feat: Add session management and global Claude terminal

Phase 1 - Accurate study status detection:
- Add is_optimization_running() to check for active processes
- Add get_accurate_study_status() with proper status logic
- Status now: not_started, running, paused, completed
- Add "paused" status styling (orange) to Home page

Phase 2 - Global Claude terminal:
- Create ClaudeTerminalContext for app-level state
- Create GlobalClaudeTerminal floating component
- Terminal persists across page navigation
- Shows green indicator when connected
- Remove inline terminal from Dashboard

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

atomizer-dashboard/frontend/src/components/ClaudeTerminal.tsx

@@ -13,6 +13,7 @@ import {
   FolderOpen
 } from 'lucide-react';
 import { useStudy } from '../context/StudyContext';
+import { useClaudeTerminal } from '../context/ClaudeTerminalContext';
 
 interface ClaudeTerminalProps {
   isExpanded?: boolean;
@@ -26,17 +27,24 @@ export const ClaudeTerminal: React.FC<ClaudeTerminalProps> = ({
   onClose
 }) => {
   const { selectedStudy } = useStudy();
+  const { setIsConnected: setGlobalConnected } = useClaudeTerminal();
   const terminalRef = useRef<HTMLDivElement>(null);
   const xtermRef = useRef<Terminal | null>(null);
   const fitAddonRef = useRef<FitAddon | null>(null);
   const wsRef = useRef<WebSocket | null>(null);
-  const [isConnected, setIsConnected] = useState(false);
+  const [isConnected, setIsConnectedLocal] = useState(false);
   const [isConnecting, setIsConnecting] = useState(false);
   const [_error, setError] = useState<string | null>(null);
   const [cliAvailable, setCliAvailable] = useState<boolean | null>(null);
   const [contextSet, setContextSet] = useState(false);
   const [settingContext, setSettingContext] = useState(false);
 
+  // Sync local connection state to global context
+  const setIsConnected = useCallback((connected: boolean) => {
+    setIsConnectedLocal(connected);
+    setGlobalConnected(connected);
+  }, [setGlobalConnected]);
+
   // Check CLI availability
   useEffect(() => {
     fetch('/api/terminal/status')
@@ -251,9 +259,13 @@ export const ClaudeTerminal: React.FC<ClaudeTerminalProps> = ({
     if (!wsRef.current || wsRef.current.readyState !== WebSocket.OPEN || !selectedStudy?.id) return;
 
     setSettingContext(true);
-    // Send context message - Claude should use CLAUDE.md and .claude/skills/ for guidance
-    const contextMessage = `Context: Working on study "${selectedStudy.id}" at studies/${selectedStudy.id}/. ` +
-      `Read .claude/skills/ for task protocols. Use atomizer conda env. Acknowledge briefly.`;
+    // Send context message with POS bootstrap instructions and study context
+    const contextMessage =
+      `You are helping with Atomizer optimization. ` +
+      `First read: .claude/skills/00_BOOTSTRAP.md for task routing. ` +
+      `Then follow the Protocol Execution Framework. ` +
+      `Study context: Working on "${selectedStudy.id}" at studies/${selectedStudy.id}/. ` +
+      `Use atomizer conda env. Acknowledge briefly.`;
     wsRef.current.send(JSON.stringify({ type: 'input', data: contextMessage + '\n' }));
 
     // Mark as done after Claude has had time to process

atomizer-dashboard/frontend/src/components/GlobalClaudeTerminal.tsx

@@ -0,0 +1,50 @@
+import React from 'react';
+import { useClaudeTerminal } from '../context/ClaudeTerminalContext';
+import { ClaudeTerminal } from './ClaudeTerminal';
+import { Terminal } from 'lucide-react';
+
+/**
+ * GlobalClaudeTerminal - A floating terminal that persists across page navigation
+ *
+ * This component renders at the App level and maintains the Claude Code session
+ * even when the user navigates between pages. It can be minimized to a floating
+ * button or expanded to a side panel.
+ */
+export const GlobalClaudeTerminal: React.FC = () => {
+  const { isOpen, setIsOpen, isExpanded, setIsExpanded, isConnected } = useClaudeTerminal();
+
+  // Floating button when terminal is closed
+  if (!isOpen) {
+    return (
+      <button
+        onClick={() => setIsOpen(true)}
+        className={`fixed bottom-6 right-6 p-4 rounded-full shadow-lg transition-all z-50 ${
+          isConnected
+            ? 'bg-green-600 hover:bg-green-500'
+            : 'bg-primary-600 hover:bg-primary-500'
+        }`}
+        title={isConnected ? 'Claude Terminal (Connected)' : 'Open Claude Terminal'}
+      >
+        <Terminal className="w-6 h-6 text-white" />
+        {isConnected && (
+          <span className="absolute -top-1 -right-1 w-3 h-3 bg-green-400 rounded-full border-2 border-dark-900 animate-pulse" />
+        )}
+      </button>
+    );
+  }
+
+  // Terminal panel
+  return (
+    <div className={`fixed z-50 transition-all duration-200 ${
+      isExpanded
+        ? 'inset-4'
+        : 'bottom-6 right-6 w-[650px] h-[500px]'
+    }`}>
+      <ClaudeTerminal
+        isExpanded={isExpanded}
+        onToggleExpand={() => setIsExpanded(!isExpanded)}
+        onClose={() => setIsOpen(false)}
+      />
+    </div>
+  );
+};