Support concurrent heartbeat runs with maxConcurrentRuns policy

Add per-agent maxConcurrentRuns (1-10) controlling how many runs
execute simultaneously. Implements agent-level start lock, optimistic
claim-then-execute flow, atomic token accounting via SQL expressions,
and proper status resolution when parallel runs finish. Updates UI
config form, live run count display, and SSE invalidation to avoid
unnecessary refetches on run event streams.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

ui/src/pages/AgentDetail.tsx

@@ -510,7 +510,14 @@ export function AgentDetail() {
                         const hb = (agent.runtimeConfig as Record<string, unknown>).heartbeat as Record<string, unknown>;
                         if (!hb.enabled) return <span className="text-muted-foreground">Disabled</span>;
                         const sec = Number(hb.intervalSec) || 300;
-                        return <span>Every {sec >= 60 ? `${Math.round(sec / 60)} min` : `${sec}s`}</span>;
+                        const maxConcurrentRuns = Math.max(1, Math.floor(Number(hb.maxConcurrentRuns) || 1));
+                        const intervalLabel = sec >= 60 ? `${Math.round(sec / 60)} min` : `${sec}s`;
+                        return (
+                          <span>
+                            Every {intervalLabel}
+                            {maxConcurrentRuns > 1 ? ` (max ${maxConcurrentRuns} concurrent)` : ""}
+                          </span>
+                        );
                       })()
                     : <span className="text-muted-foreground">Not configured</span>
                   }

ui/src/pages/Agents.tsx

@@ -90,13 +90,17 @@ export function Agents() {
     refetchInterval: 15_000,
   });
 
-  // Map agentId -> first live run (running or queued)
+  // Map agentId -> first live run + live run count
   const liveRunByAgent = useMemo(() => {
-    const map = new Map<string, { runId: string }>();
+    const map = new Map<string, { runId: string; liveCount: number }>();
     for (const r of runs ?? []) {
-      if ((r.status === "running" || r.status === "queued") && !map.has(r.agentId)) {
-        map.set(r.agentId, { runId: r.id });
+      if (r.status !== "running" && r.status !== "queued") continue;
+      const existing = map.get(r.agentId);
+      if (existing) {
+        existing.liveCount += 1;
+        continue;
       }
+      map.set(r.agentId, { runId: r.id, liveCount: 1 });
     }
     return map;
   }, [runs]);
@@ -246,6 +250,7 @@ export function Agents() {
                         <LiveRunIndicator
                           agentId={agent.id}
                           runId={liveRunByAgent.get(agent.id)!.runId}
+                          liveCount={liveRunByAgent.get(agent.id)!.liveCount}
                           navigate={navigate}
                         />
                       ) : (
@@ -257,6 +262,7 @@ export function Agents() {
                         <LiveRunIndicator
                           agentId={agent.id}
                           runId={liveRunByAgent.get(agent.id)!.runId}
+                          liveCount={liveRunByAgent.get(agent.id)!.liveCount}
                           navigate={navigate}
                         />
                       )}
@@ -319,7 +325,7 @@ function OrgTreeNode({
   depth: number;
   navigate: (path: string) => void;
   agentMap: Map<string, Agent>;
-  liveRunByAgent: Map<string, { runId: string }>;
+  liveRunByAgent: Map<string, { runId: string; liveCount: number }>;
 }) {
   const agent = agentMap.get(node.id);
 
@@ -358,6 +364,7 @@ function OrgTreeNode({
               <LiveRunIndicator
                 agentId={node.id}
                 runId={liveRunByAgent.get(node.id)!.runId}
+                liveCount={liveRunByAgent.get(node.id)!.liveCount}
                 navigate={navigate}
               />
             ) : (
@@ -369,6 +376,7 @@ function OrgTreeNode({
               <LiveRunIndicator
                 agentId={node.id}
                 runId={liveRunByAgent.get(node.id)!.runId}
+                liveCount={liveRunByAgent.get(node.id)!.liveCount}
                 navigate={navigate}
               />
             )}
@@ -402,10 +410,12 @@ function OrgTreeNode({
 function LiveRunIndicator({
   agentId,
   runId,
+  liveCount,
   navigate,
 }: {
   agentId: string;
   runId: string;
+  liveCount: number;
   navigate: (path: string) => void;
 }) {
   return (
@@ -420,7 +430,9 @@ function LiveRunIndicator({
         <span className="animate-ping absolute inline-flex h-full w-full rounded-full bg-blue-400 opacity-75" />
         <span className="relative inline-flex rounded-full h-2 w-2 bg-blue-500" />
       </span>
-      <span className="text-[11px] font-medium text-blue-400">Live</span>
+      <span className="text-[11px] font-medium text-blue-400">
+        Live{liveCount > 1 ? ` (${liveCount})` : ""}
+      </span>
     </button>
   );
 }