	searchParams: Promise<{ mode?: string }>;
+}) {
+	const { mode } = await searchParams;
 	return (
-		<NewSession initialConfigured={slotsConfigured(readSettingsRecord()?.models)} />
+		<NewSession
+			initialConfigured={slotsConfigured(readSettingsRecord()?.models)}
+			mode={mode === "practice" ? "practice" : "teach"}
+		/>
 	);
 }