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/pages/Dashboard.tsx

@@ -4,9 +4,9 @@ import { Terminal } from 'lucide-react';
 import { useOptimizationWebSocket } from '../hooks/useWebSocket';
 import { apiClient } from '../api/client';
 import { useStudy } from '../context/StudyContext';
+import { useClaudeTerminal } from '../context/ClaudeTerminalContext';
 import { Card } from '../components/common/Card';
 import { ControlPanel } from '../components/dashboard/ControlPanel';
-import { ClaudeTerminal } from '../components/ClaudeTerminal';
 import { ParetoPlot } from '../components/ParetoPlot';
 import { ParallelCoordinatesPlot } from '../components/ParallelCoordinatesPlot';
 import { ParameterImportanceChart } from '../components/ParameterImportanceChart';
@@ -64,9 +64,8 @@ export default function Dashboard() {
   // Chart library toggle: 'recharts' (faster) or 'plotly' (more interactive but slower)
   const [chartLibrary, setChartLibrary] = useState<'plotly' | 'recharts'>('recharts');
 
-  // Claude chat panel state
-  const [chatOpen, setChatOpen] = useState(false);
-  const [chatExpanded, setChatExpanded] = useState(false);
+  // Claude terminal from global context
+  const { isOpen: claudeTerminalOpen, setIsOpen: setClaudeTerminalOpen, isConnected: claudeConnected } = useClaudeTerminal();
 
   const showAlert = (type: 'success' | 'warning', message: string) => {
     const id = alertIdCounter;
@@ -353,16 +352,21 @@ export default function Dashboard() {
         <div className="flex gap-2">
           {/* Claude Code Terminal Toggle Button */}
           <button
-            onClick={() => setChatOpen(!chatOpen)}
+            onClick={() => setClaudeTerminalOpen(!claudeTerminalOpen)}
             className={`flex items-center gap-2 px-4 py-2 rounded-lg transition-colors ${
-              chatOpen
+              claudeTerminalOpen
                 ? 'bg-primary-600 text-white'
-                : 'bg-dark-700 text-dark-200 hover:bg-dark-600 hover:text-white border border-dark-600'
+                : claudeConnected
+                  ? 'bg-green-700 text-white border border-green-600'
+                  : 'bg-dark-700 text-dark-200 hover:bg-dark-600 hover:text-white border border-dark-600'
             }`}
-            title="Open Claude Code terminal"
+            title={claudeConnected ? 'Claude Terminal (Connected)' : 'Open Claude Code terminal'}
           >
             <Terminal className="w-4 h-4" />
             <span className="hidden sm:inline">Claude Code</span>
+            {claudeConnected && !claudeTerminalOpen && (
+              <span className="w-2 h-2 bg-green-400 rounded-full animate-pulse" />
+            )}
           </button>
           {selectedStudyId && (
             <StudyReportViewer studyId={selectedStudyId} />
@@ -417,10 +421,10 @@ export default function Dashboard() {
         <ControlPanel onStatusChange={refreshStudies} horizontal />
       </div>
 
-      {/* Main Layout: Charts + Claude Terminal */}
-      <div className={`grid gap-4 ${chatOpen ? 'grid-cols-12' : 'grid-cols-1'}`}>
+      {/* Main Layout: Charts (Claude Terminal is now global/floating) */}
+      <div className="grid gap-4 grid-cols-1">
         {/* Main Content - Charts stacked vertically */}
-        <main className={chatOpen ? 'col-span-7' : 'col-span-1'}>
+        <main>
           {/* Study Name Header + Metrics in one row */}
           <div className="mb-4 pb-3 border-b border-dark-600 flex items-center justify-between">
             <div>
@@ -790,17 +794,6 @@ export default function Dashboard() {
             />
           </div>
         </main>
-
-        {/* Claude Code Terminal - Right Sidebar (wider for better visibility) */}
-        {chatOpen && (
-          <aside className="col-span-5 h-[calc(100vh-12rem)] sticky top-4">
-            <ClaudeTerminal
-              isExpanded={chatExpanded}
-              onToggleExpand={() => setChatExpanded(!chatExpanded)}
-              onClose={() => setChatOpen(false)}
-            />
-          </aside>
-        )}
       </div>
     </div>
   );

atomizer-dashboard/frontend/src/pages/Home.tsx

@@ -3,6 +3,7 @@ import { useNavigate } from 'react-router-dom';
 import {
   FolderOpen,
   Play,
+  Pause,
   CheckCircle,
   Clock,
   AlertCircle,
@@ -64,6 +65,8 @@ const Home: React.FC = () => {
     switch (status) {
       case 'running':
         return <Play className="w-3.5 h-3.5" />;
+      case 'paused':
+        return <Pause className="w-3.5 h-3.5" />;
       case 'completed':
         return <CheckCircle className="w-3.5 h-3.5" />;
       case 'not_started':
@@ -81,6 +84,12 @@ const Home: React.FC = () => {
           card: 'border-green-500/30 hover:border-green-500/50',
           glow: 'shadow-green-500/10'
         };
+      case 'paused':
+        return {
+          badge: 'bg-orange-500/20 text-orange-400 border-orange-500/30',
+          card: 'border-orange-500/30 hover:border-orange-500/50',
+          glow: 'shadow-orange-500/10'
+        };
       case 'completed':
         return {
           badge: 'bg-blue-500/20 text-blue-400 border-blue-500/30',